Finite Group Databasebeta

← #10825710

Subgroups — #10825710

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
1#83999C211yesmaximal, characteristic, complemented
2#82916C212yesmaximal, complemented
3#78989C212yesmaximal, complemented
4#78626C211yesmaximal, characteristic, complemented
5C3×#2010C211yesmaximal, characteristic, complemented
6#72121C211yesmaximal, characteristic, complemented
7#71715C212yesmaximal, complemented
8#70405C212yesmaximal, complemented
9#69987C211yesmaximal, characteristic, complemented
10#68715C211yesmaximal, characteristic, complemented
11#68697C211yesmaximal, characteristic, complemented
12#23794—31nomaximal, Sylow
13C3×#575C2211yescharacteristic
14C3×#525C2212yes—
15#5822C2212yes—
16C2×#948C2212yes—
17#5791C2211yescharacteristic
18C23×C3⋊C8C2211yescharacteristic
19C2×#907C2211yescharacteristic
20C3×#421C2212yes—
21C3×#407C2211yescharacteristic
22#5236C2212yes—
23#5234C2212yes—
24#5207C2212yes—
25#5197—22no—
26#5196—22no—
27#5129C2211yescharacteristic
28#5125C2211yescharacteristic
29#5124C2212yes—
30#5122C2211yescharacteristic
31#5119C2211yescharacteristic
32#5011—24no—
33D4×C3⋊C8—24no—
34#5005C2212yes—
35#4999C2212yes—
36#4997C2212yes—
37#4824—24no—
38#4823—24no—
39#4730—22no—
40#67⋊C8—22no—
41C3×H(3,ℤ/4)C2211yescharacteristic
42C3⋊#338C2211yescharacteristic
43#4531C2212yes—
44#2955—32no—
45#2955—31no—
46#2929—31no—
47#2929—32no—
48#2098—32no—
49#2098—31no—
50#2091—31no—
51#2017—31no—
52#2010—32no—
53#2010D311yescharacteristic, complemented
54#1747—31no—
55C2×C3×#41C2312yes—
56C23×C3×C4C2311yescharacteristic
57C2×#235C2311yescharacteristic
58C3×#137—22no—
59C3×AGL(1,ℤ/8)—22no—
60C3×#135—22no—
61C2×C3×M16C2311yescharacteristic
62C2×C3×M16C2311yescharacteristic
63C3×#123—22no—
64C3×C22⋊Q8—22no—
65C3×#121—22no—
66C3×#120—22no—
67C3×C4×D4—24no—
68C3×#117C2312yes—
69#953—21no—
70#951—22no—
71#950—24no—
72#949—22no—
73#948—22no—
74#948—22no—
75#948—22no—
76#948—22no—
77#942—22no—
78#932—22no—
79#930—21no—
80#929—21no—
81C4×#67—22no—
82#926C2311yescharacteristic
83#924—21no—
84C2×#208—22no—
85C2×#208C2312yes—
86C2×#208C2312yes—
87C22×C3⋊C8—21no—
88C22×C3⋊C8—21no—
89C22×C3⋊C8—22no—
90C22×C3⋊C8—22no—
91C22×C3⋊C8—22no—
92C22×C3⋊C8C2311yescharacteristic
93C22×C3⋊C8C2311yescharacteristic
94#907—22no—
95#907—22no—
96C2×ΓL(1,𝔽25)—21no—
97C2×D3×C8—21no—
98C3×#104—24no—
99C3×C22⋊C8—22no—
100#837—24no—
101#830—24no—
102#830—24no—
103#822—24no—
104(C2×D3)⋊C8—22no—
105#819—22no—
106#815D412yescomplemented
107Dic3⋊C8—22no—
108C8×Dic3D412yescomplemented
109#805—24no—
110C4⋊(C3⋊C8)—24no—
111#803D414yescomplemented
112C4×C3⋊C8D414yescomplemented
113#575—32no—
114#575C2×D311yescharacteristic
115C2×#131—32no—
116C2×#131—31no—
117C23×C8—31no—
118#525—31no—
119#525C2×D312yes—
120#444—64no—
121#444—62no—
122#443—64no—
123#443—62no—
124#435—62no—
125#435—64no—
126C8×D4—62no—
127C8×D4—64no—
128#430—32no—
129#430—31no—
130#424—31no—
131#424—32no—
132#421—32no—
133#421—32no—
134#421C2×D312yes—
135#408—32no—
136#408—31no—
137#407—32no—
138#407C2×D311yescharacteristic
139#405—31no—
140#405—32no—
141#338—31no—
142H(3,ℤ/4)—32no—
143H(3,ℤ/4)C2×D311yescharacteristic
144C24×C3#4111yescharacteristic
145C3×#41—22no—
146C3×#41—22no—
147C3×#41—22no—
148C3×#41—22no—
149C2×C3×Q8#4112yes—
150C2×C3×D4—24no—
151C2×C3×D4—22no—
152C2×C3×D4#4112yes—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—21no—
156C22×C3×C4—21no—
157C22×C3×C4—22no—
158C22×C3×C4#4111yescharacteristic
159C22×C3×C4C2411yescharacteristic
160C2×#67—22no—
161C2×#67—21no—
162#235—22no—
163#235—22no—
164C2×D12#4111yescharacteristic
165C2×C4×D3—21no—
166C2×Dic6#4111yescharacteristic
167C3×Q16—22no—
168C3×SD16—24no—
169C3×D8—22no—
170C3×M16—22no—
171C3×M16—22no—
172C2×C3×C8—21no—
173C2×C3×C8C2×D412yes—
174C3×C4⋊C4—24no—
175C3×C4⋊C4—22no—
176C3×C22⋊C4—24no—
177C3×C22⋊C4—24no—
178C3×C22⋊C4—22no—
179C3×C42C2×D414yes—
180#217—22no—
181#217—21no—
182#216—24no—
183#215—24no—
184#214—24no—
185#213—24no—
186(C2×D3)⋊C4—22no—
187C4⋊Dic3—21no—
188Dic3⋊C4—22no—
189C4×Dic3C2×D412yes—
190#208—22no—
191#208—22no—
192#208—22no—
193#208—22no—
194#208—42no—
195#208—42no—
196C2×C3⋊C8—22no—
197C2×C3⋊C8—22no—
198C2×C3⋊C8—21no—
199C2×C3⋊C8—41no—
200C2×C3⋊C8—41no—
201C2×C3⋊C8—42no—
202C2×C3⋊C8—42no—
203C2×C3⋊C8—42no—
204C2×C3⋊C8C2×D414yes—
205C2×C3⋊C8C2×D412yes—
206C2×C3⋊C8C2×D412yes—
207C2×C3⋊C8C2×D411yescharacteristic
208C2×C3⋊C8C2×D411yescharacteristic
209ΓL(1,𝔽25)—42no—
210D3×C8—41no—
211D3×C8—41no—
212C2×#41—31no—
213C2×#41C22×D312yes—
214C23×C4C22×D311yescharacteristic
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—62no—
224#131—62no—
225#131—62no—
226#131—62no—
227#131—62no—
228#131—62no—
229C2×M16—32no—
230C2×M16—32no—
231C2×M16—61no—
232C2×M16—62no—
233C2×M16C22×D311yescharacteristic
234C2×M16C22×D311yescharacteristic
235C22×C8—31no—
236C22×C8—31no—
237C22×C8—61no—
238C22×C8—61no—
239C22×C8—62no—
240C22×C8—61no—
241C22×C8—62no—
242C22×C8—62no—
243#123—22no—
244#123—61no—
245C22⋊Q8—22no—
246C22⋊Q8—61no—
247#121—22no—
248#121—61no—
249#120—22no—
250#120—61no—
251C4×D4—24no—
252C4×D4—62no—
253#117—31no—
254#117C22×D312yes—
255#108—64no—
256#108—62no—
257C4⋊C8—62no—
258C4⋊C8—64no—
259#104—24no—
260#104—64no—
261#104—64no—
262C22⋊C8—22no—
263C22⋊C8—64no—
264C22⋊C8—64no—
265C22⋊C8—62no—
266#97—32no—
267#97—34no—
268C4×C8—32no—
269C4×C8—34no—
270C23×C3—21no—
271C23×C3—22no—
272C23×C3—22no—
273C23×C3—42no—
274C23×C3C2×#4111yescharacteristic
275C22×D3—21no—
276C3×Q8—22no—
277C3×Q8—22no—
278C3×D4—22no—
279C3×D4—22no—
280C3×D4—48no—
281C3×D4—42no—
282C3×D4—42no—
283C2×C3×C4—21no—
284C2×C3×C4—24no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—41no—
288C2×C3×C4—42no—
289C2×C3×C4—42no—
290C2×C3×C4—41no—
291C2×C3×C4—42no—
292C2×C3×C4C22×D412yes—
293C2×C3×C4C22×D411yescharacteristic
294C2×C3×C4C2×#4112yes—
295C2×C3×C4C2×#4111yescharacteristic
296#67—44no—
297#67—42no—
298C2×Dic3—22no—
299C2×Dic3—21no—
300D12—22no—
301C4×D3—41no—
302C4×D3—41no—
303Dic6—22no—
304C3×C8—22no—
305C3×C8—41no—
306C3⋊C8—24no—
307C3⋊C8—41no—
308C3⋊C8—42no—
309C3⋊C8#12711yescharacteristic
310C3⋊C8#12711yescharacteristic
311C3⋊C8#12712yes—
312C24#23711yescharacteristic
313#41—22no—
314#41—22no—
315#41—22no—
316#41—22no—
317#41—62no—
318#41—62no—
319C2×Q8—31no—
320C2×Q8#23712yes—
321C2×D4—24no—
322C2×D4—22no—
323C2×D4—31no—
324C2×D4—62no—
325C2×D4—61no—
326C2×D4#23712yes—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—61no—
333C22×C4#23711yescharacteristic
334C22×C4C23×D311yescharacteristic
335Q16—22no—
336Q16—64no—
337SD16—24no—
338SD16—64no—
339SD16—64no—
340D8—22no—
341D8—64no—
342M16—22no—
343M16—22no—
344M16—62no—
345M16—62no—
346M16—62no—
347M16—62no—
348M16—122no—
349M16—122no—
350M16—122no—
351C2×C8—21no—
352C2×C8—31no—
353C2×C8—32no—
354C2×C8—31no—
355C2×C8—32no—
356C2×C8—34no—
357C2×C8—61no—
358C2×C8—62no—
359C2×C8—62no—
360C2×C8—121no—
361C2×C8—121no—
362C2×C8—121no—
363C2×C8—121no—
364C2×C8—122no—
365C2×C8—122no—
366C2×C8—122no—
367C2×C8D3×D412yes—
368C4⋊C4—24no—
369C4⋊C4—22no—
370C4⋊C4—62no—
371C4⋊C4—61no—
372C22⋊C4—24no—
373C22⋊C4—24no—
374C22⋊C4—22no—
375C22⋊C4—62no—
376C22⋊C4—62no—
377C22⋊C4—61no—
378C42—32no—
379C42D3×D414yes—
380C22×C3—22no—
381C22×C3—24no—
382C22×C3—41no—
383C22×C3—41no—
384C22×C3—42no—
385C22×C3—44no—
386C22×C3—41no—
387C22×C3—42no—
388C22×C3#52512yes—
389C22×C3#52511yescharacteristic
390C2×D3—42no—
391C2×D3—41no—
392C3×C4—44no—
393C3×C4—41no—
394C3×C4—42no—
395C3×C4#52511yescharacteristic
396C3×C4#52512yes—
397C3×C4C2×#12711yescharacteristic
398Dic3—42no—
399Dic3—41no—
400C23—22no—
401C23—21no—
402C23—22no—
403C23—42no—
404C23—61no—
405C23C2×#23711yescharacteristic
406Q8—22no—
407Q8—22no—
408Q8—62no—
409D4—22no—
410D4—22no—
411D4—48no—
412D4—42no—
413D4—42no—
414D4—62no—
415D4—124no—
416D4—122no—
417C2×C4—24no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—21no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—62no—
427C2×C4—61no—
428C2×C4—121no—
429C2×C4—121no—
430C2×C4C2×D3×D411yescharacteristic
431C2×C4C2×D3×D412yes—
432C2×C4C2×#23712yes—
433C2×C4D3×#4111yescharacteristic
434C8—22no—
435C8—31no—
436C8—31no—
437C8—32no—
438C8—41no—
439C8—64no—
440C8—121no—
441C8—122no—
442C2×C3—21no—
443C2×C3—22no—
444C2×C3—42no—
445C2×C3—82no—
446C2×C3#239711yescharacteristic
447D3—81no—
448C22—24no—
449C22—22no—
450C22—41no—
451C22—42no—
452C22—41no—
453C22—44no—
454C22—41no—
455C22—42no—
456C22—122no—
457C22—121no—
458C22#558612yes—
459C22#560611yescharacteristic
460C4—44no—
461C4—41no—
462C4—42no—
463C4—122no—
464C4—121no—
465C4#558611yescharacteristic
466C4#560612yes—
467C4D3×#12711yescharacteristic
468C3#2379411yescharacteristic, complemented, Sylow
469C2—22no—
470C2—21no—
471C2—42no—
472C2—82no—
473C2—241no—
474C2#7666711yescharacteristic
subgroupG/H
centerC4D3×#127
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2010C2
socleC2×C3#2397
hypercenter#2010D3
hypocenterC3#23794
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #2010

Complements

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

NH ≅ G/Ntype
1. #83999C2semidirect
2. #82916C2semidirect
3. #78989C2semidirect
4. #78626C2semidirect
5. C3×#2010C2semidirect
6. #72121C2semidirect
7. #71715C2semidirect
8. #70405C2semidirect
9. #69987C2semidirect
10. #68715C2semidirect
11. #68697C2semidirect
53. #2010D3semidirect
106. #815D4semidirect
108. C8×Dic3D4semidirect
111. #803D4semidirect
112. C4×C3⋊C8D4semidirect
468. C3#23794semidirect

Maximal subgroups

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

subgroupindex
#839992
#829162
#789892
#786262
C3×#20102
#721212
#717152
#704052
#699872
#687152
#686972
#237943