Finite Group Databasebeta

← #10826085

Subgroups — #10826085

Order 768.

608 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83999C211yesmaximal, characteristic
2#79077C211yesmaximal, characteristic, complemented
3C3×#2053C211yesmaximal, characteristic
4#72117C211yesmaximal, characteristic
5#70539C211yesmaximal, characteristic, complemented
6#70294C212yesmaximal, complemented
7#70290C212yesmaximal, complemented
8#70040C212yesmaximal, complemented
9#70038C212yesmaximal, complemented
10#68701C211yesmaximal, characteristic, complemented
11#68697C211yesmaximal, characteristic, complemented
12#23834—31nomaximal, Sylow
13C3×#575C2211yescharacteristic
14C2×#948C2212yes—
15#5517C2211yescharacteristic
16C3×#424C2211yescharacteristic
17C3×#416C2212yes—
18C3×#410C2212yes—
19#5237C2212yes—
20#5236C2212yes—
21C2×#837C2212yes—
22#5197—22no—
23#5196—22no—
24#5125C2211yescharacteristic
25#5122C2211yescharacteristic
26#5118C2211yescharacteristic
27#5102C2211yescharacteristic
28C2×#815C2211yescharacteristic
29#4997C2212yes—
30#4997C2211yescharacteristic
31#4831—22no—
32#4828—22no—
33C3×H(3,ℤ/4)C2211yescharacteristic
34C3⋊#338C2211yescharacteristic
35#4552C2212yes—
36#4533C2212yes—
37#4533C2212yes—
38#4532C2212yes—
39#4531C2212yes—
40#2961—31no—
41#2955—31no—
42#2103—31no—
43#2102—31no—
44#2080—32no—
45#2053D311yescharacteristic
46#2050—31no—
47#2031—32no—
48#1765—32no—
49#1757—32no—
50#1747—31no—
51C2×C3×#41C2312yes—
52C3×#137—22no—
53C3×AGL(1,ℤ/8)—22no—
54C3×#135—22no—
55C2×C3×M16C2311yescharacteristic
56C22×C3×C8C2311yescharacteristic
57C3×#117C2312yes—
58C3×#117C2311yescharacteristic
59#948—22no—
60#948—22no—
61#948—22no—
62#948—22no—
63#926C2311yescharacteristic
64C2×C4×Dic3C2311yescharacteristic
65C2×#208—22no—
66C2×#208C2312yes—
67C2×#208C2312yes—
68C2×#208C2311yescharacteristic
69C22×C3⋊C8—22no—
70C22×C3⋊C8C2311yescharacteristic
71#890—21no—
72#888—21no—
73#878—22no—
74C4×Dic6—22no—
75C3×C4⋊C8—22no—
76C3×#103—22no—
77C3×#102—22no—
78C3×#99—24no—
79#837—24no—
80#837—22no—
81#837—22no—
82#836—22no—
83#833—22no—
84#819—22no—
85#815—21no—
86#815—21no—
87#815D412yes—
88Dic3⋊C8—22no—
89Dic3⋊C8—22no—
90C8×Dic3D412yes—
91#803—21no—
92#803—22no—
93C4×C3⋊C8—21no—
94C4×C3⋊C8—22no—
95#575C2×D311yescharacteristic
96C2×#131—32no—
97#533—31no—
98#446—62no—
99#446—62no—
100#444—62no—
101#443—62no—
102#430—31no—
103#424—31no—
104#424C2×D311yescharacteristic
105#423—31no—
106#421—32no—
107C2×#104—32no—
108#416C2×D312yes—
109#413—32no—
110#410C2×D312yes—
111#405—31no—
112#405—31no—
113#405—32no—
114C2×#97—31no—
115#344—32no—
116#344—32no—
117#341—32no—
118#339—32no—
119#338—31no—
120H(3,ℤ/4)—32no—
121H(3,ℤ/4)C2×D311yescharacteristic
122C3×#41—22no—
123C3×#41—22no—
124C3×#41—22no—
125C3×#41—22no—
126C2×C3×Q8#4112yes—
127C2×C3×D4—22no—
128C2×C3×D4#4112yes—
129C22×C3×C4—22no—
130C22×C3×C4C2411yescharacteristic
131C22×Dic3—21no—
132C2×Dic6—22no—
133C3×Q16—22no—
134C3×SD16—24no—
135C3×D8—22no—
136C3×M16—22no—
137C2×C3×C8—21no—
138C2×C3×C8—22no—
139C2×C3×C8C2×D412yes—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—22no—
142C3×C22⋊C4—22no—
143C3×C22⋊C4#4112yes—
144C3×C42—21no—
145C3×C42—22no—
146#217—21no—
147C4⋊Dic3—22no—
148C4⋊Dic3—21no—
149Dic3⋊C4—22no—
150Dic3⋊C4—22no—
151C4×Dic3—21no—
152C4×Dic3C2×D412yes—
153C4×Dic3#4111yescharacteristic
154C4×Dic3#4111yescharacteristic
155#208—22no—
156#208—22no—
157#208—22no—
158#208—22no—
159#208—22no—
160#208—42no—
161#208—42no—
162C2×C3⋊C8—21no—
163C2×C3⋊C8—22no—
164C2×C3⋊C8—22no—
165C2×C3⋊C8—42no—
166C2×C3⋊C8—42no—
167C2×C3⋊C8C2×D411yescharacteristic
168C2×C3⋊C8C2×D411yescharacteristic
169C2×C3⋊C8#4112yes—
170C2×C3⋊C8#4112yes—
171C2×#41C22×D312yes—
172#137—22no—
173AGL(1,ℤ/8)—22no—
174#135—22no—
175#131—62no—
176#131—62no—
177#131—62no—
178#131—62no—
179C2×M16—32no—
180C2×M16—32no—
181C2×M16—31no—
182C2×M16—62no—
183C2×M16C22×D311yescharacteristic
184C22×C8—31no—
185C22×C8—62no—
186C22×C8C22×D311yescharacteristic
187C4⋊Q8—61no—
188#125—61no—
189C22⋊Q8—62no—
190C4×Q8—62no—
191#117—31no—
192#117C22×D312yes—
193#117C22×D311yescharacteristic
194C2×C42—31no—
195#108—62no—
196C4⋊C8—22no—
197C4⋊C8—62no—
198C4⋊C8—62no—
199#104—62no—
200#104—62no—
201#104—64no—
202#103—22no—
203#102—22no—
204#101—62no—
205#100—62no—
206#99—24no—
207#97—32no—
208#97—62no—
209#97—61no—
210#97—61no—
211#97—61no—
212C4×C8—32no—
213C4×C8—62no—
214C4×C8—61no—
215C23×C3—22no—
216C23×C3C2×#4111yescharacteristic
217C3×Q8—22no—
218C3×Q8—22no—
219C3×D4—22no—
220C3×D4—22no—
221C3×D4—42no—
222C3×D4—42no—
223C2×C3×C4—22no—
224C2×C3×C4—22no—
225C2×C3×C4—42no—
226C2×C3×C4—42no—
227C2×C3×C4—42no—
228C2×C3×C4C22×D411yescharacteristic
229C2×C3×C4C2×#4112yes—
230C2×C3×C4C2×#4112yes—
231C2×C3×C4C2×#4111yescharacteristic
232C2×Dic3—21no—
233C2×Dic3—21no—
234C2×Dic3—22no—
235C2×Dic3—81no—
236Dic6—82no—
237C3×C8—22no—
238C3×C8—41no—
239C3⋊C8—41no—
240C3⋊C8—42no—
241C3⋊C8—42no—
242C3⋊C8#12411yescharacteristic
243C3⋊C8#12412yes—
244C3⋊C8#12411yescharacteristic
245#41—22no—
246#41—22no—
247#41—22no—
248#41—22no—
249C2×Q8—62no—
250C2×Q8#23712yes—
251C2×D4—22no—
252C2×D4#23712yes—
253C22×C4—22no—
254C22×C4—61no—
255C22×C4C23×D311yescharacteristic
256Q16—22no—
257SD16—24no—
258D8—22no—
259M16—22no—
260M16—62no—
261M16—62no—
262M16—62no—
263M16—62no—
264M16—62no—
265M16—122no—
266M16—122no—
267C2×C8—21no—
268C2×C8—22no—
269C2×C8—31no—
270C2×C8—31no—
271C2×C8—32no—
272C2×C8—32no—
273C2×C8—61no—
274C2×C8—62no—
275C2×C8—62no—
276C2×C8—122no—
277C2×C8—122no—
278C2×C8D3×D412yes—
279C4⋊C4—21no—
280C4⋊C4—22no—
281C4⋊C4—62no—
282C4⋊C4—62no—
283C4⋊C4—62no—
284C4⋊C4—61no—
285C22⋊C4—22no—
286C22⋊C4—61no—
287C22⋊C4#23712yes—
288C42—21no—
289C42—22no—
290C42—32no—
291C42—31no—
292C42—31no—
293C42—61no—
294C22×C3—41no—
295C22×C3—42no—
296C22×C3—82no—
297C22×C3#52511yescharacteristic
298C22×C3#52912yes—
299C3×C4—42no—
300C3×C4—42no—
301C3×C4—82no—
302C3×C4#52511yescharacteristic
303C3×C4C2×#12411yescharacteristic
304C3×C4#52912yes—
305Dic3—41no—
306Dic3—42no—
307Dic3—41no—
308C23—22no—
309C23C2×#23711yescharacteristic
310Q8—22no—
311Q8—22no—
312Q8—242no—
313D4—22no—
314D4—22no—
315D4—42no—
316D4—42no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—61no—
323C2×C4—61no—
324C2×C4—62no—
325C2×C4—241no—
326C2×C4C2×D3×D411yescharacteristic
327C2×C4C2×#23712yes—
328C2×C4D3×#4111yescharacteristic
329C2×C4D3×#4112yes—
330C8—22no—
331C8—31no—
332C8—32no—
333C8—31no—
334C8—41no—
335C8—122no—
336C8—122no—
337C8—121no—
338C2×C3—21no—
339C2×C3—22no—
340C2×C3—82no—
341C2×C3#239811yescharacteristic
342C22—41no—
343C22—42no—
344C22—82no—
345C22#560012yes—
346C22#560611yescharacteristic
347C4—42no—
348C4—42no—
349C4—82no—
350C4—121no—
351C4—121no—
352C4—122no—
353C4#558611yescharacteristic
354C4#564912yes—
355C4D3×#12411yescharacteristic
356C3#2383411yescharacteristic, complemented, Sylow
357C2—21no—
358C2—22no—
359C2—82no—
360C2#7689311yescharacteristic
subgroupG/H
centerC4D3×#124
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2053C2
socleC2×C3#2398
prime radicalC3×#575C22
hypercenter#2053D3
hypocenterC3#23834
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #2053

Complements

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

NH ≅ G/Ntype
2. #79077C2semidirect
5. #70539C2semidirect
6. #70294C2semidirect
7. #70290C2semidirect
8. #70040C2semidirect
9. #70038C2semidirect
10. #68701C2semidirect
11. #68697C2semidirect
356. C3#23834semidirect

Maximal subgroups

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

subgroupindex
#839992
#790772
C3×#20532
#721172
#705392
#702942
#702902
#700402
#700382
#687012
#686972
#238343