Finite Group Databasebeta

← #10876301

Subgroups — #10876301

Order 768.

720 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2901C211yesmaximal, characteristic
2#82514C211yesmaximal, characteristic, complemented
3#72146C211yesmaximal, characteristic
4#72145C211yesmaximal, characteristic
5#72027C211yesmaximal, characteristic, complemented
6#72026C211yesmaximal, characteristic, complemented
7#71644C211yesmaximal, characteristic, complemented
8#70405C212yesmaximal, complemented
9#70404C212yesmaximal, complemented
10#70378C212yesmaximal, complemented
11#70372C212yesmaximal, complemented
12#27599—31nomaximal, Sylow
13C3×#585C2211yescharacteristic
14C3×#568C2211yescharacteristic
15C3×C4×#41C2211yescharacteristic
16C3×#421—22no—
17C3×#421C2212yes—
18C2×C3×#104—22no—
19C2×C3×#104C2212yes—
20#5240C2211yescharacteristic
21#5237C2212yes—
22#5237C2211yescharacteristic
23#5236C2212yes—
24#5234C2211yescharacteristic
25#5234C2211yescharacteristic
26#5234C2212yes—
27#5233C2212yes—
28#5124C2212yescomplemented
29#5119C2212yescomplemented
30#5000—22no—
31#4999C2211yescharacteristic, complemented
32#4999C2211yescharacteristic, complemented
33#4926—22no—
34C4×#208C2211yescharacteristic, complemented
35C2×#803C2211yescharacteristic, complemented
36#4558C2212yescomplemented
37#4556C2212yescomplemented
38#4531C2212yescomplemented
39#4523C2212yescomplemented
40#2923—31no—
41#2901D311yescharacteristic
42#2098—32no—
43#2080—32no—
44#1938—32no—
45#1932—32no—
46#1868—31no—
47#1858—31no—
48#1827—31no—
49#1794—31no—
50#1791—31no—
51C3×2−1+4—22no—
52C3×2−1+4—21no—
53C3×H(5,𝔽2)—22no—
54C3×H(5,𝔽2)—22no—
55C3×H(5,𝔽2)—21no—
56C2×C3×#41—22no—
57C2×C3×#41—42no—
58C2×C3×#41C2312yes—
59C2×C3×#41C2311yescharacteristic
60C3×#131—42no—
61C2×C3×M16—22no—
62C2×C3×M16C2312yes—
63C3×C4×Q8—21no—
64C3×C4×D4—22no—
65C3×C4×D4—21no—
66C3×#117D412yes—
67C3×#117C2311yescharacteristic
68C2×C3×C42D412yes—
69C2×C3×C42C2311yescharacteristic
70#926C2312yes—
71C2×#208C2311yescharacteristic
72C2×#208C2312yes—
73C2×#208C2311yescharacteristic
74C22×C3⋊C8C2311yescharacteristic
75C22×C3⋊C8C2311yescharacteristic
76C3×#104—22no—
77C3×#104—24no—
78C3×#104—22no—
79#837—42no—
80#836—22no—
81#836—21no—
82#835—22no—
83#834—22no—
84#833—22no—
85#833—21no—
86#832—22no—
87#830—22no—
88#830—22no—
89#830—42no—
90#822—42no—
91Dic3⋊C8—42no—
92C4⋊(C3⋊C8)—22no—
93C4⋊(C3⋊C8)—22no—
94#803—21no—
95#803—21no—
96#803—21no—
97C4×C3⋊C8—21no—
98#585C2×D311yescharacteristic
99#568C2×D311yescharacteristic
100C4×#41C2×D311yescharacteristic
101#433—62no—
102#432—62no—
103#430—32no—
104#424—31no—
105#424—31no—
106#424—32no—
107#421—22no—
108#421—32no—
109#421C2×D312yes—
110C2×#104—22no—
111C2×#104C2×D312yes—
112#416—32no—
113#413—32no—
114#413—31no—
115#410—31no—
116#408—31no—
117#408—31no—
118#408—32no—
119C4×M16—31no—
120C2×#97—31no—
121#344—32no—
122#341—32no—
123#339—32no—
124H(3,ℤ/4)—32no—
125C3×#41—22no—
126C3×#41—22no—
127C3×#41—24no—
128C3×#41—22no—
129C3×#41—22no—
130C3×#41—22no—
131C3×#41—22no—
132C3×#41—44no—
133C3×#41—42no—
134C3×#41—42no—
135C3×#41—42no—
136C3×#41—42no—
137C2×C3×Q8—22no—
138C2×C3×Q8—42no—
139C2×C3×Q8#4112yes—
140C2×C3×Q8#4111yescharacteristic
141C2×C3×D4—22no—
142C2×C3×D4—22no—
143C2×C3×D4—42no—
144C2×C3×D4—44no—
145C2×C3×D4—42no—
146C2×C3×D4#4112yes—
147C2×C3×D4#4112yes—
148C2×C3×D4#4111yescharacteristic
149C22×C3×C4—22no—
150C22×C3×C4—21no—
151C22×C3×C4—22no—
152C22×C3×C4—42no—
153C22×C3×C4C2×D412yes—
154C22×C3×C4C2411yescharacteristic
155C3×M16—22no—
156C3×M16—22no—
157C3×M16—42no—
158C2×C3×C8—22no—
159C2×C3×C8—42no—
160C3×C4⋊C4—22no—
161C3×C4⋊C4—21no—
162C3×C22⋊C4—22no—
163C3×C22⋊C4—21no—
164C3×C42—22no—
165C3×C42—21no—
166C3×C42C2×D411yescharacteristic
167C3×C42C2×D411yescharacteristic
168C3×C42C2×D412yes—
169#217—22no—
170C4⋊Dic3—22no—
171C4×Dic3—22no—
172#208—22no—
173#208—22no—
174#208—22no—
175#208—22no—
176C2×C3⋊C8—21no—
177C2×C3⋊C8—21no—
178C2×C3⋊C8—21no—
179C2×C3⋊C8—21no—
180C2×C3⋊C8—22no—
181C2×C3⋊C8—22no—
182C2×C3⋊C8—22no—
1832−1+4—22no—
1842−1+4—21no—
185H(5,𝔽2)—22no—
186H(5,𝔽2)—22no—
187H(5,𝔽2)—21no—
188C2×#41—22no—
189C2×#41—42no—
190C2×#41C22×D312yes—
191C2×#41C22×D311yescharacteristic
192#131—42no—
193C2×M16—22no—
194C2×M16—32no—
195C2×M16—31no—
196C2×M16—31no—
197C2×M16C22×D312yes—
198C22×C8—31no—
199C22×C8—31no—
200C4×Q8—21no—
201C4×D4—22no—
202C4×D4—21no—
203#117—32no—
204#117#6712yes—
205#117C22×D311yescharacteristic
206C2×C42#6712yes—
207C2×C42C22×D311yescharacteristic
208#108—122no—
209C4⋊C8—62no—
210C4⋊C8—62no—
211C4⋊C8—122no—
212#104—22no—
213#104—24no—
214#104—22no—
215#104—122no—
216#103—62no—
217#102—62no—
218#101—62no—
219#101—61no—
220#100—62no—
221#100—61no—
222#99—62no—
223C22⋊C8—62no—
224C22⋊C8—62no—
225C22⋊C8—122no—
226#97—61no—
227#97—61no—
228#97—61no—
229C4×C8—61no—
230C23×C3—22no—
231C23×C3—42no—
232C23×C3#12312yes—
233C23×C3C2×#4111yescharacteristic
234C3×Q8—22no—
235C3×Q8—22no—
236C3×Q8—22no—
237C3×Q8—42no—
238C3×D4—24no—
239C3×D4—22no—
240C3×D4—22no—
241C3×D4—22no—
242C3×D4—42no—
243C3×D4—44no—
244C3×D4—42no—
245C3×D4—42no—
246C2×C3×C4—22no—
247C2×C3×C4—22no—
248C2×C3×C4—21no—
249C2×C3×C4—22no—
250C2×C3×C4—21no—
251C2×C3×C4—22no—
252C2×C3×C4—44no—
253C2×C3×C4—42no—
254C2×C3×C4—42no—
255C2×C3×C4—42no—
256C2×C3×C4—81no—
257C2×C3×C4—82no—
258C2×C3×C4#12312yes—
259C2×C3×C4C22×D411yescharacteristic
260C2×C3×C4C2×#4112yes—
261C2×C3×C4C2×#4111yescharacteristic
262C2×C3×C4H(5,𝔽2)12yes—
263C2×Dic3—42no—
264C3×C8—42no—
265C3⋊C8—41no—
266C3⋊C8—41no—
267C3⋊C8—41no—
268C3⋊C8—41no—
269C3⋊C8—42no—
270#41—22no—
271#41—22no—
272#41—24no—
273#41—22no—
274#41—22no—
275#41—22no—
276#41—22no—
277#41—44no—
278#41—42no—
279#41—42no—
280#41—42no—
281#41—42no—
282C2×Q8—22no—
283C2×Q8—42no—
284C2×Q8#23512yes—
285C2×Q8#23711yescharacteristic
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—42no—
289C2×D4—44no—
290C2×D4—42no—
291C2×D4#23512yes—
292C2×D4#23712yes—
293C2×D4#23711yescharacteristic
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—42no—
298C22×C4C2×#6712yes—
299C22×C4C23×D311yescharacteristic
300M16—22no—
301M16—22no—
302M16—42no—
303M16—62no—
304M16—62no—
305M16—62no—
306M16—62no—
307C2×C8—22no—
308C2×C8—42no—
309C2×C8—62no—
310C2×C8—62no—
311C2×C8—62no—
312C2×C8—61no—
313C2×C8—61no—
314C2×C8—61no—
315C2×C8—61no—
316C4⋊C4—22no—
317C4⋊C4—21no—
318C4⋊C4—62no—
319C22⋊C4—22no—
320C22⋊C4—21no—
321C22⋊C4—62no—
322C42—22no—
323C42—21no—
324C42—62no—
325C42C2×#6711yescharacteristic
326C42C2×#6711yescharacteristic
327C42C2×#6712yes—
328C22×C3—22no—
329C22×C3—41no—
330C22×C3—42no—
331C22×C3—42no—
332C22×C3—82no—
333C22×C3—162no—
334C22×C3C2×#12311yescharacteristic
335C22×C3#55112yes—
336C3×C4—22no—
337C3×C4—42no—
338C3×C4—41no—
339C3×C4—41no—
340C3×C4—42no—
341C3×C4#52511yescharacteristic
342C3×C4#53511yescharacteristic
343C3×C4#55112yes—
344Dic3—82no—
345C23—22no—
346C23—42no—
347C23#93512yes—
348C23C2×#23711yescharacteristic
349Q8—22no—
350Q8—22no—
351Q8—22no—
352Q8—42no—
353D4—24no—
354D4—22no—
355D4—22no—
356D4—22no—
357D4—42no—
358D4—44no—
359D4—42no—
360D4—42no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—21no—
367C2×C4—44no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—81no—
372C2×C4—82no—
373C2×C4—122no—
374C2×C4#93512yes—
375C2×C4C2×#23512yes—
376C2×C4C2×#23711yescharacteristic
377C2×C4#100412yes—
378C2×C4C22×#6711yescharacteristic
379C8—42no—
380C8—122no—
381C8—121no—
382C8—121no—
383C8—121no—
384C8—121no—
385C2×C3—22no—
386C2×C3—21no—
387C2×C3—42no—
388C2×C3—82no—
389C2×C3#248211yescharacteristic
390C22—22no—
391C22—41no—
392C22—42no—
393C22—42no—
394C22—82no—
395C22—162no—
396C22#549512yes—
397C22C2×#93511yescharacteristic
398C4—22no—
399C4—41no—
400C4—42no—
401C4—41no—
402C4—42no—
403C4—242no—
404C4#549512yes—
405C4#579111yescharacteristic
406C4#580511yescharacteristic
407C3#2759911yescharacteristic, complemented, Sylow
408C2—21no—
409C2—22no—
410C2—42no—
411C2—82no—
412C2#8211311yescharacteristic
subgroupG/H
centerC4#5805
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2901C2
socleC2×C3#2482
prime radicalC3×#585C22
hypercenter#2901D3
hypocenterC3#27599
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #2901

Complements

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

NH ≅ G/Ntype
2. #82514C2semidirect
5. #72027C2semidirect
6. #72026C2semidirect
7. #71644C2semidirect
8. #70405C2semidirect
9. #70404C2semidirect
10. #70378C2semidirect
11. #70372C2semidirect
28. #5124C22semidirect
29. #5119C22semidirect
31. #4999C22semidirect
32. #4999C22semidirect
34. C4×#208C22semidirect
35. C2×#803C22semidirect
36. #4558C22semidirect
37. #4556C22semidirect
38. #4531C22semidirect
39. #4523C22semidirect
407. C3#27599semidirect

Maximal subgroups

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

subgroupindex
C3×#29012
#825142
#721462
#721452
#720272
#720262
#716442
#704052
#704042
#703782
#703722
#275993