Finite Group Databasebeta

← #33064564

Subgroups — #33064564

Order 18688.

1170 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22621075C211yesmaximal, characteristic, complemented
2#22619939C211yesmaximal, characteristic, complemented
3#22619728C211yesmaximal, characteristic, complemented
4#22609215C214yesmaximal, complemented
5#22609082C214yesmaximal, complemented
6#22608603C211yesmaximal, characteristic, complemented
7#22608356C211yesmaximal, characteristic, complemented
8#22608322C211yesmaximal, characteristic, complemented
9C2×#547×C73C211yesmaximal, characteristic
10#17989144C2211yescharacteristic
11#17988983C2212yescomplemented
12#17988939C2212yescomplemented
13#17988917—21no—
14#17988917—22no—
15#17988887—22no—
16#17988877C2212yescomplemented
17#17988877C2212yescomplemented
18#17988869—22no—
19#17988867C2212yescomplemented
20#17988792C2212yescomplemented
21#17988738—21no—
22C4×#11924016—22no—
23#17988640C2211yescharacteristic
24#17988640C2211yescharacteristic
25#17988627—21no—
26#17988611C2211yescharacteristic
27#17988592—21no—
28#17988333C2214yescomplemented
29D4⋊#10596831C2214yescomplemented
30D4⋊#10596831C2214yescomplemented
31C22×#41×C73—21no—
32#547×C73—22no—
33#547×C73—22no—
34#547×C73—22no—
35#547×C73C2214yes—
36C2×C4⋊Q8×C73C2211yescharacteristic
37C2×#123×C73—22no—
38C2×C22⋊Q8×C73—21no—
39C2×#121×C73—21no—
40C2×C4×D4×C73C2211yescharacteristic
41C2×C4×D4×C73C2211yescharacteristic
42C22×C4⋊C4×C73—21no—
43C2×#11924016—22no—
44C2×#11924012—21no—
45#14556591C2311yescharacteristic
46#14556591C2311yescharacteristic
47#14556583—21no—
48#14556570—21no—
49#14556570—21no—
50C4×Dic146—21no—
51C4×Dic146—21no—
52#14556550—21no—
53#14556524—22no—
54#14556522—24no—
55#14556522—22no—
56#14556522—22no—
57C4⋊#10596831—21no—
58C4⋊#10596831—21no—
59C4⋊#10596831C2312yes—
60C4⋊#10596831C2312yes—
61#14556514—24no—
62#14556514—22no—
63#14556512—22no—
64#14556512—22no—
65#14556512—22no—
66C22⋊#10596831—21no—
67C22⋊#10596831—21no—
68C4×#10596831—21no—
69C4×#10596831C2312yes—
70C2×#41×C73—24no—
71C2×#41×C73—24no—
72C2×#41×C73—44no—
73C2×#41×C73—44no—
74C22×Q8×C73—21no—
75C22×D4×C73—22no—
76C22×D4×C73D411yescharacteristic
77C22×D4×C73D411yescharacteristic
78C23×C4×C73—22no—
79C23×C4×C73—21no—
80C23×C4×C73—21no—
81C4⋊Q8×C73—21no—
82C4⋊Q8×C73—21no—
83C4⋊Q8×C73D412yes—
84C4⋊Q8×C73C2312yes—
85#123×C73—28no—
86#123×C73—28no—
87C22⋊Q8×C73—24no—
88C22⋊Q8×C73—24no—
89#121×C73—24no—
90#121×C73—24no—
91C4×D4×C73—21no—
92C4×D4×C73—21no—
93C4×D4×C73—21no—
94C4×D4×C73—21no—
95C4×D4×C73D412yes—
96C4×D4×C73D412yes—
97C4×D4×C73C2312yes—
98C4×D4×C73C2312yes—
99C2×C4⋊C4×C73—22no—
100C2×C4⋊C4×C73—24no—
101C2×C4⋊C4×C73—21no—
102C2×C4⋊C4×C73—24no—
103C2×C4⋊C4×C73D411yescharacteristic
104C2×C4⋊C4×C73D411yescharacteristic
105C2×C4⋊C4×C73D412yes—
106C2×C22⋊C4×C73—22no—
107C2×C22⋊C4×C73—21no—
108C2×C22⋊C4×C73—21no—
109C2×C42×C73C2311yescharacteristic
110C2×Dic146—21no—
111C2×Dic146—21no—
112#11924016—44no—
113#11924012—42no—
114C2×#10596831—22no—
115C2×#10596831—22no—
116C2×#10596831—22no—
117C4⋊Dic73—21no—
118C4⋊Dic73—21no—
119C4⋊Dic73—21no—
120C4⋊Dic73—21no—
121#11924007—41no—
122#11924007—41no—
123C22⋊Dic73—41no—
124C4×Dic73—41no—
125C24×C73—21no—
126C24×C73—21no—
127#41×C73—48no—
128#41×C73—416no—
129#41×C73—48no—
130C2×Q8×C73—22no—
131C2×Q8×C73—22no—
132C2×Q8×C73—42no—
133C2×Q8×C73—42no—
134C2×D4×C73—28no—
135C2×D4×C73—21no—
136C2×D4×C73—21no—
137C2×D4×C73—22no—
138C2×D4×C73—42no—
139C2×D4×C73—48no—
140C2×D4×C73SD1614yescomplemented
141C2×D4×C73C2×D412yes—
142C2×D4×C73C2×D412yes—
143C22×C4×C73—28no—
144C22×C4×C73—21no—
145C22×C4×C73—21no—
146C22×C4×C73—22no—
147C22×C4×C73—22no—
148C22×C4×C73—22no—
149C22×C4×C73—22no—
150C22×C4×C73—22no—
151C22×C4×C73—44no—
152C22×C4×C73—48no—
153C22×C4×C73—42no—
154C22×C4×C73—42no—
155C22×C4×C73C2×D411yescharacteristic
156C22×C4×C73C2×D411yescharacteristic
157C22×C4×C73C2×D411yescharacteristic
158C4⋊C4×C73—28no—
159C4⋊C4×C73—21no—
160C4⋊C4×C73—22no—
161C4⋊C4×C73—21no—
162C4⋊C4×C73—22no—
163C4⋊C4×C73—22no—
164C4⋊C4×C73C2×D412yes—
165C4⋊C4×C73C2×D412yes—
166C4⋊C4×C73C2×D414yes—
167C22⋊C4×C73—28no—
168C22⋊C4×C73—22no—
169C22⋊C4×C73—22no—
170C22⋊C4×C73—22no—
171C22⋊C4×C73—22no—
172C42×C73C2×D411yescharacteristic
173C42×C73C2×D411yescharacteristic
174C42×C73C2×D411yescharacteristic
175C42×C73C2411yescharacteristic
176C23×C73—22no—
177C23×C73—24no—
178C23×C73—22no—
179C23×C73—42no—
180C23×C73—42no—
181C23×C73—42no—
182C23×C73—42no—
183C23×C73—42no—
184C23×C73#12011yescharacteristic
185Q8×C73—42no—
186Q8×C73—42no—
187Q8×C73—44no—
188D4×C73—22no—
189D4×C73—22no—
190D4×C73—22no—
191D4×C73—24no—
192D4×C73—42no—
193D4×C73—416no—
194D4×C73C2×SD1614yes—
195C2×C4×C73—22no—
196C2×C4×C73—22no—
197C2×C4×C73—22no—
198C2×C4×C73—24no—
199C2×C4×C73—41no—
200C2×C4×C73—44no—
201C2×C4×C73—42no—
202C2×C4×C73—416no—
203C2×C4×C73—42no—
204C2×C4×C73—44no—
205C2×C4×C73—42no—
206C2×C4×C73—42no—
207C2×C4×C73—44no—
208C2×C4×C73—44no—
209C2×C4×C73#12011yescharacteristic
210C2×C4×C73#12011yescharacteristic
211C2×C4×C73#12011yescharacteristic
212C2×C4×C73C2×SD1612yes—
213C2×C4×C73AGL(1,ℤ/8)12yes—
214C2×C4×C73C22×D411yescharacteristic
215C2×C4×C73C22×D411yescharacteristic
216C2×C4×C73C22×D411yescharacteristic
217C2×Dic73—41no—
218C2×Dic73—41no—
219Dic146—42no—
220Dic146—42no—
221#10596831—42no—
222#10596831—42no—
223#10596831—42no—
224C22×C73—21no—
225C22×C73—21no—
226C22×C73—24no—
227C22×C73—48no—
228C22×C73—42no—
229C22×C73—42no—
230C22×C73—42no—
231C22×C73—42no—
232C22×C73—42no—
233C22×C73—84no—
234C22×C73#45012yes—
235C22×C73C2×#12011yescharacteristic
236C4×C73—21no—
237C4×C73—21no—
238C4×C73—21no—
239C4×C73—42no—
240C4×C73—44no—
241C4×C73—42no—
242C4×C73#45012yes—
243C4×C73C22×SD1611yescharacteristic
244C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
245C4×C73#57512yes—
246Dic73—81no—
247Dic73—81no—
248#18757—731nomaximal, Sylow
249C2×C73—22no—
250C2×C73—44no—
251C2×C73—82no—
252C2×C73C2×#45011yescharacteristic
253C2×C73#300511yescharacteristic
254C2×C73#302011yescharacteristic
255C2×#547D7311yescharacteristic
256#3229—731no—
257#3101—731no—
258#3077—731no—
259#1654—734no—
260#1635—734no—
261#1546—731no—
262#1486—731no—
263#1478—731no—
264C73#1875711yescharacteristic, complemented, Sylow
265C22×#41—21no—
266#547—22no—
267#547—22no—
268#547—22no—
269#547C2×D7314yes—
270#533—731no—
271C2×C4⋊Q8C2×D7311yescharacteristic
272C2×#123—22no—
273C2×C22⋊Q8—21no—
274C2×#121—21no—
275C2×C4×D4C2×D7311yescharacteristic
276C2×C4×D4C2×D7311yescharacteristic
277C22×C4⋊C4—21no—
278#498—732no—
279#487—732no—
280#483—1461no—
281#483—1462no—
282#478—1462no—
283#476—732no—
284#476—732no—
285#475—1462no—
286#474—732no—
287#461—732no—
288#450—1461no—
289C4×SD16—1462no—
290#431—731no—
291#431—731no—
292#428—1461no—
293#423—731no—
294#418—1461no—
295#332—734no—
296D4⋊C8—734no—
297D4⋊C8—734no—
298C2×#41—24no—
299C2×#41—24no—
300C2×#41—44no—
301C2×#41—44no—
302C22×Q8—21no—
303C22×D4—22no—
304C22×D4C22⋊D7311yescharacteristic
305C22×D4C22⋊D7311yescharacteristic
306C23×C4—22no—
307C23×C4—21no—
308C23×C4—21no—
309C2×SD16—1462no—
310C2×M16—1461no—
311C4⋊Q8—21no—
312C4⋊Q8—21no—
313C4⋊Q8—731no—
314C4⋊Q8—731no—
315C4⋊Q8C22⋊D7312yes—
316C4⋊Q8C22×D7312yes—
317#125—1461no—
318#123—28no—
319#123—28no—
320C22⋊Q8—24no—
321C22⋊Q8—24no—
322C22⋊Q8—1461no—
323C22⋊Q8—1461no—
324#121—24no—
325#121—24no—
326C4×Q8—1461no—
327C4×Q8—1461no—
328C4×D4—21no—
329C4×D4—21no—
330C4×D4—21no—
331C4×D4—21no—
332C4×D4C22⋊D7312yes—
333C4×D4C22⋊D7312yes—
334C4×D4C22×D7312yes—
335C4×D4C22×D7312yes—
336#117—1461no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4C22⋊D7311yescharacteristic
342C2×C4⋊C4C22⋊D7311yescharacteristic
343C2×C4⋊C4C22⋊D7312yes—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—21no—
347C2×C42C22×D7311yescharacteristic
348#107—1462no—
349#106—1462no—
350#106—1462no—
351#106—1464no—
352C4⋊C8—732no—
353C4⋊C8—732no—
354C4⋊C8—1461no—
355C4⋊C8—1461no—
356#103—1462no—
357#103—1464no—
358#102—1462no—
359#102—1462no—
360#102—1462no—
361C22⋊C8—1461no—
362C22⋊C8—1461no—
363C4×C8—732no—
364C4×C8—1461no—
365C24—21no—
366C24—21no—
367#41—416no—
368#41—48no—
369#41—48no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—42no—
373C2×Q8—42no—
374C2×Q8—1461no—
375C2×Q8—1461no—
376C2×D4—28no—
377C2×D4—21no—
378C2×D4—21no—
379C2×D4—22no—
380C2×D4—42no—
381C2×D4—48no—
382C2×D4#1192401614yescomplemented
383C2×D4C2×C22⋊D7312yes—
384C2×D4C2×C22⋊D7312yes—
385C22×C4—28no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—44no—
394C22×C4—48no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4C2×C22⋊D7311yescharacteristic
398C22×C4C2×C22⋊D7311yescharacteristic
399C22×C4C2×C22⋊D7311yescharacteristic
400SD16—2924no—
401M16—2922no—
402C2×C8—1462no—
403C2×C8—1462no—
404C2×C8—1462no—
405C4⋊C4—28no—
406C4⋊C4—21no—
407C4⋊C4—22no—
408C4⋊C4—21no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—2921no—
412C4⋊C4—2921no—
413C4⋊C4—1461no—
414C4⋊C4—1461no—
415C4⋊C4—1461no—
416C4⋊C4—1461no—
417C4⋊C4C2×C22⋊D7312yes—
418C4⋊C4C2×C22⋊D7314yes—
419C4⋊C4C2×C22⋊D7312yes—
420C22⋊C4—28no—
421C22⋊C4—22no—
422C22⋊C4—22no—
423C22⋊C4—22no—
424C22⋊C4—22no—
425C22⋊C4—2921no—
426C42—2921no—
427C42C2×C22⋊D7311yescharacteristic
428C42C2×C22⋊D7311yescharacteristic
429C42C2×C22⋊D7311yescharacteristic
430C42C23×D7311yescharacteristic
431C23—24no—
432C23—22no—
433C23—22no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23#1455656011yescharacteristic
440Q8—44no—
441Q8—42no—
442Q8—42no—
443Q8—2922no—
444Q8—2922no—
445D4—22no—
446D4—22no—
447D4—24no—
448D4—22no—
449D4—42no—
450D4—416no—
451D4C2×#1192401614yes—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—41no—
457C2×C4—44no—
458C2×C4—416no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—2921no—
467C2×C4—2921no—
468C2×C4#1455656011yescharacteristic
469C2×C4#1455656011yescharacteristic
470C2×C4#1455656011yescharacteristic
471C2×C4C2×#1192401612yes—
472C2×C4#1455661712yes—
473C2×C4C22×C22⋊D7311yescharacteristic
474C2×C4C22×C22⋊D7311yescharacteristic
475C2×C4C22×C22⋊D7311yescharacteristic
476C8—2922no—
477C8—2922no—
478C8—2922no—
479C22—21no—
480C22—21no—
481C22—24no—
482C22—48no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—84no—
489C22#1798873812yes—
490C22C2×#1455656011yescharacteristic
491C4—21no—
492C4—21no—
493C4—21no—
494C4—42no—
495C4—42no—
496C4—44no—
497C4—5841no—
498C4—5841no—
499C4#1798873812yes—
500C4C22×#1192401611yescharacteristic
501C4C2×#1455661711yescharacteristic
502C4#1798938512yes—
503C2—22no—
504C2—44no—
505C2—82no—
506C2C2×#1798873811yescharacteristic
507C2#2261917311yescharacteristic
508C2#2261929511yescharacteristic
subgroupG/H
centerC22C2×#14556560
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
Fitting, prime radicalC2×#547×C73C2
socleC22×C73C2×#120
hypercenterC2×#547D73
hypocenterC73#18757
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. #22621075C2semidirect
2. #22619939C2semidirect
3. #22619728C2semidirect
4. #22609215C2semidirect
5. #22609082C2semidirect
6. #22608603C2semidirect
7. #22608356C2semidirect
8. #22608322C2semidirect
11. #17988983C22semidirect
12. #17988939C22semidirect
16. #17988877C22semidirect
17. #17988877C22semidirect
19. #17988867C22semidirect
20. #17988792C22semidirect
28. #17988333C22semidirect
29. D4⋊#10596831C22semidirect
30. D4⋊#10596831C22semidirect
140. C2×D4×C73SD16semidirect
264. C73#18757semidirect
382. C2×D4#11924016semidirect

Maximal subgroups

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

subgroupindex
#226210752
#226199392
#226197282
#226092152
#226090822
#226086032
#226083562
#226083222
C2×#547×C732
#1875773