Finite Group Databasebeta

← #33095537

Subgroups — #33095537

Order 18688.

970 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989215C211yesmaximal, characteristic
2#22612540C211yesmaximal, characteristic
3#22610904C211yesmaximal, characteristic, complemented
4#22610900C211yesmaximal, characteristic, complemented
5#22610896C211yesmaximal, characteristic, complemented
6#22610892C211yesmaximal, characteristic, complemented
7#22610886C211yesmaximal, characteristic, complemented
8#22610886C211yesmaximal, characteristic, complemented
9#22610882C211yesmaximal, characteristic, complemented
10#22610878C211yesmaximal, characteristic, complemented
11#22610027C211yesmaximal, characteristic
12#22610027C211yesmaximal, characteristic
13#22610022C211yesmaximal, characteristic
14#22610018C211yesmaximal, characteristic
15C2×#433×C73C211yesmaximal, characteristic
16#17989215—24no—
17#17989215C418yescomplemented
18C2×#14556591C2211yescharacteristic
19C2×#14556591C2211yescharacteristic
20C2×#14556590—21no—
21C2×#14556583C2211yescharacteristic
22C2×#14556583C2211yescharacteristic
23C2×#14556582—21no—
24C2×#14556568—21no—
25C2×#14556568—21no—
26C2×#14556550C2211yescharacteristic
27C22×C4⋊Dic73C2211yescharacteristic
28#17988480—21no—
29#17988480—21no—
30#17988473—21no—
31#17988473—21no—
32#17988457C2211yescharacteristic
33#17988453C2211yescharacteristic
34#17988447C2211yescharacteristic
35#17988447C2211yescharacteristic
36C4×C4⋊Dic73C2211yescharacteristic
37C4×C4⋊Dic73C2211yescharacteristic
38#17988350C2211yescharacteristic
39#17988350C2211yescharacteristic
40#17988350C2211yescharacteristic
41#17988350C2211yescharacteristic
42#17988350C2211yescharacteristic
43#17988350C2211yescharacteristic
44#17988350C2211yescharacteristic
45#17988350C2211yescharacteristic
46#17988350C2211yescharacteristic
47#17988350C2211yescharacteristic
48#17988350C2211yescharacteristic
49#17988350C2211yescharacteristic
50#17988350C2211yescharacteristic
51#17988350C2211yescharacteristic
52#17988350C2211yescharacteristic
53#17988350C2211yescharacteristic
54C2×#117×C73C2211yescharacteristic
55#433×C73—24no—
56C2×C4⋊C8×C73C2211yescharacteristic
57C2×C4⋊C8×C73C2211yescharacteristic
58C2×C22⋊C8×C73C2211yescharacteristic
59C2×C22⋊C8×C73C2211yescharacteristic
60C2×#97×C73C2211yescharacteristic
61C2×C4×C8×C73C2211yescharacteristic
62C22×Dic146—21no—
63C22×Dic146—21no—
64C23×Dic73—21no—
65#14556591—22no—
66#14556591—22no—
67#14556591C2×C414yes—
68#14556591C2×C414yes—
69#14556590—28no—
70#14556583—22no—
71#14556583—22no—
72#14556583C2×C414yes—
73#14556583C2×C414yes—
74#14556582—28no—
75#14556568—28no—
76#14556568—28no—
77#14556550—22no—
78#14556550C2×C414yes—
79C2×C4⋊Dic73—21no—
80C2×C4⋊Dic73—21no—
81C2×C4⋊Dic73—21no—
82C2×C4⋊Dic73—22no—
83C2×C4⋊Dic73—21no—
84C2×C4⋊Dic73—21no—
85C2×C4⋊Dic73C2×C414yes—
86C2×C4⋊Dic73C2311yescharacteristic
87C2×C4⋊Dic73C2311yescharacteristic
88C2×C4⋊Dic73C2311yescharacteristic
89C2×C4⋊Dic73C2311yescharacteristic
90C2×C4⋊Dic73C2311yescharacteristic
91C2×C4⋊Dic73C2311yescharacteristic
92C2×C4⋊Dic73C2311yescharacteristic
93C2×C4⋊Dic73C2311yescharacteristic
94C2×#11924007—21no—
95C2×#11924007—21no—
96C2×#11924007—21no—
97C2×#11924007—21no—
98C2×C22⋊Dic73—21no—
99C2×C22⋊Dic73—21no—
100C2×C22⋊Dic73—21no—
101C2×C4×Dic73—21no—
102C2×C4×Dic73—21no—
103C2×C4×Dic73—21no—
104#14556498—21no—
105#14556498—21no—
106#14556498—21no—
107#14556498—21no—
108#14556498—21no—
109#14556498—21no—
110C23×C4×C73C2311yescharacteristic
111C22×C8×C73C2311yescharacteristic
112C22×C8×C73C2311yescharacteristic
113C22×C8×C73C2311yescharacteristic
114C22×C8×C73C2311yescharacteristic
115#117×C73C2×C414yes—
116#117×C73D414yes—
117C2×C4⋊C4×C73D411yescharacteristic
118C2×C4⋊C4×C73D411yescharacteristic
119C2×C22⋊C4×C73D411yescharacteristic
120C2×C22⋊C4×C73D411yescharacteristic
121C2×C42×C73C2311yescharacteristic
122C2×C42×C73C2311yescharacteristic
123C4⋊C8×C73—22no—
124C4⋊C8×C73—22no—
125C22⋊C8×C73—22no—
126C22⋊C8×C73—22no—
127#97×C73—22no—
128C4×C8×C73—22no—
129C2×Dic146—24no—
130C2×Dic146—24no—
131C2×Dic146—81no—
132C2×Dic146—81no—
133C2×Dic146—81no—
134C2×Dic146—81no—
135C22×Dic73—21no—
136C22×Dic73—21no—
137C22×Dic73—21no—
138C22×Dic73—21no—
139C22×Dic73—24no—
140C22×Dic73—21no—
141C22×Dic73—21no—
142C22×Dic73—21no—
143C22×Dic73—21no—
144C22×Dic73—81no—
145C22×Dic73—81no—
146C4⋊Dic73—24no—
147C4⋊Dic73—24no—
148C4⋊Dic73—24no—
149C4⋊Dic73—21no—
150C4⋊Dic73—21no—
151C4⋊Dic73—21no—
152C4⋊Dic73—21no—
153C4⋊Dic73—21no—
154C4⋊Dic73—21no—
155C4⋊Dic73—21no—
156C4⋊Dic73—21no—
157C4⋊Dic73—21no—
158C4⋊Dic73—21no—
159C4⋊Dic73—21no—
160C4⋊Dic73—21no—
161C4⋊Dic73—81no—
162C4⋊Dic73—81no—
163C4⋊Dic73C22×C412yes—
164C4⋊Dic73C22×C412yes—
165C4⋊Dic73C22×C412yes—
166C4⋊Dic73C22×C412yes—
167#11924007—24no—
168#11924007—24no—
169#11924007—24no—
170#11924007—24no—
171C22⋊Dic73—24no—
172C22⋊Dic73—81no—
173C22⋊Dic73—81no—
174C4×Dic73—24no—
175C4×Dic73—81no—
176C4×Dic73—81no—
177C24×C73C2×D411yescharacteristic
178C22×C4×C73—21no—
179C22×C4×C73—21no—
180C22×C4×C73—21no—
181C22×C4×C73—21no—
182C22×C4×C73C22×C412yes—
183C22×C4×C73C2×D412yes—
184C22×C4×C73C2×D411yescharacteristic
185C22×C4×C73C2×D411yescharacteristic
186C22×C4×C73C2×D411yescharacteristic
187C22×C4×C73C2×D411yescharacteristic
188C22×C4×C73C2×D411yescharacteristic
189C22×C4×C73C2411yescharacteristic
190C2×C8×C73—21no—
191C2×C8×C73—21no—
192C2×C8×C73—21no—
193C2×C8×C73—21no—
194C2×C8×C73—21no—
195C2×C8×C73—21no—
196C2×C8×C73—21no—
197C2×C8×C73—21no—
198C2×C8×C73—21no—
199C2×C8×C73—21no—
200C2×C8×C73—21no—
201C2×C8×C73—21no—
202C4⋊C4×C73C22⋊C414yes—
203C4⋊C4×C73C22⋊C414yes—
204C22⋊C4×C73C22⋊C414yes—
205C22⋊C4×C73C22⋊C414yes—
206C42×C73C22×C412yes—
207C42×C73C22×C412yes—
208C42×C73C2×D412yes—
209C42×C73C2×D412yes—
210C23×C73—21no—
211C23×C73—21no—
212C23×C73—21no—
213C23×C73—21no—
214C23×C73—21no—
215C23×C73—21no—
216C23×C73C2×C22⋊C412yes—
217C23×C73C22×D411yescharacteristic
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—21no—
225C2×C4×C73—21no—
226C2×C4×C73—21no—
227C2×C4×C73—21no—
228C2×C4×C73—22no—
229C2×C4×C73—22no—
230C2×C4×C73—22no—
231C2×C4×C73—22no—
232C2×C4×C73C2×C22⋊C412yes—
233C2×C4×C73C2×C22⋊C412yes—
234C2×C4×C73C2×C22⋊C412yes—
235C2×C4×C73C2×C22⋊C412yes—
236C2×C4×C73C2×C22⋊C412yes—
237C2×C4×C73C23×C411yescharacteristic
238C2×C4×C73C22×D411yescharacteristic
239C2×C4×C732−1+411yescharacteristic
240C2×C4×C732−1+411yescharacteristic
241C2×C4×C732−1+411yescharacteristic
242C2×C4×C732−1+411yescharacteristic
243C2×Dic73—22no—
244C2×Dic73—22no—
245C2×Dic73—22no—
246C2×Dic73—22no—
247C2×Dic73—41no—
248C2×Dic73—41no—
249C2×Dic73—41no—
250C2×Dic73—41no—
251C2×Dic73—41no—
252C2×Dic73—41no—
253C2×Dic73—41no—
254C2×Dic73—41no—
255C2×Dic73—41no—
256C2×Dic73—41no—
257C2×Dic73—41no—
258C2×Dic73—41no—
259C2×Dic73—41no—
260C2×Dic73—41no—
261C2×Dic73—41no—
262C2×Dic73—41no—
263C2×Dic73—41no—
264C2×Dic73—41no—
265C2×Dic73—41no—
266C2×Dic73—41no—
267C2×Dic73—84no—
268Dic146—84no—
269Dic146—84no—
270C8×C73—41no—
271C8×C73—41no—
272C8×C73—41no—
273C8×C73—41no—
274C22×C73—22no—
275C22×C73—22no—
276C22×C73—22no—
277C22×C73—41no—
278C22×C73—41no—
279C22×C73—41no—
280C22×C73—41no—
281C22×C73C22×C22⋊C411yescharacteristic
282C22×C73#53611yescharacteristic
283C22×C73#53611yescharacteristic
284C22×C73#57611yescharacteristic
285C22×C73#57711yescharacteristic
286C22×C73#57711yescharacteristic
287C22×C73#57811yescharacteristic
288C4×C73—22no—
289C4×C73—22no—
290C4×C73—22no—
291C4×C73—22no—
292C4×C73—22no—
293C4×C73#51911yescharacteristic
294C4×C73#51911yescharacteristic
295C4×C73#53611yescharacteristic
296C4×C73#53611yescharacteristic
297Dic73—42no—
298Dic73—42no—
299Dic73—42no—
300Dic73—42no—
301Dic73—81no—
302Dic73—81no—
303Dic73—81no—
304Dic73—81no—
305#21843—731nomaximal, Sylow
306C2×C73—42no—
307C2×C73#231011yescharacteristic
308C2×C73#289711yescharacteristic
309C2×C73#289811yescharacteristic
310C2×C73#324111yescharacteristic
311C2×C73#324211yescharacteristic
312C2×C73#324311yescharacteristic
313C2×C73#324411yescharacteristic
314C2×#544—731no—
315C2×#433D7311yescharacteristic
316#2314—731no—
317#1993—731no—
318#1992—731no—
319#1991—731no—
320#1990—731no—
321#1989—731no—
322#1989—731no—
323#1988—731no—
324#1987—731no—
325#1825—731no—
326#1825—731no—
327#1824—731no—
328#1823—731no—
329C73#2184311yescharacteristic, complemented, Sylow
330#544—738no—
331#544—1464no—
332C2×C4⋊Q8—731no—
333C2×C4⋊Q8—731no—
334C2×C4⋊Q8—1461no—
335C2×#125—731no—
336C2×#125—731no—
337C2×#125—1461no—
338C2×C22⋊Q8—1461no—
339C2×C22⋊Q8—1461no—
340C2×#117—731no—
341C2×#117C2×D7311yescharacteristic
342C22×C4⋊C4—731no—
343#433—24no—
344C2×C4⋊C8C2×D7311yescharacteristic
345C2×C4⋊C8C2×D7311yescharacteristic
346C2×C22⋊C8C2×D7311yescharacteristic
347C2×C22⋊C8C2×D7311yescharacteristic
348C2×#97C2×D7311yescharacteristic
349C2×C4×C8C2×D7311yescharacteristic
350#388—1461no—
351#388—1461no—
352#387—1461no—
353#387—1461no—
354#384—731no—
355#382—731no—
356#380—731no—
357#380—731no—
358C4×C4⋊C4—731no—
359C4×C4⋊C4—731no—
360#340—731no—
361#340—731no—
362#340—731no—
363#340—731no—
364#340—731no—
365#340—731no—
366#340—731no—
367#340—731no—
368#340—731no—
369#340—731no—
370#340—731no—
371#340—731no—
372#340—731no—
373#340—731no—
374#340—731no—
375#340—731no—
376C22×Q8—1461no—
377C22×Q8—1461no—
378C23×C4—1461no—
379C23×C4C22×D7311yescharacteristic
380C22×C8C22×D7311yescharacteristic
381C22×C8C22×D7311yescharacteristic
382C22×C8C22×D7311yescharacteristic
383C22×C8C22×D7311yescharacteristic
384C4⋊Q8—734no—
385C4⋊Q8—734no—
386C4⋊Q8—1468no—
387C4⋊Q8—1462no—
388C4⋊Q8—1462no—
389#125—734no—
390#125—734no—
391#125—1468no—
392#125—1462no—
393#125—1462no—
394C22⋊Q8—1468no—
395C22⋊Q8—1468no—
396#117—734no—
397#117—1462no—
398#117C4×D7314yes—
399#117C22⋊D7314yes—
400C2×C4⋊C4—734no—
401C2×C4⋊C4—731no—
402C2×C4⋊C4—731no—
403C2×C4⋊C4—731no—
404C2×C4⋊C4—731no—
405C2×C4⋊C4—731no—
406C2×C4⋊C4—731no—
407C2×C4⋊C4—731no—
408C2×C4⋊C4—731no—
409C2×C4⋊C4—1461no—
410C2×C4⋊C4—1461no—
411C2×C4⋊C4—1462no—
412C2×C4⋊C4—1461no—
413C2×C4⋊C4—1461no—
414C2×C4⋊C4—1461no—
415C2×C4⋊C4—1461no—
416C2×C4⋊C4—1461no—
417C2×C4⋊C4—1461no—
418C2×C4⋊C4—1461no—
419C2×C4⋊C4D29211yescharacteristic
420C2×C4⋊C4D29211yescharacteristic
421C2×C22⋊C4—1461no—
422C2×C22⋊C4—1461no—
423C2×C22⋊C4—1461no—
424C2×C22⋊C4D29211yescharacteristic
425C2×C22⋊C4D29211yescharacteristic
426C2×C42—1461no—
427C2×C42—1461no—
428C2×C42—1461no—
429C2×C42C22×D7311yescharacteristic
430C2×C42C22×D7311yescharacteristic
431C4⋊C8—22no—
432C4⋊C8—22no—
433C22⋊C8—22no—
434C22⋊C8—22no—
435#97—22no—
436C4×C8—22no—
437#95—1461no—
438#95—1461no—
439#95—1461no—
440#95—1461no—
441#95—1461no—
442#95—1461no—
443C24C2×D29211yescharacteristic
444C2×Q8—5841no—
445C2×Q8—5841no—
446C2×Q8—5841no—
447C2×Q8—5841no—
448C2×Q8—1464no—
449C2×Q8—1464no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—5841no—
455C22×C4—5841no—
456C22×C4—1461no—
457C22×C4—1461no—
458C22×C4—1461no—
459C22×C4—1461no—
460C22×C4—1461no—
461C22×C4—1461no—
462C22×C4—1461no—
463C22×C4—1461no—
464C22×C4—1464no—
465C22×C4C2×C4×D7312yes—
466C22×C4C2×C22⋊D7312yes—
467C22×C4C2×D29211yescharacteristic
468C22×C4C2×D29211yescharacteristic
469C22×C4C2×D29211yescharacteristic
470C22×C4C2×D29211yescharacteristic
471C22×C4C2×D29211yescharacteristic
472C22×C4C23×D7311yescharacteristic
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C4⋊C4—5841no—
486C4⋊C4—5841no—
487C4⋊C4—732no—
488C4⋊C4—732no—
489C4⋊C4—732no—
490C4⋊C4—732no—
491C4⋊C4—1461no—
492C4⋊C4—1461no—
493C4⋊C4—1461no—
494C4⋊C4—1461no—
495C4⋊C4—1461no—
496C4⋊C4—1461no—
497C4⋊C4—1461no—
498C4⋊C4—1461no—
499C4⋊C4—1464no—
500C4⋊C4—1464no—
501C4⋊C4—1464no—
502C4⋊C4—1464no—
503C4⋊C4—1464no—
504C4⋊C4—1464no—
505C4⋊C4—1464no—
506C4⋊C4—1461no—
507C4⋊C4—1461no—
508C4⋊C4—1461no—
509C4⋊C4—1461no—
510C4⋊C4#1192400514yes—
511C4⋊C4#1192400514yes—
512C22⋊C4—5841no—
513C22⋊C4—5841no—
514C22⋊C4—1464no—
515C22⋊C4#1192400514yes—
516C22⋊C4#1192400514yes—
517C42—5841no—
518C42—5841no—
519C42—1464no—
520C42C2×C4×D7312yes—
521C42C2×C4×D7312yes—
522C42C2×C22⋊D7312yes—
523C42C2×C22⋊D7312yes—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23C2×#1192400512yes—
531C23C22×D29211yescharacteristic
532Q8—5844no—
533Q8—5844no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—2921no—
549C2×C4—2921no—
550C2×C4—2921no—
551C2×C4—2921no—
552C2×C4—2921no—
553C2×C4—2921no—
554C2×C4—2921no—
555C2×C4—2921no—
556C2×C4—2921no—
557C2×C4—2921no—
558C2×C4—2921no—
559C2×C4—2921no—
560C2×C4—2921no—
561C2×C4—2921no—
562C2×C4—2921no—
563C2×C4—2921no—
564C2×C4—2921no—
565C2×C4—2921no—
566C2×C4—2921no—
567C2×C4—2921no—
568C2×C4—5844no—
569C2×C4—1462no—
570C2×C4—1462no—
571C2×C4—1462no—
572C2×C4—1462no—
573C2×C4C2×#1192400512yes—
574C2×C4C2×#1192400512yes—
575C2×C4C2×#1192400512yes—
576C2×C4C2×#1192400512yes—
577C2×C4C2×#1192400512yes—
578C2×C4C22×C4×D7311yescharacteristic
579C2×C4C22×C22⋊D7311yescharacteristic
580C2×C4#1455663811yescharacteristic
581C2×C4#1455663811yescharacteristic
582C2×C4#1455663811yescharacteristic
583C2×C4#1455663811yescharacteristic
584C8—41no—
585C8—41no—
586C8—41no—
587C8—41no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22C22×#1192400511yescharacteristic
596C22#1798916311yescharacteristic
597C22#1798916311yescharacteristic
598C22#1798939211yescharacteristic
599C22#1798939911yescharacteristic
600C22#1798939911yescharacteristic
601C22#1798940411yescharacteristic
602C4—22no—
603C4—22no—
604C4—22no—
605C4—22no—
606C4—22no—
607C4—2922no—
608C4—2922no—
609C4—2922no—
610C4—2922no—
611C4—5841no—
612C4—5841no—
613C4—5841no—
614C4—5841no—
615C4#1798906411yescharacteristic
616C4#1798906411yescharacteristic
617C4#1798915811yescharacteristic
618C4#1798915811yescharacteristic
619C2—42no—
620C2#2261251511yescharacteristic
621C2#2261845811yescharacteristic
622C2#2261846511yescharacteristic
623C2#2262118611yescharacteristic
624C2#2262120111yescharacteristic
625C2#2262121611yescharacteristic
626C2#2262123111yescharacteristic
subgroupG/H
centerC23C22×D292
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC2×#433×C73C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
cosocleC22×C4×C73C24
hypercenterC2×#433D73
hypocenterC73#21843
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#433

Complements

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

NH ≅ G/Ntype
3. #22610904C2semidirect
4. #22610900C2semidirect
5. #22610896C2semidirect
6. #22610892C2semidirect
7. #22610886C2semidirect
8. #22610886C2semidirect
9. #22610882C2semidirect
10. #22610878C2semidirect
17. #17989215C4semidirect
329. C73#21843semidirect

Maximal subgroups

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

subgroupindex
C2×#179892152
#226125402
#226109042
#226109002
#226108962
#226108922
#226108862
#226108822
#226108782
#226100272
#226100222
#226100182
C2×#433×C732
#2184373