Finite Group Databasebeta

← #33137906

Subgroups — #33137906

Order 18688.

754 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988887C211yesmaximal, characteristic
2C2×#17988718C211yesmaximal, characteristic, complemented
3#22615319C211yesmaximal, characteristic, complemented
4#22611432C211yesmaximal, characteristic, complemented
5#22611422C211yesmaximal, characteristic, complemented
6#22611319C211yesmaximal, characteristic
7#22611305C211yesmaximal, characteristic
8#22610895C211yesmaximal, characteristic, complemented
9#22610697C211yesmaximal, characteristic, complemented
10#22610671C211yesmaximal, characteristic, complemented
11#22610666C211yesmaximal, characteristic, complemented
12#22610631C211yesmaximal, characteristic
13#22610631C211yesmaximal, characteristic
14C4×#14556512C211yesmaximal, characteristic
15C73×#2580C211yesmaximal, characteristic
16C2×#14556583C2211yescharacteristic
17#17988887—24no—
18#17988718—24no—
19C2×#14556524C2211yescharacteristic
20C2×#14556524C2211yescharacteristic
21C2×#14556522C2211yescharacteristic
22C2×#14556522C2211yescharacteristic
23C2×C4⋊#10596831C2211yescharacteristic
24C2×C4⋊#10596831C2211yescharacteristic
25C2×C4⋊#10596831C2211yescharacteristic
26C2×#14556512C2211yescharacteristic
27C2×#14556512C2211yescharacteristic
28C2×#14556512C2211yescharacteristic
29C2×#14556512C2211yescharacteristic
30C2×#14556502C2211yescharacteristic
31C2×#14556502C2211yescharacteristic
32C2×C4×#10596831C2211yescharacteristic
33#17988518—21no—
34#17988506—21no—
35#17988485C2211yescharacteristic
36#17988485C2211yescharacteristic
37#17988485C2211yescharacteristic
38#17988485C2211yescharacteristic
39#17988482—21no—
40#17988482—21no—
41C4×C4⋊Dic73C2211yescharacteristic
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50C2×C4×Q8×C73C2211yescharacteristic
51C2×C4×D4×C73C2211yescharacteristic
52#399×C73—21no—
53#393×C73—21no—
54#390×C73C2211yescharacteristic
55#390×C73C2211yescharacteristic
56#389×C73C2211yescharacteristic
57#389×C73C2211yescharacteristic
58#385×C73—21no—
59#385×C73—21no—
60#382×C73C2211yescharacteristic
61C22×#10596831C2311yescharacteristic
62C22×#10596831C2311yescharacteristic
63C22×#10596831C2311yescharacteristic
64C22×#10596831C2311yescharacteristic
65#14556583—22no—
66#14556583—22no—
67C2×C4⋊Dic73C2311yescharacteristic
68C2×C4⋊Dic73C2311yescharacteristic
69C2×C4⋊Dic73C2311yescharacteristic
70C2×C4⋊Dic73C2311yescharacteristic
71C2×#11924007—21no—
72C2×#11924007—21no—
73C2×#11924007—21no—
74C2×#11924007—21no—
75C2×#11924007—21no—
76C2×#11924007—21no—
77C2×C4×Dic73—21no—
78#14556524—22no—
79#14556524—22no—
80#14556522—22no—
81#14556522—22no—
82C4⋊#10596831—22no—
83C4⋊#10596831—22no—
84C4⋊#10596831D414yes—
85#14556512—22no—
86#14556512—22no—
87#14556512—22no—
88#14556512Q814yescomplemented
89#14556502—22no—
90#14556502—22no—
91C4×#10596831—22no—
92#14556498—21no—
93#14556498—21no—
94#14556498—21no—
95#14556498—21no—
96#14556498—21no—
97C22×Q8×C73D411yescharacteristic
98C22×D4×C73C2311yescharacteristic
99C23×C4×C73—21no—
100C4×Q8×C73—22no—
101C4×Q8×C73—22no—
102C4×D4×C73—22no—
103C4×D4×C73—22no—
104C2×C4⋊C4×C73—21no—
105C2×C4⋊C4×C73—21no—
106C2×C4⋊C4×C73D411yescharacteristic
107C2×C4⋊C4×C73D411yescharacteristic
108C2×C4⋊C4×C73D411yescharacteristic
109C2×C4⋊C4×C73C2311yescharacteristic
110C2×C4⋊C4×C73C2311yescharacteristic
111C2×C4⋊C4×C73C2311yescharacteristic
112C2×C22⋊C4×C73—21no—
113C2×C22⋊C4×C73—21no—
114C2×C22⋊C4×C73—21no—
115C2×C22⋊C4×C73—21no—
116C2×C22⋊C4×C73—21no—
117C2×C42×C73C2311yescharacteristic
118C2×C42×C73C2311yescharacteristic
119C2×C42×C73C2311yescharacteristic
120#95×C73—21no—
121#95×C73—21no—
122#95×C73—21no—
123#95×C73—21no—
124C22×Dic73—21no—
125C22×Dic73—21no—
126C22×Dic73—21no—
127C22×Dic73—21no—
128C2×#10596831—21no—
129C2×#10596831—21no—
130C2×#10596831—21no—
131C2×#10596831—21no—
132C2×#10596831—21no—
133C2×#10596831—21no—
134C2×#10596831—21no—
135C2×#10596831—21no—
136C2×#10596831—21no—
137C2×#10596831C2×D412yes—
138C2×#10596831C2×D412yes—
139C2×#10596831C2×Q812yes—
140C4⋊Dic73—21no—
141C4⋊Dic73—21no—
142C4⋊Dic73—21no—
143C4⋊Dic73—21no—
144C4⋊Dic73—21no—
145C4⋊Dic73—21no—
146C4⋊Dic73C2×Q812yes—
147C4⋊Dic73#4112yes—
148#11924007—42no—
149#11924007—42no—
150#11924007—42no—
151#11924007—42no—
152#11924007—81no—
153#11924007—81no—
154C4×Dic73—81no—
155C24×C73—21no—
156C2×Q8×C73—22no—
157C2×Q8×C73—41no—
158C2×Q8×C73—41no—
159C2×D4×C73—21no—
160C2×D4×C73—21no—
161C2×D4×C73—21no—
162C2×D4×C73—21no—
163C2×D4×C73—21no—
164C2×D4×C73C2×Q812yes—
165C22×C4×C73—21no—
166C22×C4×C73—21no—
167C22×C4×C73—21no—
168C22×C4×C73—42no—
169C22×C4×C73—81no—
170C22×C4×C73—81no—
171C22×C4×C73C2×D411yescharacteristic
172C22×C4×C73C2×D411yescharacteristic
173C22×C4×C73C2×D411yescharacteristic
174C22×C4×C73C2×D411yescharacteristic
175C22×C4×C73C2×D411yescharacteristic
176C22×C4×C73C2×D411yescharacteristic
177C22×C4×C73C2411yescharacteristic
178C4⋊C4×C73—22no—
179C4⋊C4×C73—22no—
180C4⋊C4×C73—22no—
181C4⋊C4×C73—21no—
182C4⋊C4×C73—21no—
183C4⋊C4×C73—21no—
184C4⋊C4×C73—21no—
185C4⋊C4×C73—21no—
186C4⋊C4×C73—21no—
187C4⋊C4×C73—81no—
188C4⋊C4×C73—81no—
189C22⋊C4×C73—42no—
190C22⋊C4×C73—42no—
191C22⋊C4×C73—42no—
192C22⋊C4×C73—81no—
193C22⋊C4×C73—81no—
194C42×C73—21no—
195C42×C73—21no—
196C42×C73—21no—
197C42×C73—21no—
198C42×C73C2×D412yes—
199C42×C73#4112yes—
200C23×C73—22no—
201C23×C73—41no—
202C23×C73—41no—
203C23×C73—41no—
204C23×C73—41no—
205C23×C73—41no—
206C23×C73—41no—
207C23×C73C22×D411yescharacteristic
208Q8×C73—44no—
209D4×C73—22no—
210D4×C73—22no—
211D4×C73—41no—
212D4×C73—41no—
213C2×C4×C73—21no—
214C2×C4×C73—21no—
215C2×C4×C73—21no—
216C2×C4×C73—21no—
217C2×C4×C73—21no—
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—21no—
229C2×C4×C73—41no—
230C2×C4×C73—41no—
231C2×C4×C73—41no—
232C2×C4×C73—41no—
233C2×C4×C73—41no—
234C2×C4×C73—41no—
235C2×C4×C73—41no—
236C2×C4×C73—41no—
237C2×C4×C73—41no—
238C2×C4×C73—84no—
239C2×C4×C73#12112yes—
240C2×C4×C73#12112yes—
241C2×C4×C73C22×D411yescharacteristic
242C2×C4×C73C22×Q811yescharacteristic
243C2×C4×C73C2×#4111yescharacteristic
244C2×C4×C73C2×#4111yescharacteristic
245C2×C4×C732−1+411yescharacteristic
246C2×C4×C732−1+411yescharacteristic
247C2×Dic73—22no—
248C2×Dic73—22no—
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—
259#10596831—22no—
260#10596831—22no—
261#10596831—22no—
262#10596831—41no—
263C22×C73—42no—
264C22×C73—42no—
265C22×C73—42no—
266C22×C73—81no—
267C22×C73—81no—
268C22×C73—81no—
269C22×C73—81no—
270C22×C73C2×#12111yescharacteristic
271C22×C73#54811yescharacteristic
272C22×C73D4×Q811yescharacteristic
273C22×C73#57511yescharacteristic
274C22×C73#57511yescharacteristic
275C22×C73#57611yescharacteristic
276C22×C73#57711yescharacteristic
277C4×C73—22no—
278C4×C73—22no—
279C4×C73—41no—
280C4×C73—41no—
281C4×C73—41no—
282C4×C73—41no—
283C4×C73—81no—
284C4×C73—81no—
285C4×C73—81no—
286C4×C73#53311yescharacteristic
287C4×C73#54111yescharacteristic
288C4×C73#54711yescharacteristic
289C4×C73D4×Q811yescharacteristic
290Dic73—42no—
291Dic73—42no—
292Dic73—81no—
293Dic73—81no—
294#25475—731nomaximal, Sylow
295C2×C73—82no—
296C2×C73#242911yescharacteristic
297C2×C73#305411yescharacteristic
298C2×C73#306211yescharacteristic
299C2×C73#337011yescharacteristic
300C2×C73#337111yescharacteristic
301C2×C73#338811yescharacteristic
302C2×C73#338911yescharacteristic
303C2×#478—731no—
304C2×#446—731no—
305#2584—731no—
306#2580D7311yescharacteristic
307#2088—731no—
308#2086—731no—
309#2067—731no—
310#2065—731no—
311#1991—731no—
312#1949—731no—
313#1944—731no—
314#1943—731no—
315#1936—731no—
316#1936—731no—
317C4×#102—731no—
318C73#2547511yescharacteristic, complemented, Sylow
319C2×#125—731no—
320C2×C4×Q8C2×D7311yescharacteristic
321C2×C4×D4C2×D7311yescharacteristic
322#478—1464no—
323#446—1464no—
324C2×#107—731no—
325C2×#107—731no—
326C2×#106—731no—
327C2×#106—731no—
328C2×C4⋊C8—731no—
329C2×C4⋊C8—731no—
330C2×C4⋊C8—731no—
331C2×#102—731no—
332C2×#102—731no—
333C2×#102—731no—
334C2×#102—731no—
335C2×#97—731no—
336C2×#97—731no—
337C2×C4×C8—731no—
338#399—21no—
339#398—1461no—
340#395—1461no—
341#393—21no—
342#390C2×D7311yescharacteristic
343#390C2×D7311yescharacteristic
344#389—731no—
345#389—731no—
346#389—731no—
347#389—731no—
348#389—1461no—
349#389—1461no—
350#389C2×D7311yescharacteristic
351#389C2×D7311yescharacteristic
352#385—21no—
353#385—21no—
354#382C2×D7311yescharacteristic
355C4×C4⋊C4—731no—
356#340—731no—
357#340—731no—
358#340—731no—
359#340—731no—
360#340—731no—
361#340—731no—
362#340—731no—
363#340—731no—
364C22×Q8C22⋊D7311yescharacteristic
365C22×D4C22×D7311yescharacteristic
366C23×C4—21no—
367C22×C8—731no—
368C22×C8—731no—
369C22×C8—731no—
370C22×C8—731no—
371#125—1462no—
372#125—1462no—
373C4×Q8—22no—
374C4×Q8—22no—
375C4×D4—22no—
376C4×D4—22no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—731no—
380C2×C4⋊C4—731no—
381C2×C4⋊C4—731no—
382C2×C4⋊C4—731no—
383C2×C4⋊C4—1461no—
384C2×C4⋊C4—1461no—
385C2×C4⋊C4—1461no—
386C2×C4⋊C4—1461no—
387C2×C4⋊C4—1461no—
388C2×C4⋊C4—1461no—
389C2×C4⋊C4C22⋊D7311yescharacteristic
390C2×C4⋊C4C22⋊D7311yescharacteristic
391C2×C4⋊C4C22⋊D7311yescharacteristic
392C2×C4⋊C4C22×D7311yescharacteristic
393C2×C4⋊C4C22×D7311yescharacteristic
394C2×C4⋊C4C22×D7311yescharacteristic
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—21no—
399C2×C22⋊C4—21no—
400C2×C42—1461no—
401C2×C42C22×D7311yescharacteristic
402C2×C42C22×D7311yescharacteristic
403C2×C42C22×D7311yescharacteristic
404#107—1462no—
405#107—1462no—
406#106—1462no—
407#106—1462no—
408C4⋊C8—734no—
409C4⋊C8—1462no—
410C4⋊C8—1462no—
411#102—734no—
412#102—1462no—
413#102—1462no—
414#102—1462no—
415#97—1462no—
416#97—1462no—
417C4×C8—1462no—
418#95—21no—
419#95—21no—
420#95—21no—
421#95—21no—
422#95—1461no—
423#95—1461no—
424#95—1461no—
425#95—1461no—
426#95—1461no—
427C24—21no—
428C2×Q8—22no—
429C2×Q8—41no—
430C2×Q8—41no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4Q8×D7312yes—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—42no—
441C22×C4—81no—
442C22×C4—81no—
443C22×C4—1461no—
444C22×C4—1461no—
445C22×C4—1461no—
446C22×C4—1461no—
447C22×C4C2×C22⋊D7311yescharacteristic
448C22×C4C2×C22⋊D7311yescharacteristic
449C22×C4C2×C22⋊D7311yescharacteristic
450C22×C4C2×C22⋊D7311yescharacteristic
451C22×C4C2×C22⋊D7311yescharacteristic
452C22×C4C2×C22⋊D7311yescharacteristic
453C22×C4C23×D7311yescharacteristic
454C2×C8—732no—
455C2×C8—732no—
456C2×C8—732no—
457C2×C8—1461no—
458C2×C8—1461no—
459C2×C8—1461no—
460C2×C8—1461no—
461C2×C8—1461no—
462C2×C8—1461no—
463C2×C8—1461no—
464C2×C8—1461no—
465C2×C8—1461no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—81no—
476C4⋊C4—81no—
477C4⋊C4—2922no—
478C4⋊C4—2922no—
479C4⋊C4—2922no—
480C4⋊C4—2922no—
481C4⋊C4—5841no—
482C4⋊C4—5841no—
483C4⋊C4—732no—
484C4⋊C4—732no—
485C4⋊C4—1461no—
486C4⋊C4—1461no—
487C4⋊C4—1461no—
488C4⋊C4—1461no—
489C4⋊C4—1461no—
490C4⋊C4—1461no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C22⋊C4—42no—
494C22⋊C4—81no—
495C22⋊C4—81no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—21no—
500C42—5841no—
501C42D4×D7312yes—
502C42#1192402712yes—
503C23—22no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23C22×C22⋊D7311yescharacteristic
511Q8—44no—
512D4—22no—
513D4—22no—
514D4—41no—
515D4—41no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—84no—
542C2×C4—2921no—
543C2×C4—2921no—
544C2×C4—2921no—
545C2×C4—2921no—
546C2×C4—2921no—
547C2×C4—2921no—
548C2×C4—2921no—
549C2×C4—2921no—
550C2×C4—2921no—
551C2×C4—2921no—
552C2×C4—1462no—
553C2×C4—1462no—
554C2×C4#1455656312yes—
555C2×C4#1455656312yes—
556C2×C4C2×D4×D7311yescharacteristic
557C2×C4C2×Q8×D7311yescharacteristic
558C2×C4#41×D7311yescharacteristic
559C2×C4C2×#1192402711yescharacteristic
560C2×C4#1455663811yescharacteristic
561C2×C4#1455663911yescharacteristic
562C8—2921no—
563C8—1462no—
564C8—1462no—
565C8—1462no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—81no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22C2×#1455656311yescharacteristic
574C22#1798924411yescharacteristic
575C22Q8×C22⋊D7311yescharacteristic
576C22#1798938511yescharacteristic
577C22#1798938511yescharacteristic
578C22#1798939011yescharacteristic
579C22#1798939811yescharacteristic
580C4—22no—
581C4—22no—
582C4—41no—
583C4—41no—
584C4—41no—
585C4—41no—
586C4—81no—
587C4—81no—
588C4—81no—
589C4—2922no—
590C4—2922no—
591C4—5841no—
592C4—5841no—
593C4#1798913911yescharacteristic
594C4#1798919711yescharacteristic
595C4#1798923911yescharacteristic
596C4#1798925011yescharacteristic
597C2—82no—
598C2#2261353211yescharacteristic
599C2#2261953711yescharacteristic
600C2#2261960711yescharacteristic
601C2#2262267811yescharacteristic
602C2#2262269311yescharacteristic
603C2#2262287611yescharacteristic
604C2#2262288511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2580C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
hypercenter#2580D73
hypocenterC73#25475
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2580

Complements

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

NH ≅ G/Ntype
2. C2×#17988718C2semidirect
3. #22615319C2semidirect
4. #22611432C2semidirect
5. #22611422C2semidirect
8. #22610895C2semidirect
9. #22610697C2semidirect
10. #22610671C2semidirect
11. #22610666C2semidirect
88. #14556512Q8semidirect
318. C73#25475semidirect

Maximal subgroups

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

subgroupindex
C2×#179888872
C2×#179887182
#226153192
#226114322
#226114222
#226113192
#226113052
#226108952
#226106972
#226106712
#226106662
#226106312
C4×#145565122
C73×#25802
#2547573