Finite Group Databasebeta

← #10825720

Subgroups — #10825720

Order 768.

908 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2930C211yesmaximal, characteristic, complemented
2#83792C212yesmaximal, complemented
3#83756C212yesmaximal, complemented
4#78579C211yesmaximal, characteristic, complemented
5#71715C212yesmaximal, complemented
6#70549C212yesmaximal, complemented
7#70013C211yesmaximal, characteristic, complemented
8#70005C211yesmaximal, characteristic, complemented
9#69993C211yesmaximal, characteristic, complemented
10#69987C211yesmaximal, characteristic, complemented
11#69895C211yesmaximal, characteristic, complemented
12#23795—31nomaximal, Sylow
13C2×C3×#131C2211yescharacteristic
14C22×C3×M16C2211yescharacteristic
15C3×#525C2211yescharacteristic
16#5822C2212yes—
17#5791C2212yes—
18#5748C2211yescharacteristic
19C2×#901C2212yes—
20C3×#436—22no—
21C3×#435—22no—
22C3×#424C2211yescharacteristic
23C3×#408C2211yescharacteristic
24C3×#407C2211yescharacteristic
25C3×#405C2211yescharacteristic
26#5236C2212yes—
27#5207C2212yes—
28#5145—24no—
29#5144—24no—
30#5139C2212yes—
31#5128—24no—
32#5127—24no—
33#5125C2211yescharacteristic
34#5124C2212yes—
35#5110C2212yes—
36#5102C2212yes—
37#5101C2212yes—
38#5005C2212yes—
39#4717—22no—
40#4700—22no—
41C3⋊#338C2211yescharacteristic
42#4557C2211yescharacteristic
43#4531C2212yes—
44#2955—31no—
45#2955—32no—
46#2930—32no—
47#2930D311yescharacteristic, complemented
48#2098—32no—
49#2098—31no—
50#2091—31no—
51#2017—31no—
52#2010—31no—
53#2010—32no—
54#1747—31no—
55C2×C3×#41C2311yescharacteristic
56C23×C3×C4C2311yescharacteristic
57C2×#235C2312yes—
58C3×#131—22no—
59C3×#131—22no—
60C2×C3×M16—22no—
61C2×C3×M16—22no—
62C2×C3×M16—21no—
63C2×C3×M16—21no—
64C2×C3×M16—21no—
65C2×C3×M16C2311yescharacteristic
66C2×C3×M16C2311yescharacteristic
67C22×C3×C8—21no—
68C22×C3×C8C2311yescharacteristic
69C22×C3×C8C2311yescharacteristic
70C3×#123—21no—
71C3×C22⋊Q8—21no—
72C3×#121—21no—
73C3×#120—21no—
74C3×C4×D4—22no—
75C3×#117C2311yescharacteristic
76#953—22no—
77#951—22no—
78#950—24no—
79#949—22no—
80#942—22no—
81#932—22no—
82#930—22no—
83#929—22no—
84C4×#67—24no—
85#926C2312yes—
86#924—22no—
87C2×#208C2312yes—
88C2×#208C2312yes—
89#909—22no—
90#908—22no—
91#904—22no—
92#901—22no—
93#901—22no—
94#901—22no—
95#901—22no—
96C2×ΓL(1,𝔽25)—22no—
97C2×D3×C8—22no—
98C3×C4⋊C8—22no—
99C3×C22⋊C8—22no—
100C3×C22⋊C8—22no—
101C3×#97D412yescomplemented
102C3×C4×C8D412yescomplemented
103#837—24no—
104#830—24no—
105#825—24no—
106#822—24no—
107(C2×D3)⋊C8—24no—
108#819—22no—
109#815D414yescomplemented
110Dic3⋊C8—24no—
111C8×Dic3D414yescomplemented
112#805—24no—
113#575—32no—
114#575—31no—
115C2×#131—32no—
116C2×#131C2×D311yescharacteristic
117C22×M16C2×D311yescharacteristic
118#525—32no—
119#525C2×D311yescharacteristic
120#444—64no—
121#444—62no—
122#443—64no—
123#443—62no—
124#436—22no—
125#436—64no—
126#435—22no—
127#435—64no—
128#430—32no—
129#430—31no—
130#424—32no—
131#424C2×D311yescharacteristic
132#421—32no—
133#421—32no—
134#421—32no—
135#408—32no—
136#408C2×D311yescharacteristic
137#407—32no—
138#407C2×D311yescharacteristic
139#405—32no—
140#405C2×D311yescharacteristic
141#338—31no—
142H(3,ℤ/4)—31no—
143H(3,ℤ/4)—32no—
144C24×C3#4111yescharacteristic
145C3×#41—22no—
146C3×#41—22no—
147C2×C3×Q8#4111yescharacteristic
148C2×C3×D4—21no—
149C2×C3×D4—22no—
150C2×C3×D4#4111yescharacteristic
151C22×C3×C4—22no—
152C22×C3×C4—21no—
153C22×C3×C4—21no—
154C22×C3×C4—22no—
155C22×C3×C4—21no—
156C22×C3×C4#4111yescharacteristic
157C22×C3×C4C2411yescharacteristic
158C2×#67—22no—
159C2×#67—24no—
160#235—22no—
161#235—22no—
162#235—22no—
163#235—22no—
164C2×D12#4112yes—
165C2×C4×D3—22no—
166C2×Dic6#4112yes—
167C3×M16—22no—
168C3×M16—22no—
169C3×M16—41no—
170C3×M16—42no—
171C3×M16—41no—
172C3×M16—41no—
173C3×M16—41no—
174C2×C3×C8—22no—
175C2×C3×C8—21no—
176C2×C3×C8—21no—
177C2×C3×C8—42no—
178C2×C3×C8C2×D412yes—
179C2×C3×C8C2×D412yes—
180C2×C3×C8C2×D412yes—
181C2×C3×C8C2×D411yescharacteristic
182C2×C3×C8C2×D411yescharacteristic
183C3×C4⋊C4—22no—
184C3×C4⋊C4—21no—
185C3×C22⋊C4—22no—
186C3×C22⋊C4—22no—
187C3×C22⋊C4—21no—
188C3×C42C2×D412yes—
189#217—24no—
190#217—22no—
191#216—24no—
192#215—24no—
193#214—24no—
194#213—24no—
195(C2×D3)⋊C4—24no—
196C4⋊Dic3—22no—
197Dic3⋊C4—24no—
198C4×Dic3C2×D414yes—
199#208—22no—
200#208—22no—
201#208—22no—
202#208—22no—
203C2×C3⋊C8—22no—
204C2×C3⋊C8C2×D414yes—
205Dic12—22no—
206D24—22no—
207#204—24no—
208ΓL(1,𝔽25)—42no—
209ΓL(1,𝔽25)—42no—
210D3×C8—42no—
211D3×C8—42no—
212C2×#41—32no—
213C2×#41C22×D311yescharacteristic
214C23×C4C22×D311yescharacteristic
215#137—62no—
216#137—62no—
217#137—62no—
218AGL(1,ℤ/8)—62no—
219AGL(1,ℤ/8)—62no—
220AGL(1,ℤ/8)—62no—
221#135—64no—
222#135—62no—
223#131—22no—
224#131—22no—
225#131—62no—
226#131—62no—
227#131—62no—
228#131—62no—
229C2×M16—22no—
230C2×M16—22no—
231C2×M16—21no—
232C2×M16—21no—
233C2×M16—21no—
234C2×M16—32no—
235C2×M16—32no—
236C2×M16—62no—
237C2×M16C22×D311yescharacteristic
238C2×M16C22×D311yescharacteristic
239C22×C8—21no—
240C22×C8—62no—
241C22×C8C22×D311yescharacteristic
242C22×C8C22×D311yescharacteristic
243#123—21no—
244#123—62no—
245C22⋊Q8—21no—
246C22⋊Q8—62no—
247#121—21no—
248#121—62no—
249#120—21no—
250#120—62no—
251C4×D4—22no—
252C4×D4—64no—
253#117—32no—
254#117C22×D311yescharacteristic
255#108—64no—
256#108—62no—
257C4⋊C8—22no—
258C4⋊C8—64no—
259#104—64no—
260#104—64no—
261#104—64no—
262C22⋊C8—22no—
263C22⋊C8—22no—
264C22⋊C8—64no—
265C22⋊C8—64no—
266#97—34no—
267#97#6712yescomplemented
268C4×C8—34no—
269C4×C8#6712yescomplemented
270C23×C3—22no—
271C23×C3—21no—
272C23×C3—21no—
273C23×C3—81no—
274C23×C3C2×#4111yescharacteristic
275C22×D3—22no—
276C3×Q8—22no—
277C3×D4—22no—
278C3×D4—42no—
279C3×D4—82no—
280C2×C3×C4—22no—
281C2×C3×C4—22no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—42no—
285C2×C3×C4—41no—
286C2×C3×C4—41no—
287C2×C3×C4—41no—
288C2×C3×C4—41no—
289C2×C3×C4C22×D412yes—
290C2×C3×C4C22×D411yescharacteristic
291C2×C3×C4C2×#4111yescharacteristic
292C2×C3×C4C2×#4112yes—
293#67—42no—
294#67—42no—
295#67—84no—
296C2×Dic3—24no—
297C2×Dic3—22no—
298D12—22no—
299D12—22no—
300C4×D3—42no—
301C4×D3—42no—
302Dic6—22no—
303Dic6—22no—
304C3×C8—22no—
305C3×C8—41no—
306C3×C8—41no—
307C3×C8#12711yescharacteristic
308C3×C8#12712yes—
309C3×C8#12711yescharacteristic
310C3⋊C8—24no—
311C3⋊C8—42no—
312C24#23511yescharacteristic
313#41—22no—
314#41—22no—
315#41—62no—
316#41—62no—
317#41—62no—
318#41—62no—
319C2×Q8—32no—
320C2×Q8#23511yescharacteristic
321C2×D4—21no—
322C2×D4—22no—
323C2×D4—32no—
324C2×D4—62no—
325C2×D4—64no—
326C2×D4#23511yescharacteristic
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—62no—
333C22×C4#23511yescharacteristic
334C22×C4C23×D311yescharacteristic
335Q16—62no—
336Q16—64no—
337SD16—64no—
338SD16—64no—
339SD16—64no—
340D8—62no—
341D8—64no—
342M16—22no—
343M16—22no—
344M16—42no—
345M16—41no—
346M16—41no—
347M16—41no—
348M16—41no—
349M16—62no—
350M16—62no—
351M16—62no—
352M16—62no—
353M16—122no—
354M16—122no—
355C2×C8—22no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—34no—
359C2×C8—42no—
360C2×C8—62no—
361C2×C8—122no—
362C2×C8—122no—
363C2×C8D3×D412yes—
364C2×C8D3×D412yes—
365C2×C8C2×#6712yes—
366C2×C8C2×#6711yescharacteristic
367C2×C8C2×#6711yescharacteristic
368C4⋊C4—22no—
369C4⋊C4—21no—
370C4⋊C4—64no—
371C4⋊C4—62no—
372C22⋊C4—22no—
373C22⋊C4—22no—
374C22⋊C4—21no—
375C22⋊C4—64no—
376C22⋊C4—64no—
377C22⋊C4—62no—
378C42—34no—
379C42C2×#6712yes—
380C22×C3—41no—
381C22×C3—42no—
382C22×C3—41no—
383C22×C3—42no—
384C22×C3—42no—
385C22×C3—41no—
386C22×C3—41no—
387C22×C3—41no—
388C22×C3#52511yescharacteristic
389C22×C3#52512yes—
390C2×D3—44no—
391C2×D3—42no—
392C3×C4—42no—
393C3×C4—41no—
394C3×C4—41no—
395C3×C4#52512yes—
396C3×C4#52511yescharacteristic
397C3×C4C2×#12711yescharacteristic
398Dic3—44no—
399Dic3—42no—
400C23—21no—
401C23—21no—
402C23—22no—
403C23—62no—
404C23—81no—
405C23C2×#23511yescharacteristic
406Q8—22no—
407Q8—62no—
408Q8—62no—
409D4—22no—
410D4—42no—
411D4—62no—
412D4—62no—
413D4—82no—
414D4—122no—
415D4—122no—
416D4—244no—
417C2×C4—22no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—64no—
427C2×C4—62no—
428C2×C4—122no—
429C2×C4—122no—
430C2×C4C2×#23511yescharacteristic
431C2×C4C2×D3×D412yes—
432C2×C4D3×#4112yes—
433C2×C4C22×#6711yescharacteristic
434C8—22no—
435C8—41no—
436C8—41no—
437C8—64no—
438C8—122no—
439C8#94012yes—
440C8#94011yescharacteristic
441C8#94011yescharacteristic
442C2×C3—22no—
443C2×C3—21no—
444C2×C3—81no—
445C2×C3—81no—
446C2×C3#239711yescharacteristic
447D3—82no—
448C22—41no—
449C22—41no—
450C22—42no—
451C22—42no—
452C22—41no—
453C22—42no—
454C22—41no—
455C22—41no—
456C22—124no—
457C22—122no—
458C22#554712yes—
459C22#579111yescharacteristic
460C4—42no—
461C4—41no—
462C4—41no—
463C4—124no—
464C4—122no—
465C4#554712yes—
466C4C2×#94011yescharacteristic
467C4#582811yescharacteristic
468C3#2379511yescharacteristic, complemented, Sylow
469C2—21no—
470C2—22no—
471C2—81no—
472C2—81no—
473C2—242no—
474C2#8234711yescharacteristic
subgroupG/H
centerC4C2×#940
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2930C2
socleC2×C3#2397
hypercenter#2930D3
hypocenterC3#23795
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C8 ◁ #2930

Complements

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

NH ≅ G/Ntype
1. C3×#2930C2semidirect
2. #83792C2semidirect
3. #83756C2semidirect
4. #78579C2semidirect
5. #71715C2semidirect
6. #70549C2semidirect
7. #70013C2semidirect
8. #70005C2semidirect
9. #69993C2semidirect
10. #69987C2semidirect
11. #69895C2semidirect
47. #2930D3semidirect
101. C3×#97D4semidirect
102. C3×C4×C8D4semidirect
109. #815D4semidirect
111. C8×Dic3D4semidirect
267. #97#67semidirect
269. C4×C8#67semidirect
468. C3#23795semidirect

Maximal subgroups

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

subgroupindex
C3×#29302
#837922
#837562
#785792
#717152
#705492
#700132
#700052
#699932
#699872
#698952
#237953