Finite Group Databasebeta

← #10896453

Subgroups — #10896453

Order 768.

604 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83910C211yesmaximal, characteristic
2#78543C211yesmaximal, characteristic, complemented
3C3×#1976C211yesmaximal, characteristic
4#72085C211yesmaximal, characteristic
5#70663C211yesmaximal, characteristic, complemented
6#70657C211yesmaximal, characteristic, complemented
7#70405C212yesmaximal, complemented
8#70398C212yesmaximal, complemented
9#70044C212yesmaximal, complemented
10#70038C212yesmaximal, complemented
11#69901C211yesmaximal, characteristic, complemented
12#29271—31nomaximal, Sylow
13C2×C3×#135C2211yescharacteristic
14#5819C2212yes—
15#5517C2211yescharacteristic
16C3×#427C2211yescharacteristic
17C3×#421C2212yes—
18C3×#410C2212yes—
19C3×#405C2211yescharacteristic
20#5236C2212yes—
21C2×#837C2212yes—
22#5234C2212yes—
23#5197—22no—
24#5196—22no—
25#5124C2212yes—
26#5119C2211yescharacteristic
27#5118C2211yescharacteristic
28C2×#819C2211yescharacteristic
29C2×#819C2211yescharacteristic
30#5101C2211yescharacteristic
31C2×#815C2211yescharacteristic
32#4999C2211yescharacteristic
33#4999C2212yes—
34#4701—22no—
35#4679—22no—
36#4533C2212yes—
37#4533C2212yes—
38#4532C2212yes—
39#4531C2212yes—
40#2961—31no—
41#2957—31no—
42#2098—32no—
43#2031—32no—
44#1976D311yescharacteristic
45#1975—31no—
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×M16C2311yescharacteristic
60C22×C3×C8C2311yescharacteristic
61C3×#117C2312yes—
62C3×#117C2311yescharacteristic
63#948—24no—
64#948—42no—
65#926C2311yescharacteristic
66C2×C4×Dic3C2311yescharacteristic
67C2×#208—42no—
68C2×#208C2312yes—
69C2×#208C2312yes—
70C2×#208C2311yescharacteristic
71C22×C3⋊C8C2311yescharacteristic
72#890—21no—
73#888—21no—
74#878—22no—
75C4×Dic6—22no—
76C3×#107—21no—
77C3×#106—21no—
78C3×#104—42no—
79C3×#99—24no—
80C3×#97—21no—
81C3×C4×C8—21no—
82#837—24no—
83#837—22no—
84#837—22no—
85#830—42no—
86#822—42no—
87#819—21no—
88#819—21no—
89#819—21no—
90#819—21no—
91#815—21no—
92#815—21no—
93#815D412yes—
94Dic3⋊C8—22no—
95Dic3⋊C8—22no—
96C8×Dic3D412yes—
97C4⋊(C3⋊C8)—22no—
98C4⋊(C3⋊C8)—42no—
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×#108—31no—
109#427C2×D311yescharacteristic
110#424—31no—
111#424—31no—
112#424—32no—
113#423—31no—
114#421—32no—
115#421C2×D312yes—
116C2×#104—32no—
117#410C2×D312yes—
118#408—32no—
119#405—31no—
120#405C2×D311yescharacteristic
121C2×#97—31no—
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—22no—
134C22×C3×C4C2411yescharacteristic
135C22×Dic3—21no—
136C2×Dic6—22no—
137C3×Q16—21no—
138C3×Q16—21no—
139C3×SD16—22no—
140C3×SD16—22no—
141C3×D8—21no—
142C3×D8—21no—
143C3×M16—21no—
144C3×M16—21no—
145C2×C3×C8—21no—
146C2×C3×C8C2×D411yescharacteristic
147C2×C3×C8C2×D411yescharacteristic
148C2×C3×C8C2×Q811yescharacteristic
149C2×C3×C8C2×Q811yescharacteristic
150C2×C3×C8#4111yescharacteristic
151C2×C3×C8#4111yescharacteristic
152C3×C4⋊C4—21no—
153C3×C4⋊C4—22no—
154C3×C22⋊C4—22no—
155C3×C22⋊C4#4112yes—
156C3×C42—21no—
157C3×C42—22no—
158#217—21no—
159C4⋊Dic3—22no—
160C4⋊Dic3—21no—
161Dic3⋊C4—22no—
162Dic3⋊C4—22no—
163C4×Dic3—21no—
164C4×Dic3C2×D412yes—
165C4×Dic3#4111yescharacteristic
166C4×Dic3#4111yescharacteristic
167#208—21no—
168#208—21no—
169#208—22no—
170#208—22no—
171#208—22no—
172#208—22no—
173#208—44no—
174C2×C3⋊C8—21no—
175C2×C3⋊C8—21no—
176C2×C3⋊C8—22no—
177C2×C3⋊C8—22no—
178C2×C3⋊C8—21no—
179C2×C3⋊C8—44no—
180C2×C3⋊C8C2×D412yes—
181C2×#41C22×D312yes—
182#135—21no—
183#135—21no—
184#135—21no—
185#135—21no—
186C2×Q16Dic611yescharacteristic, complemented
187C2×SD16Dic612yescomplemented
188C2×D8Dic611yescharacteristic, complemented
189#131—64no—
190#131—122no—
191C2×M16—31no—
192C2×M16—32no—
193C2×M16—32no—
194C2×M16—122no—
195C2×M16C22×D311yescharacteristic
196C22×C8—31no—
197C22×C8C22×D311yescharacteristic
198C4⋊Q8—61no—
199#125—61no—
200C22⋊Q8—62no—
201C4×Q8—62no—
202#117—31no—
203#117C22×D312yes—
204#117C22×D311yescharacteristic
205C2×C42—31no—
206#108—61no—
207#108—61no—
208#108—61no—
209#108—61no—
210#108—122no—
211#107—21no—
212#106—21no—
213C4⋊C8—62no—
214C4⋊C8—62no—
215C4⋊C8—62no—
216C4⋊C8—122no—
217#104—42no—
218#104—62no—
219#104—62no—
220#104—64no—
221#99—24no—
222C22⋊C8—122no—
223#97—21no—
224#97—32no—
225#97—61no—
226#97—61no—
227C4×C8—21no—
228C4×C8—32no—
229C23×C3—22no—
230C23×C3C2×#4111yescharacteristic
231C3×Q8—22no—
232C3×Q8—22no—
233C3×D4—22no—
234C3×D4—22no—
235C3×D4—42no—
236C3×D4—42no—
237C2×C3×C4—22no—
238C2×C3×C4—22no—
239C2×C3×C4—42no—
240C2×C3×C4—42no—
241C2×C3×C4—42no—
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—21no—
249C2×Dic3—21no—
250C2×Dic3—22no—
251C2×Dic3—81no—
252Dic6—82no—
253C3×C8—41no—
254C3×C8C22⋊Q811yescharacteristic
255C3×C8C22⋊Q811yescharacteristic
256C3×C8C22⋊Q811yescharacteristic
257C3×C8C22⋊Q811yescharacteristic
258C3⋊C8—22no—
259C3⋊C8—41no—
260C3⋊C8—42no—
261C3⋊C8—42no—
262#41—22no—
263#41—22no—
264#41—22no—
265#41—22no—
266C2×Q8—62no—
267C2×Q8C2×Dic612yes—
268C2×D4—22no—
269C2×D4C2×Dic612yes—
270C22×C4—22no—
271C22×C4—61no—
272C22×C4C23×D311yescharacteristic
273Q16—21no—
274Q16—21no—
275SD16—22no—
276SD16—22no—
277D8—21no—
278D8—21no—
279M16—21no—
280M16—21no—
281M16—61no—
282M16—61no—
283M16—62no—
284M16—62no—
285M16—62no—
286M16—62no—
287M16—124no—
288C2×C8—21no—
289C2×C8—32no—
290C2×C8—61no—
291C2×C8—61no—
292C2×C8—62no—
293C2×C8—61no—
294C2×C8—62no—
295C2×C8—124no—
296C2×C8C2×Dic611yescharacteristic
297C2×C8C2×Dic611yescharacteristic
298C2×C8D3×D411yescharacteristic
299C2×C8D3×D411yescharacteristic
300C2×C8#23711yescharacteristic
301C2×C8#23711yescharacteristic
302C4⋊C4—21no—
303C4⋊C4—22no—
304C4⋊C4—62no—
305C4⋊C4—62no—
306C4⋊C4—62no—
307C4⋊C4—61no—
308C22⋊C4—22no—
309C22⋊C4—61no—
310C22⋊C4#23712yes—
311C42—21no—
312C42—22no—
313C42—32no—
314C42—31no—
315C42—31no—
316C42—61no—
317C22×C3—41no—
318C22×C3—42no—
319C22×C3—82no—
320C22×C3#52511yescharacteristic
321C22×C3#55411yescharacteristic
322C22×C3#55911yescharacteristic
323C3×C4—42no—
324C3×C4—42no—
325C3×C4—82no—
326C3×C4C2×C22⋊Q811yescharacteristic
327C3×C4#53511yescharacteristic
328C3×C4#55411yescharacteristic
329C3×C4#55911yescharacteristic
330Dic3—41no—
331Dic3—42no—
332Dic3—41no—
333C23—22no—
334C23C2×#23711yescharacteristic
335Q8—22no—
336Q8—22no—
337Q8—242no—
338D4—22no—
339D4—22no—
340D4—42no—
341D4—42no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—61no—
348C2×C4—61no—
349C2×C4—62no—
350C2×C4—241no—
351C2×C4C22×Dic611yescharacteristic
352C2×C4C2×D3×D411yescharacteristic
353C2×C4C2×#23711yescharacteristic
354C2×C4#100411yescharacteristic
355C2×C4D3×#4111yescharacteristic
356C2×C4#100911yescharacteristic
357C8—41no—
358C8—62no—
359C8—122no—
360C8—121no—
361C8—122no—
362C8#87811yescharacteristic
363C8#87811yescharacteristic
364C8#87811yescharacteristic
365C8#87811yescharacteristic
366C2×C3—21no—
367C2×C3—21no—
368C2×C3—21no—
369C2×C3—82no—
370C2×C3#258011yescharacteristic
371C22—41no—
372C22—42no—
373C22—82no—
374C22#554311yescharacteristic
375C22#560611yescharacteristic
376C22#571311yescharacteristic
377C4—42no—
378C4—42no—
379C4—82no—
380C4—121no—
381C4—121no—
382C4—122no—
383C4C2×#87811yescharacteristic
384C4#553911yescharacteristic
385C4#560711yescharacteristic
386C4#568111yescharacteristic
387C3#2927111yescharacteristic, complemented, Sylow
388C2—21no—
389C2—21no—
390C2—21no—
391C2—82no—
392C2#7636411yescharacteristic
subgroupG/H
centerC4#5607
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1976C2
socleC2×C3#2580
prime radicalC2×C3×#135C22
hypercenter#1976D3
hypocenterC3#29271
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1976

Complements

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

NH ≅ G/Ntype
2. #78543C2semidirect
5. #70663C2semidirect
6. #70657C2semidirect
7. #70405C2semidirect
8. #70398C2semidirect
9. #70044C2semidirect
10. #70038C2semidirect
11. #69901C2semidirect
56. C2×C3×Q16Q8semidirect
57. C2×C3×SD16Q8semidirect
58. C2×C3×D8Q8semidirect
186. C2×Q16Dic6semidirect
187. C2×SD16Dic6semidirect
188. C2×D8Dic6semidirect
387. C3#29271semidirect

Maximal subgroups

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

subgroupindex
#839102
#785432
C3×#19762
#720852
#706632
#706572
#704052
#703982
#700442
#700382
#699012
#292713