Finite Group Databasebeta

← #33073851

Subgroups — #33073851

Order 18688.

848 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620869C211yesmaximal, characteristic, complemented
2#22620853C211yesmaximal, characteristic, complemented
3#22620478C211yesmaximal, characteristic, complemented
4#22620384C211yesmaximal, characteristic
5C2×#17988934C211yesmaximal, characteristic, complemented
6#22609558C212yesmaximal, complemented
7#22609556C212yesmaximal, complemented
8#22609534C212yesmaximal, complemented
9#22609522C212yesmaximal, complemented
10C8×#11924012C211yesmaximal, characteristic, complemented
11C73×#3167C211yesmaximal, characteristic, complemented
12C2×#14556616—22no—
13C2×#14556591C2211yescharacteristic
14C2×#14556588C2211yescharacteristic
15#17988997—22no—
16#17988990C2212yescomplemented
17#17988983C2212yescomplemented
18#17988977—22no—
19#17988963C2212yes—
20#17988961C2212yes—
21#17988961C2212yes—
22#17988959C2212yescomplemented
23#17988939C2212yescomplemented
24#17988935C2212yes—
25#17988934—22no—
26#17988934C2212yes—
27#17988934C2212yes—
28#17988792—22no—
29#17988787—22no—
30#17988737—22no—
31#17988725—22no—
32#17988640C2211yescharacteristic
33#17988630—22no—
34C2×#14556513—22no—
35C4×#11924012C2211yescharacteristic
36#17988309C2212yescomplemented
37C8×#10596831C2212yescomplemented
38#533×C73C2211yescharacteristic
39#532×C73C2211yescharacteristic
40#498×C73C2211yescharacteristic, complemented
41#492×C73C2211yescharacteristic, complemented
42#487×C73C2211yescharacteristic, complemented
43#486×C73C2211yescharacteristic, complemented
44#465×C73—22no—
45C4×SD16×C73—22no—
46C2×C4×C8×C73C2211yescharacteristic
47C22×Dic146—21no—
48C22×D292—21no—
49C22×D292—21no—
50C22×D292—21no—
51#14556616—48no—
52C2×#11924016—24no—
53C2×#11924015—24no—
54C2×#11924015—24no—
55C2×#11924013—24no—
56C2×#11924012D412yes—
57#14556591—21no—
58#14556591—21no—
59#14556591—21no—
60#14556591C2312yes—
61#14556588—21no—
62#14556588—21no—
63#14556588—21no—
64#14556588C2312yes—
65C2×C4⋊Dic73—21no—
66C2×C4⋊Dic73—21no—
67#14556524—24no—
68#14556522—24no—
69C4⋊#10596831D412yes—
70#14556514—24no—
71#14556513—24no—
72#14556513—24no—
73#14556511—24no—
74C22⋊#10596831D412yes—
75#14556502D412yes—
76C4×#10596831C2312yes—
77C4×#10596831C2312yes—
78C2×#41×C73—21no—
79C2×SD16×C73—22no—
80C22×C8×C73D412yes—
81C4⋊Q8×C73C2311yescharacteristic
82C4⋊Q8×C73C2311yescharacteristic
83C4⋊Q8×C73C2311yescharacteristic
84#127×C73C2311yescharacteristic
85#125×C73—21no—
86#124×C73—21no—
87C22⋊Q8×C73—21no—
88C22⋊Q8×C73—21no—
89#121×C73—21no—
90#121×C73—21no—
91C4×Q8×C73—21no—
92C4×Q8×C73—21no—
93C4×D4×C73—21no—
94C4×D4×C73—21no—
95#117×C73—21no—
96C2×C42×C73C2311yescharacteristic
97#106×C73—22no—
98#103×C73—22no—
99#102×C73—22no—
100C4×C8×C73D412yes—
101C4×C8×C73C2311yescharacteristic
102C4×C8×C73C2311yescharacteristic
103C23×D73—41no—
104C2×Dic146—22no—
105C2×Dic146—41no—
106C2×Dic146—82no—
107C2×D292—22no—
108C2×D292—22no—
109C2×D292—22no—
110C2×D292—44no—
111C2×D292—44no—
112C2×D292—41no—
113C2×D292—41no—
114C2×D292—41no—
115C2×D292—82no—
116C22×Dic73—41no—
117#11924016—84no—
118#11924015—48no—
119#11924015—84no—
120#11924013—48no—
121#11924012SD1618yescomplemented
122C2×#10596831C2×D414yes—
123C2×#10596831C2×D414yes—
124C4⋊Dic73—22no—
125C4⋊Dic73—22no—
126C4⋊Dic73—41no—
127C4⋊Dic73—41no—
128#41×C73—44no—
129C2×Q8×C73—21no—
130C2×Q8×C73—21no—
131C2×Q8×C73—21no—
132C2×D4×C73—21no—
133C2×D4×C73—21no—
134C2×D4×C73—21no—
135C2×D4×C73—41no—
136C22×C4×C73—41no—
137C22×C4×C73C2×D411yescharacteristic
138C22×C4×C73C2×D411yescharacteristic
139C22×C4×C73C2×D411yescharacteristic
140SD16×C73—44no—
141C2×C8×C73—42no—
142C2×C8×C73C2×D412yes—
143C2×C8×C73C2×D412yes—
144C4⋊C4×C73—21no—
145C4⋊C4×C73—21no—
146C4⋊C4×C73—21no—
147C4⋊C4×C73—21no—
148C4⋊C4×C73—21no—
149C4⋊C4×C73—21no—
150C4⋊C4×C73—41no—
151C4⋊C4×C73—41no—
152C22⋊C4×C73—41no—
153C22⋊C4×C73—41no—
154C22⋊C4×C73—41no—
155C42×C73—41no—
156C42×C73C2×D411yescharacteristic
157C42×C73C2×D411yescharacteristic
158C42×C73C2×D411yescharacteristic
159C42×C73C2411yescharacteristic
160C23×C73—41no—
161C23×C73#12711yescharacteristic
162C22×D73—42no—
163C22×D73—82no—
164C22×D73—82no—
165C22×D73—82no—
166Q8×C73—42no—
167Q8×C73—42no—
168Q8×C73—42no—
169D4×C73—42no—
170D4×C73—42no—
171D4×C73—42no—
172D4×C73—82no—
173C2×C4×C73—22no—
174C2×C4×C73—22no—
175C2×C4×C73—41no—
176C2×C4×C73—41no—
177C2×C4×C73—41no—
178C2×C4×C73—82no—
179C2×C4×C73#12711yescharacteristic
180C2×C4×C73#12711yescharacteristic
181C2×C4×C73#12711yescharacteristic
182C2×C4×C73C2×SD1614yes—
183C2×C4×C73C22×D411yescharacteristic
184C2×C4×C73C22×D411yescharacteristic
185C2×C4×C73C22×D411yescharacteristic
186C2×Dic73—42no—
187C2×Dic73—82no—
188D292—44no—
189D292—44no—
190D292—44no—
191D292—82no—
192D292—82no—
193D292—82no—
194Dic146—44no—
195Dic146—82no—
196C8×C73—22no—
197C8×C73—22no—
198#10596831—24no—
199#10596831C2×SD1618yes—
200C22×C73—21no—
201C22×C73—21no—
202C22×C73—81no—
203C22×C73—81no—
204C22×C73—81no—
205C22×C73#49212yes—
206C22×C73C2×#12711yescharacteristic
207C2×D73—82no—
208C2×D73—82no—
209C2×D73—82no—
210C2×D73—164no—
211C4×C73—21no—
212C4×C73—21no—
213C4×C73—81no—
214C4×C73—81no—
215C4×C73—81no—
216C4×C73#49212yes—
217C4×C73C22×SD1612yes—
218C4×C73#57512yes—
219C4×C73#57612yes—
220Dic73—82no—
221#19623—731nomaximal, Sylow
222C2×C73—22no—
223C2×C73—161no—
224C2×C73C2×#49211yescharacteristic
225C2×C73#315311yescharacteristic
226C2×C73#315511yescharacteristic
227D73—162no—
228#3208—731no—
229#3206—731no—
230#3167D7311yescharacteristic, complemented
231#3165—731no—
232#3151—731no—
233C2×#486—731no—
234#1715—732no—
235#1714—732no—
236#1708—732no—
237#1706—732no—
238C8×M16—731no—
239C73#1962311yescharacteristic, complemented, Sylow
240C2×AGL(1,ℤ/8)—1462no—
241#533C2×D7311yescharacteristic
242#532C2×D7311yescharacteristic
243C2×C4⋊Q8—731no—
244C2×#127—731no—
245#501—1462no—
246#500—732no—
247#498—732no—
248#498C2×D7311yescharacteristic, complemented
249#496—1462no—
250#493—732no—
251#492—732no—
252#492—732no—
253#492—732no—
254#492C2×D7311yescharacteristic, complemented
255#487—732no—
256#487C2×D7311yescharacteristic, complemented
257#486—732no—
258#486—732no—
259#486—732no—
260#486—1462no—
261#486C2×D7311yescharacteristic, complemented
262#465—22no—
263#461—1462no—
264#460—1462no—
265#450—1462no—
266#448—1462no—
267C4×SD16—22no—
268#431—731no—
269#428—1462no—
270C2×#102—1462no—
271C4×M16—731no—
272C2×C4×C8C2×D7311yescharacteristic
273#322—732no—
274C82—732no—
275C2×#41—21no—
276C22×Q8—1461no—
277C22×D4—1461no—
278C22×D4—1461no—
279C22×D4—1461no—
280AGL(1,ℤ/8)—2928no—
281C2×SD16—22no—
282C2×SD16—1464no—
283C2×SD16—1464no—
284C2×SD16—1464no—
285C2×D8—1464no—
286C2×M16—732no—
287C22×C8C22⋊D7312yes—
288C4⋊Q8—732no—
289C4⋊Q8—1461no—
290C4⋊Q8—1461no—
291C4⋊Q8—1461no—
292C4⋊Q8C22×D7311yescharacteristic
293C4⋊Q8C22×D7311yescharacteristic
294C4⋊Q8C22×D7311yescharacteristic
295#127—732no—
296#127—1461no—
297#127—1461no—
298#127—1461no—
299#127C22×D7311yescharacteristic
300#125—21no—
301#124—21no—
302C22⋊Q8—21no—
303C22⋊Q8—21no—
304#121—21no—
305#121—21no—
306C4×Q8—21no—
307C4×Q8—21no—
308C4×D4—21no—
309C4×D4—21no—
310#117—21no—
311C2×C4⋊C4—1461no—
312C2×C4⋊C4—1461no—
313C2×C42C22×D7311yescharacteristic
314#107—1464no—
315#106—22no—
316#106—1464no—
317C4⋊C8—732no—
318#103—22no—
319#103—1464no—
320#102—22no—
321#102—1464no—
322#102—1464no—
323#102—1464no—
324C22⋊C8—732no—
325#97—732no—
326C4×C8—732no—
327C4×C8—732no—
328C4×C8C22⋊D7312yes—
329C4×C8C22×D7311yescharacteristic
330C4×C8C22×D7311yescharacteristic
331C24—2921no—
332#41—44no—
333C2×Q8—21no—
334C2×Q8—21no—
335C2×Q8—21no—
336C2×Q8—2921no—
337C2×Q8—5842no—
338C2×Q8—1462no—
339C2×D4—21no—
340C2×D4—21no—
341C2×D4—21no—
342C2×D4—41no—
343C2×D4—2924no—
344C2×D4—2921no—
345C2×D4—2921no—
346C2×D4—2921no—
347C2×D4—2924no—
348C2×D4—5842no—
349C2×D4—1462no—
350C2×D4—1462no—
351C2×D4—1462no—
352C22×C4—41no—
353C22×C4—2921no—
354C22×C4D4×D7311yescharacteristic
355C22×C4D4×D7311yescharacteristic
356C22×C4C2×C22⋊D7311yescharacteristic
357SD16—44no—
358SD16—2928no—
359SD16—5844no—
360SD16—5844no—
361D8—2928no—
362M16—738no—
363C2×C8—42no—
364C2×C8—734no—
365C2×C8—734no—
366C2×C8C2×C22⋊D7312yes—
367C2×C8C2×C22⋊D7312yes—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—2921no—
377C4⋊C4—2921no—
378C4⋊C4—1462no—
379C4⋊C4—1462no—
380C22⋊C4—41no—
381C22⋊C4—41no—
382C22⋊C4—41no—
383C42—41no—
384C42D4×D7311yescharacteristic
385C42D4×D7311yescharacteristic
386C42C2×C22⋊D7311yescharacteristic
387C42C23×D7311yescharacteristic
388C23—41no—
389C23—2922no—
390C23—5842no—
391C23—5842no—
392C23—5842no—
393C23#1455658711yescharacteristic
394Q8—42no—
395Q8—42no—
396Q8—42no—
397Q8—2924no—
398Q8—5842no—
399D4—42no—
400D4—42no—
401D4—42no—
402D4—82no—
403D4—2924no—
404D4—2924no—
405D4—2924no—
406D4—5842no—
407D4—5842no—
408D4—5842no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—82no—
415C2×C4—2922no—
416C2×C4—5842no—
417C2×C4#1455658711yescharacteristic
418C2×C4#1455658711yescharacteristic
419C2×C4#1455658711yescharacteristic
420C2×C4SD16×D7314yes—
421C2×C4C2×D4×D7311yescharacteristic
422C2×C4C2×D4×D7311yescharacteristic
423C2×C4C22×C22⋊D7311yescharacteristic
424C8—22no—
425C8—22no—
426C8—738no—
427C8—1464no—
428C22—21no—
429C22—21no—
430C22—81no—
431C22—81no—
432C22—81no—
433C22—5842no—
434C22—5842no—
435C22—5842no—
436C22—11684no—
437C22#1798895812yes—
438C22C2×#1455658711yescharacteristic
439C4—21no—
440C4—21no—
441C4—81no—
442C4—81no—
443C4—81no—
444C4—5842no—
445C4#1798895812yes—
446C4C2×SD16×D7312yes—
447C4#1798938512yes—
448C4#1798938912yes—
449C2—22no—
450C2—161no—
451C2—11682no—
452C2C2×#1798895811yescharacteristic
453C2#2262039911yescharacteristic
454C2#2262041811yescharacteristic
subgroupG/H
centerC22C2×#14556587
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
FittingC73×#3167C2
socleC22×C73C2×#127
prime radical#22620384C2
hypercenter#3167D73
hypocenterC73#19623
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3167

Complements

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

NH ≅ G/Ntype
1. #22620869C2semidirect
2. #22620853C2semidirect
3. #22620478C2semidirect
5. C2×#17988934C2semidirect
6. #22609558C2semidirect
7. #22609556C2semidirect
8. #22609534C2semidirect
9. #22609522C2semidirect
10. C8×#11924012C2semidirect
11. C73×#3167C2semidirect
16. #17988990C22semidirect
17. #17988983C22semidirect
22. #17988959C22semidirect
23. #17988939C22semidirect
36. #17988309C22semidirect
37. C8×#10596831C22semidirect
40. #498×C73C22semidirect
41. #492×C73C22semidirect
42. #487×C73C22semidirect
43. #486×C73C22semidirect
121. #11924012SD16semidirect
230. #3167D73semidirect
239. C73#19623semidirect
248. #498C2×D73semidirect
254. #492C2×D73semidirect
256. #487C2×D73semidirect
261. #486C2×D73semidirect

Maximal subgroups

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

subgroupindex
#226208692
#226208532
#226204782
#226203842
C2×#179889342
#226095582
#226095562
#226095342
#226095222
C8×#119240122
C73×#31672
#1962373