Finite Group Databasebeta

← #33064822

Subgroups — #33064822

Order 18688.

1170 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22621082C211yesmaximal, characteristic, complemented
2#22619939C211yesmaximal, characteristic, complemented
3#22619728C211yesmaximal, characteristic, complemented
4#22609166C214yesmaximal, complemented
5#22609082C214yesmaximal, complemented
6#22608616C211yesmaximal, characteristic, complemented
7#22608356C211yesmaximal, characteristic, complemented
8#22608322C211yesmaximal, characteristic, complemented
9C2×#547×C73C211yesmaximal, characteristic
10#17989144C2211yescharacteristic
11#17988990C2212yescomplemented
12#17988939C2212yescomplemented
13#17988922—22no—
14#17988917—21no—
15#17988887—22no—
16#17988879—22no—
17#17988877C2212yescomplemented
18#17988877C2212yescomplemented
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#14556524—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—48no—
73C22×Q8×C73—21no—
74C22×D4×C73—22no—
75C22×D4×C73D411yescharacteristic
76C22×D4×C73D411yescharacteristic
77C23×C4×C73—22no—
78C23×C4×C73—21no—
79C23×C4×C73—21no—
80C4⋊Q8×C73—21no—
81C4⋊Q8×C73—21no—
82C4⋊Q8×C73D412yes—
83C4⋊Q8×C73C2312yes—
84#123×C73—28no—
85#123×C73—28no—
86C22⋊Q8×C73—24no—
87C22⋊Q8×C73—24no—
88#121×C73—24no—
89#121×C73—24no—
90C4×D4×C73—21no—
91C4×D4×C73—21no—
92C4×D4×C73—21no—
93C4×D4×C73—21no—
94C4×D4×C73D412yes—
95C4×D4×C73D412yes—
96C4×D4×C73C2312yes—
97C4×D4×C73C2312yes—
98C2×C4⋊C4×C73—22no—
99C2×C4⋊C4×C73—24no—
100C2×C4⋊C4×C73—21no—
101C2×C4⋊C4×C73—24no—
102C2×C4⋊C4×C73D411yescharacteristic
103C2×C4⋊C4×C73D411yescharacteristic
104C2×C4⋊C4×C73D412yes—
105C2×C22⋊C4×C73—22no—
106C2×C22⋊C4×C73—21no—
107C2×C22⋊C4×C73—21no—
108C2×C42×C73C2311yescharacteristic
109C2×Dic146—21no—
110C2×Dic146—21no—
111#11924016—44no—
112#11924012—42no—
113C2×#10596831—22no—
114C2×#10596831—22no—
115C2×#10596831—22no—
116C4⋊Dic73—21no—
117C4⋊Dic73—21no—
118C4⋊Dic73—21no—
119C4⋊Dic73—21no—
120#11924007—41no—
121#11924007—41no—
122C22⋊Dic73—41no—
123C4×Dic73—41no—
124C24×C73—21no—
125C24×C73—21no—
126#41×C73—416no—
127#41×C73—48no—
128#41×C73—48no—
129C2×Q8×C73—22no—
130C2×Q8×C73—22no—
131C2×Q8×C73—44no—
132C2×D4×C73—24no—
133C2×D4×C73—28no—
134C2×D4×C73—21no—
135C2×D4×C73—21no—
136C2×D4×C73—22no—
137C2×D4×C73—22no—
138C2×D4×C73—48no—
139C2×D4×C73C2×D412yes—
140C2×D4×C73C2×D412yes—
141C22×C4×C73—28no—
142C22×C4×C73—21no—
143C22×C4×C73—21no—
144C22×C4×C73—22no—
145C22×C4×C73—22no—
146C22×C4×C73—22no—
147C22×C4×C73—22no—
148C22×C4×C73—22no—
149C22×C4×C73—44no—
150C22×C4×C73—48no—
151C22×C4×C73—42no—
152C22×C4×C73—42no—
153C22×C4×C73C2×D411yescharacteristic
154C22×C4×C73C2×D411yescharacteristic
155C22×C4×C73C2×D411yescharacteristic
156C4⋊C4×C73—28no—
157C4⋊C4×C73—21no—
158C4⋊C4×C73—22no—
159C4⋊C4×C73—21no—
160C4⋊C4×C73—22no—
161C4⋊C4×C73—22no—
162C4⋊C4×C73C2×D412yes—
163C4⋊C4×C73C2×D412yes—
164C4⋊C4×C73C2×D414yes—
165C22⋊C4×C73—28no—
166C22⋊C4×C73—22no—
167C22⋊C4×C73—22no—
168C22⋊C4×C73—22no—
169C22⋊C4×C73—22no—
170C42×C73C2×D411yescharacteristic
171C42×C73C2×D411yescharacteristic
172C42×C73C2×D411yescharacteristic
173C42×C73C2411yescharacteristic
174C23×C73—24no—
175C23×C73—22no—
176C23×C73—22no—
177C23×C73—42no—
178C23×C73—42no—
179C23×C73—42no—
180C23×C73—42no—
181C23×C73—42no—
182C23×C73#12011yescharacteristic
183Q8×C73—44no—
184Q8×C73—44no—
185D4×C73—24no—
186D4×C73—22no—
187D4×C73—22no—
188D4×C73—22no—
189D4×C73—416no—
190D4×C73—42no—
191D4×C73AGL(1,ℤ/8)14yes—
192C2×C4×C73—22no—
193C2×C4×C73—22no—
194C2×C4×C73—22no—
195C2×C4×C73—24no—
196C2×C4×C73—41no—
197C2×C4×C73—44no—
198C2×C4×C73—42no—
199C2×C4×C73—44no—
200C2×C4×C73—416no—
201C2×C4×C73—42no—
202C2×C4×C73—42no—
203C2×C4×C73—44no—
204C2×C4×C73—42no—
205C2×C4×C73—44no—
206C2×C4×C73#12011yescharacteristic
207C2×C4×C73#12011yescharacteristic
208C2×C4×C73#12011yescharacteristic
209C2×C4×C73#13512yes—
210C2×C4×C73#13712yes—
211C2×C4×C73C22×D411yescharacteristic
212C2×C4×C73C22×D411yescharacteristic
213C2×C4×C73C22×D411yescharacteristic
214C2×Dic73—41no—
215C2×Dic73—41no—
216Dic146—42no—
217Dic146—42no—
218#10596831—42no—
219#10596831—42no—
220#10596831—42no—
221C22×C73—21no—
222C22×C73—21no—
223C22×C73—24no—
224C22×C73—48no—
225C22×C73—42no—
226C22×C73—42no—
227C22×C73—42no—
228C22×C73—42no—
229C22×C73—42no—
230C22×C73—84no—
231C22×C73#45212yes—
232C22×C73C2×#12011yescharacteristic
233C4×C73—21no—
234C4×C73—21no—
235C4×C73—21no—
236C4×C73—42no—
237C4×C73—42no—
238C4×C73—44no—
239C4×C73#45212yes—
240C4×C73C2×#13511yescharacteristic
241C4×C73C2×AGL(1,ℤ/8)12yes—
242C4×C73C2×#13711yescharacteristic
243Dic73—81no—
244Dic73—81no—
245#18783—731nomaximal, Sylow
246C2×C73—22no—
247C2×C73—44no—
248C2×C73—82no—
249C2×C73C2×#45211yescharacteristic
250C2×C73#300311yescharacteristic
251C2×C73#302011yescharacteristic
252C2×#547D7311yescharacteristic
253#3230—731no—
254#3101—731no—
255#3077—731no—
256#1647—734no—
257#1635—734no—
258#1549—731no—
259#1486—731no—
260#1478—731no—
261C73#1878311yescharacteristic, complemented, Sylow
262C22×#41—21no—
263#547—22no—
264#547—22no—
265#547—22no—
266#547C2×D7314yes—
267#533—731no—
268C2×C4⋊Q8C2×D7311yescharacteristic
269C2×#123—22no—
270C2×C22⋊Q8—21no—
271C2×#121—21no—
272C2×C4×D4C2×D7311yescharacteristic
273C2×C4×D4C2×D7311yescharacteristic
274C22×C4⋊C4—21no—
275#500—732no—
276#487—732no—
277#484—1462no—
278#483—1461no—
279#478—1462no—
280#477—1462no—
281#476—732no—
282#476—732no—
283#474—732no—
284#461—732no—
285#450—1461no—
286C4×SD16—1462no—
287#431—731no—
288#431—731no—
289#428—1461no—
290#423—731no—
291#418—1461no—
292#332—734no—
293D4⋊C8—734no—
294D4⋊C8—734no—
295C2×#41—24no—
296C2×#41—24no—
297C2×#41—48no—
298C22×Q8—21no—
299C22×D4—22no—
300C22×D4C22⋊D7311yescharacteristic
301C22×D4C22⋊D7311yescharacteristic
302C23×C4—22no—
303C23×C4—21no—
304C23×C4—21no—
305C2×SD16—1462no—
306C2×M16—1461no—
307C4⋊Q8—21no—
308C4⋊Q8—21no—
309C4⋊Q8—731no—
310C4⋊Q8—731no—
311C4⋊Q8C22⋊D7312yes—
312C4⋊Q8C22×D7312yes—
313#125—1461no—
314#123—28no—
315#123—28no—
316C22⋊Q8—24no—
317C22⋊Q8—24no—
318C22⋊Q8—1461no—
319C22⋊Q8—1461no—
320#121—24no—
321#121—24no—
322C4×Q8—1461no—
323C4×Q8—1461no—
324C4×D4—21no—
325C4×D4—21no—
326C4×D4—21no—
327C4×D4—21no—
328C4×D4C22⋊D7312yes—
329C4×D4C22⋊D7312yes—
330C4×D4C22×D7312yes—
331C4×D4C22×D7312yes—
332#117—1461no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—24no—
337C2×C4⋊C4C22⋊D7311yescharacteristic
338C2×C4⋊C4C22⋊D7311yescharacteristic
339C2×C4⋊C4C22⋊D7312yes—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C42C22×D7311yescharacteristic
344#107—1462no—
345#107—1464no—
346#106—1462no—
347#106—1462no—
348C4⋊C8—732no—
349C4⋊C8—732no—
350C4⋊C8—1461no—
351C4⋊C8—1461no—
352#103—1462no—
353#103—1464no—
354#102—1462no—
355#102—1462no—
356#102—1462no—
357C22⋊C8—1461no—
358C22⋊C8—1461no—
359C4×C8—732no—
360C4×C8—1461no—
361C24—21no—
362C24—21no—
363#41—416no—
364#41—48no—
365#41—48no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—44no—
369C2×Q8—1461no—
370C2×Q8—1461no—
371C2×D4—24no—
372C2×D4—28no—
373C2×D4—21no—
374C2×D4—21no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—48no—
378C2×D4C2×C22⋊D7312yes—
379C2×D4C2×C22⋊D7312yes—
380C22×C4—28no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—44no—
389C22×C4—48no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4C2×C22⋊D7311yescharacteristic
393C22×C4C2×C22⋊D7311yescharacteristic
394C22×C4C2×C22⋊D7311yescharacteristic
395SD16—2924no—
396M16—2922no—
397C2×C8—1462no—
398C2×C8—1462no—
399C2×C8—1462no—
400C4⋊C4—28no—
401C4⋊C4—21no—
402C4⋊C4—22no—
403C4⋊C4—21no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—2921no—
407C4⋊C4—2921no—
408C4⋊C4—1461no—
409C4⋊C4—1461no—
410C4⋊C4—1461no—
411C4⋊C4—1461no—
412C4⋊C4C2×C22⋊D7312yes—
413C4⋊C4C2×C22⋊D7314yes—
414C4⋊C4C2×C22⋊D7312yes—
415C22⋊C4—28no—
416C22⋊C4—22no—
417C22⋊C4—22no—
418C22⋊C4—22no—
419C22⋊C4—22no—
420C22⋊C4—2921no—
421C42—2921no—
422C42C2×C22⋊D7311yescharacteristic
423C42C2×C22⋊D7311yescharacteristic
424C42C2×C22⋊D7311yescharacteristic
425C42C23×D7311yescharacteristic
426C23—24no—
427C23—22no—
428C23—22no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23#1455656011yescharacteristic
435Q8—44no—
436Q8—44no—
437Q8—2922no—
438Q8—2922no—
439D4—24no—
440D4—22no—
441D4—22no—
442D4—22no—
443D4—416no—
444D4—42no—
445D4#1455661714yes—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—41no—
451C2×C4—44no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—416no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—2921no—
461C2×C4—2921no—
462C2×C4#1455656011yescharacteristic
463C2×C4#1455656011yescharacteristic
464C2×C4#1455656011yescharacteristic
465C2×C4#1455661112yes—
466C2×C4#1455662112yes—
467C2×C4C22×C22⋊D7311yescharacteristic
468C2×C4C22×C22⋊D7311yescharacteristic
469C2×C4C22×C22⋊D7311yescharacteristic
470C8—2922no—
471C8—2922no—
472C8—2922no—
473C22—21no—
474C22—21no—
475C22—24no—
476C22—48no—
477C22—42no—
478C22—42no—
479C22—42no—
480C22—42no—
481C22—42no—
482C22—84no—
483C22#1798874912yes—
484C22C2×#1455656011yescharacteristic
485C4—21no—
486C4—21no—
487C4—21no—
488C4—42no—
489C4—42no—
490C4—44no—
491C4—5841no—
492C4—5841no—
493C4#1798874912yes—
494C4C2×#1455661111yescharacteristic
495C4C2×#1455661712yes—
496C4C2×#1455662111yescharacteristic
497C2—22no—
498C2—44no—
499C2—82no—
500C2C2×#1798874911yescharacteristic
501C2#2261915811yescharacteristic
502C2#2261929511yescharacteristic
subgroupG/H
centerC22C2×#14556560
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
Fitting, prime radicalC2×#547×C73C2
socleC22×C73C2×#120
hypercenterC2×#547D73
hypocenterC73#18783
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. #22621082C2semidirect
2. #22619939C2semidirect
3. #22619728C2semidirect
4. #22609166C2semidirect
5. #22609082C2semidirect
6. #22608616C2semidirect
7. #22608356C2semidirect
8. #22608322C2semidirect
11. #17988990C22semidirect
12. #17988939C22semidirect
17. #17988877C22semidirect
18. #17988877C22semidirect
19. #17988867C22semidirect
20. #17988792C22semidirect
28. #17988333C22semidirect
29. D4⋊#10596831C22semidirect
30. D4⋊#10596831C22semidirect
261. C73#18783semidirect

Maximal subgroups

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

subgroupindex
#226210822
#226199392
#226197282
#226091662
#226090822
#226086162
#226083562
#226083222
C2×#547×C732
#1878373