Finite Group Databasebeta

← #33102678

Subgroups — #33102678

Order 18688.

1096 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#41×Dic146C211yesmaximal, characteristic
2#22618882C211yesmaximal, characteristic, complemented
3C4×#14556556C211yesmaximal, characteristic, complemented
4#22610907C211yesmaximal, characteristic, complemented
5#22610906C212yesmaximal, complemented
6#22610874C211yesmaximal, characteristic, complemented
7#22610873C212yesmaximal, complemented
8#22609949C212yesmaximal
9#104×Dic73C212yesmaximal
10C4×#14556518C211yesmaximal, characteristic
11C73×#1806C211yesmaximal, characteristic
12Q8×Dic146C411yescharacteristic, complemented
13#17989316C411yescharacteristic, complemented
14#17989313C412yescomplemented
15#17989296—22no—
16#17989296C412yescomplemented
17#17989294—21no—
18#17989294C411yescharacteristic, complemented
19D4×Dic146—21no—
20D4×Dic146C411yescharacteristic, complemented
21#17989144—21no—
22#17989144C2211yescharacteristic
23#17989140—22no—
24#17989140C2212yes—
25#41×Dic73C2212yes—
26C2×C4×Dic146—21no—
27C2×C4×Dic146C2211yescharacteristic
28C2×#14556556—21no—
29#17988716—22no—
30#17988714—21no—
31#17988711—21no—
32#17988610C2212yes—
33C2×#14556518C2211yescharacteristic
34C2×#14556518C2211yescharacteristic
35C4×#11924012C2211yescharacteristic
36M16×Dic73C2212yes—
37C4⋊C4×Dic73—21no—
38C4×#11924007C2212yes—
39#17988438—22no—
40C42×Dic73C2211yescharacteristic
41C42×Dic73C2212yes—
42#17988348C2212yes—
43#17988348C2212yes—
44#17988347C2212yes—
45#17988347C2212yes—
46#17988346C2211yescharacteristic
47#17988346C2211yescharacteristic
48C4×#41×C73C2211yescharacteristic
49#423×C73C2211yescharacteristic
50C2×#104×C73C2212yes—
51C4×C4⋊C4×C73C2211yescharacteristic
52#339×C73C2212yes—
53C22×Dic146—21no—
54C22×Dic146D411yescharacteristic
55C2×#11924012C2311yescharacteristic
56C2×#11924012C2311yescharacteristic
57#14556591—21no—
58#14556591—21no—
59#14556591C2×C411yescharacteristic
60#14556591C2×C411yescharacteristic
61#14556590—22no—
62#14556590—22no—
63#14556590C2×C412yes—
64#14556590C2×C412yes—
65#14556583—22no—
66#14556583C2×C412yes—
67#14556582—22no—
68#14556582—22no—
69#14556582C2×C412yes—
70#14556582C2×C412yes—
71#14556570—21no—
72#14556570—21no—
73#14556570—22no—
74#14556570—21no—
75#14556570—21no—
76#14556568—22no—
77#14556568—22no—
78#14556568—22no—
79#14556568—22no—
80#14556568—22no—
81#14556568—22no—
82Q8×Dic73C2×C412yes—
83Q8×Dic73C2×C412yes—
84C4×Dic146—21no—
85C4×Dic146—21no—
86C4×Dic146—22no—
87C4×Dic146—22no—
88C4×Dic146—22no—
89C4×Dic146—21no—
90C4×Dic146—21no—
91C4×Dic146C2×C411yescharacteristic
92C4×Dic146C2×C411yescharacteristic
93C4×Dic146C2×C412yes—
94C4×Dic146D412yes—
95C4×Dic146D411yescharacteristic
96C4×Dic146D411yescharacteristic
97#14556556—28no—
98D4×Dic73—22no—
99D4×Dic73—22no—
100D4×Dic73C2×C412yes—
101D4×Dic73C2×C412yes—
102#14556550—22no—
103#14556550C2312yes—
104C2×C4⋊Dic73—21no—
105C2×C4⋊Dic73D411yescharacteristic
106C2×#11924007—22no—
107C2×#11924007—22no—
108C2×#11924007D412yes—
109C2×C4×Dic73—22no—
110C2×C4×Dic73—22no—
111C2×C4×Dic73—22no—
112C2×C4×Dic73—21no—
113C2×C4×Dic73—21no—
114C2×C4×Dic73C2311yescharacteristic
115C2×C4×Dic73C2311yescharacteristic
116C2×C4×Dic73C2312yes—
117#14556520—22no—
118#14556520—22no—
119#14556518—22no—
120#14556518Q814yescomplemented
121#14556502—21no—
122#14556501—22no—
123C4×#10596831—21no—
124C8×Dic73—22no—
125#14556498—22no—
126C2×#41×C73C2311yescharacteristic
127C2×M16×C73C2312yes—
128C4×Q8×C73C2×C411yescharacteristic
129C4×Q8×C73C2×C411yescharacteristic
130C4×D4×C73—21no—
131C4×D4×C73—21no—
132C4×D4×C73C2×C411yescharacteristic
133C4×D4×C73C2×C411yescharacteristic
134#117×C73—21no—
135#117×C73C2311yescharacteristic
136C2×C4⋊C4×C73—21no—
137C2×C42×C73—21no—
138C2×C42×C73C2312yes—
139C2×C42×C73C2311yescharacteristic
140C4⋊C8×C73—21no—
141C4⋊C8×C73—21no—
142#104×C73—24no—
143#95×C73—21no—
144C2×Dic146—24no—
145C2×Dic146—21no—
146C2×Dic146—22no—
147C2×Dic146—21no—
148C2×Dic146—22no—
149C2×Dic146—44no—
150C2×Dic146C22⋊C412yes—
151C2×Dic146C22⋊C411yescharacteristic
152C2×Dic146C22⋊C411yescharacteristic
153C22×Dic73—21no—
154C22×Dic73—21no—
155C22×Dic73—22no—
156C22×Dic73—22no—
157C22×Dic73C2×D412yes—
158C22×Dic73C2×D412yes—
159#11924012—21no—
160#11924012—21no—
161#11924012—21no—
162#11924012C2×Q812yescomplemented
163C2×#10596831—21no—
164C2×#10596831—21no—
165C4⋊Dic73—21no—
166C4⋊Dic73—21no—
167C4⋊Dic73—22no—
168C4⋊Dic73—22no—
169C4⋊Dic73—22no—
170C4⋊Dic73—22no—
171C4⋊Dic73C22⋊C411yescharacteristic
172C4⋊Dic73C22⋊C412yes—
173C4⋊Dic73C22⋊C411yescharacteristic
174C4⋊Dic73C22×C412yes—
175C4⋊Dic73C22×C412yes—
176#11924007—28no—
177#11924007—22no—
178#11924007—22no—
179#11924007—22no—
180#11924007—22no—
181#11924007—22no—
182#11924007—22no—
183#11924007C22⋊C412yes—
184#11924007C22⋊C412yes—
185#11924007C22⋊C412yes—
186#11924007C22⋊C412yes—
187C22⋊Dic73—22no—
188C22⋊Dic73—22no—
189C22⋊Dic73—22no—
190C4×Dic73—28no—
191C4×Dic73—24no—
192C4×Dic73—22no—
193C4×Dic73—21no—
194C4×Dic73—22no—
195C4×Dic73—22no—
196C4×Dic73—21no—
197C4×Dic73—21no—
198C4×Dic73—44no—
199C4×Dic73—42no—
200C4×Dic73C22×C412yes—
201C4×Dic73C22×C412yes—
202C4×Dic73C2×D412yes—
203C4×Dic73C2×D412yes—
204C4×Dic73C2×Q812yescomplemented
205#41×C73—21no—
206#41×C73—21no—
207#41×C73C2×Q812yes—
208#41×C73#4112yes—
209C2×Q8×C73C22×C411yescharacteristic
210C2×D4×C73—21no—
211C2×D4×C73C22×C411yescharacteristic
212C22×C4×C73—21no—
213C22×C4×C73—22no—
214C22×C4×C73—21no—
215C22×C4×C73—21no—
216C22×C4×C73C2×D411yescharacteristic
217C22×C4×C73C2×D411yescharacteristic
218C22×C4×C73C2411yescharacteristic
219M16×C73—22no—
220M16×C73—22no—
221C2×C8×C73—22no—
222C4⋊C4×C73—24no—
223C4⋊C4×C73—21no—
224C4⋊C4×C73—21no—
225C4⋊C4×C73C22×C411yescharacteristic
226C4⋊C4×C73C22×C411yescharacteristic
227C22⋊C4×C73—21no—
228C22⋊C4×C73—21no—
229C22⋊C4×C73—21no—
230C42×C73—21no—
231C42×C73—21no—
232C42×C73—21no—
233C42×C73—22no—
234C42×C73—22no—
235C42×C73C22×C411yescharacteristic
236C42×C73C22×C411yescharacteristic
237C42×C73C2×D411yescharacteristic
238C42×C73C2×D411yescharacteristic
239C23×C73—21no—
240C23×C73C22×D411yescharacteristic
241Q8×C73—21no—
242Q8×C73C4×Q812yescomplemented
243D4×C73—22no—
244D4×C73—22no—
245D4×C73—21no—
246D4×C73C4×Q812yescomplemented
247C2×C4×C73—21no—
248C2×C4×C73—22no—
249C2×C4×C73—22no—
250C2×C4×C73—22no—
251C2×C4×C73—24no—
252C2×C4×C73—21no—
253C2×C4×C73—21no—
254C2×C4×C73—41no—
255C2×C4×C73—42no—
256C2×C4×C73—42no—
257C2×C4×C73—42no—
258C2×C4×C73—81no—
259C2×C4×C73C2×C22⋊C411yescharacteristic
260C2×C4×C73C2×C22⋊C411yescharacteristic
261C2×C4×C73C2×C22⋊C411yescharacteristic
262C2×C4×C73C2×C22⋊C411yescharacteristic
263C2×C4×C73C23×C411yescharacteristic
264C2×C4×C73C22×D411yescharacteristic
265C2×C4×C73C22×Q811yescharacteristic
266C2×C4×C73C2×#4111yescharacteristic
267C2×C4×C732−1+411yescharacteristic
268C2×C4×C732−1+411yescharacteristic
269C2×Dic73—22no—
270C2×Dic73—22no—
271C2×Dic73—22no—
272C2×Dic73—24no—
273C2×Dic73—22no—
274C2×Dic73—22no—
275C2×Dic73—22no—
276C2×Dic73—42no—
277C2×Dic73—41no—
278C2×Dic73—41no—
279C2×Dic73—44no—
280C2×Dic73—82no—
281C2×Dic73C2×C22⋊C412yes—
282C2×Dic73C2×C22⋊C412yes—
283C2×Dic73C2×C22⋊C412yes—
284C2×Dic73C2×C22⋊C412yes—
285Dic146—22no—
286Dic146—24no—
287Dic146—42no—
288Dic146—44no—
289Dic146—82no—
290Dic146—81no—
291Dic146#10414yescomplemented
292C8×C73—42no—
293#10596831—22no—
294#10596831—41no—
295C22×C73—21no—
296C22×C73—21no—
297C22×C73—22no—
298C22×C73—41no—
299C22×C73—41no—
300C22×C73C22×C22⋊C411yescharacteristic
301C22×C73#54811yescharacteristic
302C22×C73D4×Q811yescharacteristic
303C4×C73—22no—
304C4×C73—21no—
305C4×C73—21no—
306C4×C73—21no—
307C4×C73—44no—
308C4×C73—41no—
309C4×C73—41no—
310C4×C73C2×#10412yes—
311C4×C73C2×C4×Q811yescharacteristic
312C4×C73#51911yescharacteristic
313C4×C73#54811yescharacteristic
314C4×C73D4×Q811yescharacteristic
315Dic73—24no—
316Dic73—22no—
317Dic73—22no—
318Dic73—22no—
319Dic73—42no—
320Dic73—42no—
321Dic73—42no—
322Dic73C2×#10414yes—
323Q8×#104—731nomaximal, Sylow
324C2×C73—21no—
325C2×C73—21no—
326C2×C73—42no—
327C2×C73Q8×C22⋊C411yescharacteristic
328C2×C73C22×#10411yescharacteristic
329C2×C73#290111yescharacteristic
330Q8×#41—731no—
331Q8×M16—731no—
332C42×Q8—731no—
333#1994—731no—
334#1994—732no—
335#1986—731no—
336#1986—732no—
337#1806—732no—
338#1806D7311yescharacteristic
339C4×#104—732no—
340C4×#104—731no—
341C73Q8×#10411yescharacteristic, complemented, Sylow
342Q82—731no—
343#557—732no—
344#557—731no—
345#554—731no—
346#554—732no—
347#554—1461no—
348#554—1462no—
349D4×Q8—731no—
350D4×Q8—1461no—
351#533—732no—
352#533—731no—
353#533—1461no—
354#533—1462no—
355C4×#41—732no—
356C4×#41C2×D7311yescharacteristic
357C2×C4×Q8—731no—
358C2×C4×Q8—1461no—
359C2×C4×Q8—1461no—
360#446—1461no—
361#446—1462no—
362C8×Q8—1461no—
363#423—732no—
364#423C2×D7311yescharacteristic
365C2×#104—731no—
366C2×#104—731no—
367C2×#104C2×D7312yes—
368C4×M16—732no—
369C4×M16—731no—
370C4×C4⋊C4—732no—
371C4×C4⋊C4—1462no—
372C4×C4⋊C4—1461no—
373C4×C4⋊C4C2×D7311yescharacteristic
374C43—732no—
375C43—731no—
376#339—732no—
377#339—732no—
378#339—731no—
379#339—731no—
380#339—732no—
381#339—732no—
382#339C2×D7312yes—
383C2×#41C22×D7311yescharacteristic
384C22×Q8—731no—
385C22×Q8—1461no—
386C2×M16—731no—
387C2×M16—731no—
388C2×M16C22×D7312yes—
389C4⋊Q8—732no—
390C4⋊Q8—732no—
391C4⋊Q8—731no—
392C4⋊Q8—731no—
393C4⋊Q8—1461no—
394C4⋊Q8—1461no—
395C4⋊Q8—1462no—
396C4⋊Q8—1462no—
397#125—732no—
398#125—732no—
399#125—732no—
400#125—1462no—
401#125—1462no—
402#125—1462no—
403C22⋊Q8—1462no—
404C22⋊Q8—1462no—
405C22⋊Q8—1462no—
406C22⋊Q8—1462no—
407C22⋊Q8—1462no—
408C22⋊Q8—1462no—
409C22⋊Q8—1461no—
410C22⋊Q8—1462no—
411C22⋊Q8—1461no—
412C22⋊Q8—1461no—
413C22⋊Q8—1461no—
414C4×Q8—731no—
415C4×Q8—731no—
416C4×Q8—732no—
417C4×Q8—731no—
418C4×Q8—732no—
419C4×Q8—731no—
420C4×Q8—732no—
421C4×Q8—732no—
422C4×Q8—1462no—
423C4×Q8—1462no—
424C4×Q8—1462no—
425C4×Q8—1461no—
426C4×Q8—1461no—
427C4×Q8—1468no—
428C4×Q8—1461no—
429C4×Q8—1461no—
430C4×Q8C4×D7311yescharacteristic
431C4×Q8C4×D7311yescharacteristic
432C4×D4—21no—
433C4×D4—21no—
434C4×D4—732no—
435C4×D4—732no—
436C4×D4—1462no—
437C4×D4—1462no—
438C4×D4C4×D7311yescharacteristic
439C4×D4C4×D7311yescharacteristic
440#117—21no—
441#117—732no—
442#117—1462no—
443#117C22×D7311yescharacteristic
444C2×C4⋊C4—21no—
445C2×C4⋊C4—732no—
446C2×C4⋊C4—731no—
447C2×C4⋊C4—1462no—
448C2×C4⋊C4—1461no—
449C2×C4⋊C4—1462no—
450C2×C42—21no—
451C2×C42—732no—
452C2×C42—731no—
453C2×C42—731no—
454C2×C42—1462no—
455C2×C42—1461no—
456C2×C42—1462no—
457C2×C42—1462no—
458C2×C42—1461no—
459C2×C42C22×D7312yes—
460C2×C42C22×D7311yescharacteristic
461C4⋊C8—21no—
462C4⋊C8—21no—
463C4⋊C8—1462no—
464C4⋊C8—1462no—
465#104—24no—
466#104—734no—
467#104—1462no—
468#97—1462no—
469#97—1461no—
470C4×C8—1462no—
471C4×C8—1461no—
472#95—21no—
473#95—1462no—
474#41—21no—
475#41—21no—
476#41Q8×D7312yes—
477#41#1192402712yes—
478C2×Q8—2924no—
479C2×Q8—731no—
480C2×Q8—732no—
481C2×Q8—731no—
482C2×Q8—1462no—
483C2×Q8—1462no—
484C2×Q8—1464no—
485C2×Q8—1461no—
486C2×Q8—1461no—
487C2×Q8C2×C4×D7311yescharacteristic
488C2×D4—21no—
489C2×D4C2×C4×D7311yescharacteristic
490C22×C4—21no—
491C22×C4—22no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—732no—
495C22×C4—732no—
496C22×C4—1462no—
497C22×C4—1461no—
498C22×C4—1462no—
499C22×C4—1461no—
500C22×C4D4×D7311yescharacteristic
501C22×C4D4×D7311yescharacteristic
502C22×C4C23×D7311yescharacteristic
503M16—22no—
504M16—22no—
505M16—732no—
506M16—1461no—
507M16—1461no—
508M16—1461no—
509C2×C8—22no—
510C2×C8—1461no—
511C2×C8—1461no—
512C4⋊C4—24no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—732no—
516C4⋊C4—732no—
517C4⋊C4—732no—
518C4⋊C4—732no—
519C4⋊C4—731no—
520C4⋊C4—732no—
521C4⋊C4—731no—
522C4⋊C4—732no—
523C4⋊C4—732no—
524C4⋊C4—1462no—
525C4⋊C4—1462no—
526C4⋊C4—1462no—
527C4⋊C4—1462no—
528C4⋊C4—1462no—
529C4⋊C4—1462no—
530C4⋊C4—1462no—
531C4⋊C4—1462no—
532C4⋊C4—1468no—
533C4⋊C4—1462no—
534C4⋊C4—1462no—
535C4⋊C4—1461no—
536C4⋊C4—1461no—
537C4⋊C4C2×C4×D7311yescharacteristic
538C4⋊C4C2×C4×D7311yescharacteristic
539C22⋊C4—21no—
540C22⋊C4—21no—
541C22⋊C4—21no—
542C22⋊C4—1462no—
543C22⋊C4—1462no—
544C22⋊C4—1462no—
545C42—21no—
546C42—21no—
547C42—21no—
548C42—22no—
549C42—22no—
550C42—2922no—
551C42—2924no—
552C42—732no—
553C42—732no—
554C42—732no—
555C42—732no—
556C42—732no—
557C42—1462no—
558C42—1462no—
559C42—1464no—
560C42—1468no—
561C42—1461no—
562C42—1461no—
563C42—1462no—
564C42—1461no—
565C42C2×C4×D7311yescharacteristic
566C42C2×C4×D7311yescharacteristic
567C42D4×D7311yescharacteristic
568C42D4×D7311yescharacteristic
569C23—21no—
570C23C2×D4×D7311yescharacteristic
571Q8—21no—
572Q8—2924no—
573Q8—2922no—
574Q8—5841no—
575Q8—5842no—
576Q8—734no—
577Q8—1464no—
578Q8—1462no—
579Q8#1455655612yescomplemented
580D4—22no—
581D4—22no—
582D4—21no—
583D4#1455655612yescomplemented
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—81no—
596C2×C4—2921no—
597C2×C4—2921no—
598C2×C4—2924no—
599C2×C4—2922no—
600C2×C4—5842no—
601C2×C4—732no—
602C2×C4—732no—
603C2×C4—732no—
604C2×C4—732no—
605C2×C4—1462no—
606C2×C4—1462no—
607C2×C4—1462no—
608C2×C4—1464no—
609C2×C4—1462no—
610C2×C4—1462no—
611C2×C4—1462no—
612C2×C4C22⋊C4×D7311yescharacteristic
613C2×C4C22⋊C4×D7311yescharacteristic
614C2×C4C22⋊C4×D7311yescharacteristic
615C2×C4C22⋊C4×D7311yescharacteristic
616C2×C4C22×C4×D7311yescharacteristic
617C2×C4C2×D4×D7311yescharacteristic
618C2×C4C2×Q8×D7311yescharacteristic
619C2×C4C2×#1192402711yescharacteristic
620C2×C4#1455663811yescharacteristic
621C2×C4#1455663811yescharacteristic
622C8—42no—
623C8—2921no—
624C8—1462no—
625C22—21no—
626C22—21no—
627C22—22no—
628C22—41no—
629C22—41no—
630C22C2×C22⋊C4×D7311yescharacteristic
631C22#1798924311yescharacteristic
632C22#1798925011yescharacteristic
633C4—22no—
634C4—21no—
635C4—21no—
636C4—21no—
637C4—44no—
638C4—41no—
639C4—41no—
640C4—2922no—
641C4—2922no—
642C4—2922no—
643C4—734no—
644C4—1464no—
645C4—1462no—
646C4—1462no—
647C4—1462no—
648C4#104×D7312yes—
649C4C2×#1455655611yescharacteristic
650C4#1798906411yescharacteristic
651C4#1798924311yescharacteristic
652C4#1798925011yescharacteristic
653C2—21no—
654C2—21no—
655C2—42no—
656C2#2261276311yescharacteristic
657C2C2×#104×D7311yescharacteristic
658C2#2261847711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1806C2
socleC22×C73C22×C22⋊C4
prime radicalC2×#41×C73C23
cosocleC22×C4×C73C24
hypercenter#1806D73
hypocenterC73Q8×#104
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1806

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #22618882C2semidirect
3. C4×#14556556C2semidirect
4. #22610907C2semidirect
5. #22610906C2semidirect
6. #22610874C2semidirect
7. #22610873C2semidirect
12. Q8×Dic146C4semidirect
13. #17989316C4semidirect
14. #17989313C4semidirect
16. #17989296C4semidirect
18. #17989294C4semidirect
20. D4×Dic146C4semidirect
120. #14556518Q8semidirect
162. #11924012C2×Q8semidirect
204. C4×Dic73C2×Q8semidirect
242. Q8×C73C4×Q8semidirect
246. D4×C73C4×Q8semidirect
291. Dic146#104semidirect
341. C73Q8×#104semidirect
579. Q8#14556556semidirect
583. D4#14556556semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#41×Dic1462
#226188822
C4×#145565562
#226109072
#226109062
#226108742
#226108732
#226099492
#104×Dic732
C4×#145565182
C73×#18062
Q8×#10473