Finite Group Databasebeta

← #10825711

Subgroups — #10825711

Order 768.

930 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×#2955C211yesmaximal, characteristic, complemented
2#83794C212yesmaximal, complemented
3#83792C212yesmaximal, complemented
4#78553C211yesmaximal, characteristic, complemented
5#72121C211yesmaximal, characteristic, complemented
6#72118C211yesmaximal, characteristic, complemented
7#70738C211yesmaximal, characteristic, complemented
8#70729C211yesmaximal, characteristic, complemented
9#70406C212yesmaximal, complemented
10#70405C212yesmaximal, complemented
11#69895C211yesmaximal, characteristic, complemented
12#23794—31nomaximal, Sylow
13C3×#575C2211yescharacteristic
14C2×C3×#131C2212yes—
15#5828C2212yes—
16#5822C2212yes—
17C2×#901C2211yescharacteristic
18C22×D3×C8C2211yescharacteristic
19#5524C2211yescharacteristic
20C3×#444—22no—
21C3×#443—22no—
22C3×#430C2211yescharacteristic
23C3×#421C2212yes—
24C3×#405C2211yescharacteristic
25#5236C2212yes—
26#5234C2212yes—
27#5145—24no—
28#5144—24no—
29#5139C2212yes—
30#5129C2211yescharacteristic
31#5127—24no—
32#5126C2212yes—
33#5126C2211yescharacteristic
34#5124C2212yes—
35#5119C2212yes—
36C8×#67—24no—
37#5101C2212yes—
38#4999C2211yescharacteristic
39#4706—22no—
40C8×D12—22no—
41C3⋊#338C2211yescharacteristic
42#4557C2211yescharacteristic
43#4531C2212yes—
44#2955—32no—
45#2955D311yescharacteristic, complemented
46#2929—31no—
47#2929—32no—
48#2098—31no—
49#2098—32no—
50#2091—31no—
51#2017—31no—
52#2010—32no—
53#2010—31no—
54#1747—31no—
55C2×C3×#41C2312yes—
56C2×#235C2311yescharacteristic
57C22×C4×D3C2311yescharacteristic
58C3×#137—22no—
59C3×AGL(1,ℤ/8)—22no—
60C3×#135—22no—
61C3×#131—22no—
62C3×#131—22no—
63C3×#131—22no—
64C3×#131—22no—
65C2×C3×M16—22no—
66C2×C3×M16C2312yes—
67C2×C3×M16C2311yescharacteristic
68C22×C3×C8—22no—
69C22×C3×C8C2311yescharacteristic
70C3×#117C2311yescharacteristic
71#951—22no—
72#950—24no—
73#949—22no—
74#946—22no—
75#942—22no—
76#938—22no—
77#937—22no—
78#935—22no—
79#932—22no—
80C4×#67—24no—
81#926C2312yes—
82C2×#208C2311yescharacteristic
83C2×#208C2312yes—
84C22×C3⋊C8C2311yescharacteristic
85#901—22no—
86#901—22no—
87C2×ΓL(1,𝔽25)—21no—
88C2×D3×C8—22no—
89C2×D3×C8—21no—
90C2×D3×C8—21no—
91C2×D3×C8—21no—
92C2×D3×C8—22no—
93#897—21no—
94#895—21no—
95#886—21no—
96#882—21no—
97C4×D12—22no—
98C3×#108—22no—
99C3×#104—24no—
100C3×#97D412yescomplemented
101C3×C4×C8D412yescomplemented
102#837—24no—
103#830—24no—
104#825—24no—
105#822—24no—
106(C2×D3)⋊C8—22no—
107(C2×D3)⋊C8—22no—
108(C2×D3)⋊C8—24no—
109#815D414yescomplemented
110Dic3⋊C8—24no—
111C8×Dic3D414yescomplemented
112C4⋊(C3⋊C8)—22no—
113#575—32no—
114#575C2×D311yescharacteristic
115C2×#131—31no—
116C2×#131C2×D312yes—
117C23×C8—31no—
118#525—32no—
119#525—31no—
120#444—22no—
121#444—64no—
122#443—22no—
123#443—64no—
124#435—64no—
125#435—62no—
126C8×D4—64no—
127C8×D4—62no—
128#430—32no—
129#430C2×D311yescharacteristic
130#424—32no—
131#424—31no—
132#421—32no—
133#421—32no—
134#421C2×D312yes—
135#408—32no—
136#408—31no—
137#407—32no—
138#407—31no—
139#405—32no—
140#405C2×D311yescharacteristic
141#338—31no—
142H(3,ℤ/4)—31no—
143H(3,ℤ/4)—32no—
144C23×D3#4111yescharacteristic
145C3×#41—22no—
146C3×#41—22no—
147C3×#41—22no—
148C3×#41—22no—
149C2×C3×Q8#4112yes—
150C2×C3×D4—22no—
151C2×C3×D4#4112yes—
152C22×C3×C4—22no—
153C22×C3×C4C2411yescharacteristic
154C2×#67—24no—
155C2×#67—21no—
156C22×Dic3#4111yescharacteristic
157#235—22no—
158#235—22no—
159C2×D12—22no—
160C2×D12#4111yescharacteristic
161C2×C4×D3—21no—
162C2×C4×D3—21no—
163C2×C4×D3—22no—
164C2×C4×D3—21no—
165C2×C4×D3—22no—
166C2×Dic6#4111yescharacteristic
167C3×Q16—22no—
168C3×SD16—24no—
169C3×D8—22no—
170C3×M16—22no—
171C3×M16—22no—
172C3×M16—22no—
173C3×M16—42no—
174C3×M16—42no—
175C2×C3×C8—22no—
176C2×C3×C8—42no—
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—21no—
184C3×C22⋊C4—21no—
185C3×C42C2×D412yes—
186#217—24no—
187#217—22no—
188#216—24no—
189#215—24no—
190#214—24no—
191#213—24no—
192(C2×D3)⋊C4—22no—
193(C2×D3)⋊C4—22no—
194(C2×D3)⋊C4—24no—
195C4⋊Dic3—22no—
196C4⋊Dic3—22no—
197Dic3⋊C4—24no—
198C4×Dic3C2×D414yes—
199#208—22no—
200#208—22no—
201#208—22no—
202C2×C3⋊C8—21no—
203C2×C3⋊C8—21no—
204C2×C3⋊C8—22no—
205C2×C3⋊C8C2×D414yes—
206ΓL(1,𝔽25)—42no—
207D3×C8—41no—
208D3×C8—42no—
209D3×C8—41no—
210D3×C8—41no—
211D3×C8—41no—
212C2×#41—31no—
213C2×#41C22×D312yes—
214C23×C4—31no—
215#137—22no—
216#137—62no—
217#137—62no—
218AGL(1,ℤ/8)—22no—
219AGL(1,ℤ/8)—62no—
220AGL(1,ℤ/8)—62no—
221#135—22no—
222#135—64no—
223#131—22no—
224#131—22no—
225#131—22no—
226#131—22no—
227#131—62no—
228#131—62no—
229C2×M16—22no—
230C2×M16—32no—
231C2×M16—31no—
232C2×M16—61no—
233C2×M16C22×D312yes—
234C2×M16C22×D311yescharacteristic
235C22×C8—22no—
236C22×C8—31no—
237C22×C8—62no—
238C22×C8—62no—
239C22×C8—61no—
240C22×C8—61no—
241C22×C8—61no—
242C22×C8C22×D311yescharacteristic
243#123—61no—
244#123—62no—
245C22⋊Q8—61no—
246C22⋊Q8—62no—
247#121—61no—
248#121—62no—
249#120—61no—
250#120—62no—
251C4×D4—64no—
252C4×D4—62no—
253#117—32no—
254#117C22×D311yescharacteristic
255#108—22no—
256#108—64no—
257C4⋊C8—64no—
258C4⋊C8—62no—
259#104—24no—
260#104—64no—
261#104—64no—
262C22⋊C8—64no—
263C22⋊C8—64no—
264C22⋊C8—62no—
265C22⋊C8—62no—
266#97—34no—
267#97#6712yescomplemented
268C4×C8—34no—
269C4×C8#6712yescomplemented
270C23×C3—22no—
271C23×C3C2×#4111yescharacteristic
272C22×D3—21no—
273C22×D3—22no—
274C22×D3—21no—
275C22×D3—42no—
276C3×Q8—22no—
277C3×Q8—22no—
278C3×D4—22no—
279C3×D4—22no—
280C3×D4—42no—
281C3×D4—42no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—42no—
285C2×C3×C4—42no—
286C2×C3×C4C22×D412yes—
287C2×C3×C4C22×D411yescharacteristic
288C2×C3×C4C2×#4112yes—
289C2×C3×C4C2×#4111yescharacteristic
290#67—48no—
291#67—42no—
292C2×Dic3—24no—
293C2×Dic3—22no—
294C2×Dic3—21no—
295C2×Dic3—21no—
296D12—22no—
297D12—44no—
298C4×D3—41no—
299C4×D3—42no—
300C4×D3—41no—
301C4×D3—41no—
302C4×D3—41no—
303Dic6—22no—
304C3×C8—22no—
305C3×C8—42no—
306C3×C8#12711yescharacteristic
307C3×C8#12712yes—
308C3×C8#12711yescharacteristic
309C3⋊C8—24no—
310C3⋊C8—41no—
311C3⋊C8—41no—
312C24—31no—
313#41—22no—
314#41—22no—
315#41—22no—
316#41—22no—
317#41—62no—
318#41—62no—
319C2×Q8—31no—
320C2×Q8#23512yes—
321C2×D4—22no—
322C2×D4—31no—
323C2×D4—62no—
324C2×D4—64no—
325C2×D4—61no—
326C2×D4#23512yes—
327C22×C4—22no—
328C22×C4—31no—
329C22×C4—62no—
330C22×C4—62no—
331C22×C4—61no—
332C22×C4—61no—
333C22×C4—61no—
334C22×C4C23×D311yescharacteristic
335Q16—22no—
336Q16—64no—
337SD16—24no—
338SD16—64no—
339SD16—64no—
340D8—22no—
341D8—64no—
342M16—22no—
343M16—22no—
344M16—22no—
345M16—42no—
346M16—42no—
347M16—62no—
348M16—62no—
349M16—62no—
350M16—122no—
351C2×C8—22no—
352C2×C8—34no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—62no—
356C2×C8—61no—
357C2×C8—61no—
358C2×C8—122no—
359C2×C8—121no—
360C2×C8—121no—
361C2×C8—121no—
362C2×C8—121no—
363C2×C8D3×D412yes—
364C2×C8D3×D412yes—
365C2×C8C2×#6712yes—
366C2×C8C2×#6711yescharacteristic
367C2×C8C2×#6711yescharacteristic
368C4⋊C4—21no—
369C4⋊C4—64no—
370C4⋊C4—62no—
371C4⋊C4—62no—
372C22⋊C4—21no—
373C22⋊C4—64no—
374C22⋊C4—62no—
375C22⋊C4—62no—
376C22⋊C4—64no—
377C22⋊C4—62no—
378C42—34no—
379C42C2×#6712yes—
380C22×C3—41no—
381C22×C3—44no—
382C22×C3—42no—
383C22×C3#52512yes—
384C22×C3#52511yescharacteristic
385C2×D3—22no—
386C2×D3—24no—
387C2×D3—42no—
388C2×D3—41no—
389C2×D3—41no—
390C2×D3—42no—
391C2×D3—41no—
392C3×C4—42no—
393C3×C4—42no—
394C3×C4#52511yescharacteristic
395C3×C4#52512yes—
396C3×C4C2×#12711yescharacteristic
397Dic3—44no—
398Dic3—41no—
399Dic3—41no—
400C23—22no—
401C23—62no—
402C23—61no—
403C23—61no—
404C23—122no—
405C23D3×#4111yescharacteristic
406Q8—22no—
407Q8—22no—
408Q8—62no—
409D4—22no—
410D4—22no—
411D4—42no—
412D4—42no—
413D4—62no—
414D4—124no—
415D4—128no—
416D4—122no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—64no—
422C2×C4—62no—
423C2×C4—61no—
424C2×C4—61no—
425C2×C4—121no—
426C2×C4—121no—
427C2×C4—122no—
428C2×C4—121no—
429C2×C4—121no—
430C2×C4C2×#23512yes—
431C2×C4C2×D3×D412yes—
432C2×C4D3×#4111yescharacteristic
433C2×C4C22×#6711yescharacteristic
434C8—22no—
435C8—42no—
436C8—64no—
437C8—121no—
438C8—121no—
439C8#94012yes—
440C8#94011yescharacteristic
441C8#94011yescharacteristic
442C2×C3—22no—
443C2×C3—21no—
444C2×C3—82no—
445C2×C3#239711yescharacteristic
446D3—42no—
447D3—81no—
448C22—41no—
449C22—44no—
450C22—42no—
451C22—62no—
452C22—64no—
453C22—121no—
454C22—122no—
455C22—122no—
456C22—121no—
457C22—121no—
458C22#554712yes—
459C22#582811yescharacteristic
460C4—42no—
461C4—42no—
462C4—124no—
463C4—121no—
464C4—121no—
465C4#554712yes—
466C4#579111yescharacteristic
467C4C2×#94011yescharacteristic
468C3#2379411yescharacteristic, complemented, Sylow
469C2—21no—
470C2—22no—
471C2—82no—
472C2—122no—
473C2—241no—
474C2#8234711yescharacteristic
subgroupG/H
centerC8#940
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2955C2
socleC2×C3#2397
hypercenter#2955D3
hypocenterC3#23794
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C8 ◁ C22×C8 ◁ #2955

Complements

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

NH ≅ G/Ntype
1. C3×#2955C2semidirect
2. #83794C2semidirect
3. #83792C2semidirect
4. #78553C2semidirect
5. #72121C2semidirect
6. #72118C2semidirect
7. #70738C2semidirect
8. #70729C2semidirect
9. #70406C2semidirect
10. #70405C2semidirect
11. #69895C2semidirect
45. #2955D3semidirect
100. C3×#97D4semidirect
101. C3×C4×C8D4semidirect
109. #815D4semidirect
111. C8×Dic3D4semidirect
267. #97#67semidirect
269. C4×C8#67semidirect
468. C3#23794semidirect

Maximal subgroups

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

subgroupindex
C3×#29552
#837942
#837922
#785532
#721212
#721182
#707382
#707292
#704062
#704052
#698952
#237943