Finite Group Databasebeta

← #10896455

Subgroups — #10896455

Order 768.

604 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83910C211yesmaximal, characteristic
2#78480C211yesmaximal, characteristic, complemented
3C3×#1975C211yesmaximal, characteristic
4#72086C211yesmaximal, characteristic
5#70707C211yesmaximal, characteristic, complemented
6#70698C211yesmaximal, characteristic, complemented
7#70375C212yesmaximal, complemented
8#70368C212yesmaximal, complemented
9#70005C212yesmaximal, complemented
10#70002C212yesmaximal, complemented
11#69844C211yesmaximal, characteristic, complemented
12#29271—31nomaximal, Sylow
13C2×C3×#135C2211yescharacteristic
14#5819C2212yes—
15#5470C2211yescharacteristic
16C2×C3×#108C2211yescharacteristic
17C2×C3×#104C2212yes—
18C3×#408C2212yes—
19C2×C3×#97C2211yescharacteristic
20#5240C2212yes—
21#5236C2212yes—
22#5236C2212yes—
23#5197—22no—
24#5196—22no—
25#5125C2212yes—
26C2×#819C2211yescharacteristic
27#5106C2211yescharacteristic
28#5102C2211yescharacteristic
29#5102C2211yescharacteristic
30#5102C2212yes—
31#5101C2211yescharacteristic
32#5101C2211yescharacteristic
33#4924C2211yescharacteristic
34#4701—22no—
35#4701—22no—
36#4559C2212yes—
37#4559C2212yes—
38#4557C2212yes—
39#4523C2212yes—
40#2961—31no—
41#2957—31no—
42#2098—32no—
43#2031—32no—
44#1976—31no—
45#1975D311yescharacteristic
46#1953—31no—
47#1946—31no—
48#1868—32no—
49#1861—32no—
50#1833—31no—
51C2×C3×#41C2312yes—
52C3×#135—21no—
53C3×#135—21no—
54C3×#135—21no—
55C3×#135—21no—
56C2×C3×Q16Q811yescharacteristic, complemented
57C2×C3×SD16Q812yescomplemented
58C2×C3×D8Q811yescharacteristic, complemented
59C2×C3×M16C2312yes—
60C22×C3×C8C2311yescharacteristic
61C22×C3×C8C2311yescharacteristic
62C2×C3×C42C2311yescharacteristic
63#948—24no—
64#948—42no—
65#926C2312yes—
66#926C2311yescharacteristic
67#926C2311yescharacteristic
68#924—22no—
69C2×#208—42no—
70C2×#208C2311yescharacteristic
71C2×#208C2312yes—
72C2×#208C2311yescharacteristic
73#870—21no—
74#869—21no—
75C4×Dic6—22no—
76C3×#108—21no—
77C3×#108—21no—
78C3×#104—22no—
79C3×#104—22no—
80C3×C22⋊C8—42no—
81C3×#97—21no—
82C3×#97—21no—
83#837—24no—
84#837—42no—
85#834—24no—
86#819—21no—
87#819—21no—
88#819—42no—
89#818—21no—
90#817—21no—
91#815—21no—
92#815D412yes—
93Dic3⋊C8—22no—
94Dic3⋊C8—22no—
95Dic3⋊C8—42no—
96C8×Dic3—21no—
97C8×Dic3D412yes—
98C4⋊(C3⋊C8)—22no—
99C2×#135C2×D311yescharacteristic
100#568—32no—
101#533—31no—
102#446—62no—
103#446—62no—
104#444—62no—
105#443—62no—
106#430—32no—
107C2×#108—31no—
108C2×#108C2×D311yescharacteristic
109#427—31no—
110#424—31no—
111#424—31no—
112#424—32no—
113#423—31no—
114#421—32no—
115#421—32no—
116C2×#104C2×D312yes—
117#410—32no—
118#408C2×D312yes—
119#405—31no—
120#405—31no—
121C2×#97C2×D311yescharacteristic
122#344—32no—
123#344—32no—
124#339—32no—
125H(3,ℤ/4)—32no—
126C3×#41—22no—
127C3×#41—22no—
128C3×#41—22no—
129C3×#41—22no—
130C2×C3×Q8C2×Q812yes—
131C2×C3×D4—22no—
132C2×C3×D4C2×Q812yes—
133C22×C3×C4—21no—
134C22×C3×C4—22no—
135C22×C3×C4C2411yescharacteristic
136C2×Dic6—22no—
137C3×Q16—21no—
138C3×Q16—21no—
139C3×SD16—22no—
140C3×SD16—22no—
141C3×D8—21no—
142C3×D8—21no—
143C3×M16—22no—
144C3×M16—22no—
145C2×C3×C8—21no—
146C2×C3×C8—21no—
147C2×C3×C8—22no—
148C2×C3×C8—21no—
149C2×C3×C8C2×D411yescharacteristic
150C2×C3×C8C2×D411yescharacteristic
151C2×C3×C8C2×Q811yescharacteristic
152C2×C3×C8C2×Q811yescharacteristic
153C2×C3×C8#4111yescharacteristic
154C2×C3×C8#4111yescharacteristic
155C3×C42—21no—
156C3×C42#4111yescharacteristic
157C3×C42#4111yescharacteristic
158#217—21no—
159#217—22no—
160#217#4112yes—
161C4⋊Dic3—22no—
162C4⋊Dic3—21no—
163C4⋊Dic3—21no—
164C4⋊Dic3—22no—
165Dic3⋊C4—24no—
166C4×Dic3—21no—
167C4×Dic3—22no—
168C4×Dic3C2×D412yes—
169#208—21no—
170#208—21no—
171#208—22no—
172#208—21no—
173#208—22no—
174#208—21no—
175#208—44no—
176C2×C3⋊C8—21no—
177C2×C3⋊C8—22no—
178C2×C3⋊C8—44no—
179C2×C3⋊C8C2×D412yes—
180C2×#41C22×D312yes—
181#135—21no—
182#135—21no—
183#135—21no—
184#135—21no—
185C2×Q16Dic611yescharacteristic, complemented
186C2×SD16Dic612yescomplemented
187C2×D8Dic611yescharacteristic, complemented
188#131—64no—
189#131—122no—
190C2×M16—31no—
191C2×M16—32no—
192C2×M16—31no—
193C2×M16—122no—
194C2×M16C22×D312yes—
195C22×C8C22×D311yescharacteristic
196C22×C8C22×D311yescharacteristic
197C4⋊Q8—61no—
198#125—61no—
199C22⋊Q8—62no—
200C4×Q8—62no—
201#117—31no—
202#117—31no—
203#117—32no—
204C2×C42C22×D311yescharacteristic
205#108—21no—
206#108—21no—
207#108—61no—
208#108—61no—
209#108—122no—
210#107—61no—
211#106—61no—
212C4⋊C8—62no—
213C4⋊C8—62no—
214C4⋊C8—62no—
215C4⋊C8—122no—
216#104—22no—
217#104—22no—
218#104—64no—
219#104—122no—
220#99—64no—
221C22⋊C8—42no—
222#97—21no—
223#97—21no—
224#97—32no—
225#97—61no—
226C4×C8—32no—
227C4×C8—61no—
228C23×C3—22no—
229C23×C3C2×#4111yescharacteristic
230C3×Q8—22no—
231C3×Q8—22no—
232C3×D4—22no—
233C3×D4—22no—
234C3×D4—42no—
235C3×D4—42no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—22no—
239C2×C3×C4—42no—
240C2×C3×C4—42no—
241C2×C3×C4—81no—
242C2×C3×C4C22×D411yescharacteristic
243C2×C3×C4C22×Q811yescharacteristic
244C2×C3×C4C2×#4111yescharacteristic
245C2×C3×C4C2×#4111yescharacteristic
246C2×C3×C4H(5,𝔽2)11yescharacteristic
247C2×C3×C4H(5,𝔽2)11yescharacteristic
248C2×Dic3—22no—
249C2×Dic3—22no—
250C2×Dic3—42no—
251Dic6—82no—
252C3×C8—41no—
253C3×C8—42no—
254C3×C8C22⋊Q811yescharacteristic
255C3×C8C22⋊Q811yescharacteristic
256C3×C8C22⋊Q811yescharacteristic
257C3×C8C22⋊Q811yescharacteristic
258C3⋊C8—22no—
259C3⋊C8—42no—
260C3⋊C8—41no—
261#41—22no—
262#41—22no—
263#41—22no—
264#41—22no—
265C2×Q8—62no—
266C2×Q8C2×Dic612yes—
267C2×D4—22no—
268C2×D4C2×Dic612yes—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4C23×D311yescharacteristic
272Q16—21no—
273Q16—21no—
274SD16—22no—
275SD16—22no—
276D8—21no—
277D8—21no—
278M16—22no—
279M16—22no—
280M16—61no—
281M16—61no—
282M16—62no—
283M16—61no—
284M16—62no—
285M16—61no—
286M16—124no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—22no—
290C2×C8—21no—
291C2×C8—32no—
292C2×C8—61no—
293C2×C8—62no—
294C2×C8—124no—
295C2×C8C2×Dic611yescharacteristic
296C2×C8C2×Dic611yescharacteristic
297C2×C8D3×D411yescharacteristic
298C2×C8D3×D411yescharacteristic
299C2×C8#23711yescharacteristic
300C2×C8#23711yescharacteristic
301C4⋊C4—64no—
302C4⋊C4—62no—
303C4⋊C4—61no—
304C4⋊C4—61no—
305C4⋊C4—62no—
306C22⋊C4—32no—
307C22⋊C4—61no—
308C22⋊C4—62no—
309C42—21no—
310C42—32no—
311C42—61no—
312C42—62no—
313C42#23711yescharacteristic
314C42#23711yescharacteristic
315C22×C3—41no—
316C22×C3—42no—
317C22×C3—82no—
318C22×C3#52511yescharacteristic
319C22×C3#55411yescharacteristic
320C22×C3#55911yescharacteristic
321C3×C4—41no—
322C3×C4—41no—
323C3×C4—42no—
324C3×C4C2×C22⋊Q811yescharacteristic
325C3×C4#53511yescharacteristic
326C3×C4#55411yescharacteristic
327C3×C4#55911yescharacteristic
328Dic3—42no—
329Dic3—42no—
330Dic3—82no—
331C23—22no—
332C23D3×#4111yescharacteristic
333Q8—22no—
334Q8—22no—
335Q8—242no—
336D4—22no—
337D4—22no—
338D4—42no—
339D4—42no—
340C2×C4—22no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—62no—
346C2×C4—62no—
347C2×C4—81no—
348C2×C4—122no—
349C2×C4C22×Dic611yescharacteristic
350C2×C4C2×D3×D411yescharacteristic
351C2×C4C2×#23711yescharacteristic
352C2×C4C2×#23711yescharacteristic
353C2×C4#100411yescharacteristic
354C2×C4#100911yescharacteristic
355C8—42no—
356C8—41no—
357C8—62no—
358C8—122no—
359C8—121no—
360C8#87811yescharacteristic
361C8#87811yescharacteristic
362C8#87811yescharacteristic
363C8#87811yescharacteristic
364C2×C3—21no—
365C2×C3—21no—
366C2×C3—21no—
367C2×C3—82no—
368C2×C3#258011yescharacteristic
369C22—41no—
370C22—42no—
371C22—82no—
372C22#553911yescharacteristic
373C22#560611yescharacteristic
374C22#568111yescharacteristic
375C4—41no—
376C4—41no—
377C4—42no—
378C4—122no—
379C4—122no—
380C4—242no—
381C4C2×#87811yescharacteristic
382C4#554311yescharacteristic
383C4#560711yescharacteristic
384C4#571311yescharacteristic
385C3#2927111yescharacteristic, complemented, Sylow
386C2—21no—
387C2—21no—
388C2—21no—
389C2—82no—
390C2#7636411yescharacteristic
subgroupG/H
centerC4#5607
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1975C2
socleC2×C3#2580
prime radicalC2×C3×#135C22
hypercenter#1975D3
hypocenterC3#29271
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1975

Complements

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

NH ≅ G/Ntype
2. #78480C2semidirect
5. #70707C2semidirect
6. #70698C2semidirect
7. #70375C2semidirect
8. #70368C2semidirect
9. #70005C2semidirect
10. #70002C2semidirect
11. #69844C2semidirect
56. C2×C3×Q16Q8semidirect
57. C2×C3×SD16Q8semidirect
58. C2×C3×D8Q8semidirect
185. C2×Q16Dic6semidirect
186. C2×SD16Dic6semidirect
187. C2×D8Dic6semidirect
385. C3#29271semidirect

Maximal subgroups

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

subgroupindex
#839102
#784802
C3×#19752
#720862
#707072
#706982
#703752
#703682
#700052
#700022
#698442
#292713