Finite Group Databasebeta

← #10692591

Subgroups — #10692591

Order 768.

534 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 109 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79288C211yesmaximal, characteristic
2#73802C211yesmaximal, characteristic
3#70778C211yesmaximal, characteristic, complemented
4#70527C211yesmaximal, characteristic
5#70506C211yesmaximal, characteristic, complemented
6C3×#1303C211yesmaximal, characteristic, complemented
7#67620C211yesmaximal, characteristic, complemented
8#9401—31nomaximal, Sylow
9C2×#875—21no—
10C2×C4×D12C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#423C2211yescharacteristic
14C3×C4×C4⋊C4C2211yescharacteristic
15C2×#823—21no—
16#4987—22no—
17#4929—21no—
18#4925C2211yescharacteristic
19#4924C2211yescharacteristic
20#4838—22no—
21#4726—22no—
22#4688—22no—
23#4674C414yes—
24#4554—21no—
25#4486—22no—
26#4482—22no—
27#2975—31no—
28#2547—31no—
29#1800—31no—
30#1546—31no—
31#1546—31no—
32#1303—31no—
33#1303D311yescharacteristic, complemented
34C22×D12D411yescharacteristic
35C22×C4×D3D412yes—
36C22×Dic6—21no—
37C2×C3×M16—21no—
38C2×C3×C4⋊C4—21no—
39C2×C3×C42—21no—
40C2×C3×C42C2311yescharacteristic
41C2×(C2×D3)⋊C4—24no—
42C2×(C2×D3)⋊C4C2×C412yes—
43C2×C4⋊Dic3—21no—
44C2×C4⋊Dic3C2×C412yes—
45C2×C4⋊Dic3D411yescharacteristic
46C2×#208—21no—
47C2×ΓL(1,𝔽25)—24no—
48#875—42no—
49#875—42no—
50C4×D12—21no—
51C4×D12—21no—
52C4×D12—21no—
53C4×D12—21no—
54#869—21no—
55#869—21no—
56#869—21no—
57#869—21no—
58C3×C4⋊C8—21no—
59C3×C4⋊C8—21no—
60C3×C4⋊C8D412yes—
61C3×C22⋊C8—21no—
62C3×C4×C8—21no—
63C3×#95C2×C412yes—
64C3⋊#95—24no—
65#830—21no—
66#823—42no—
67(C2×D3)⋊C8—24no—
68C4⋊(C3⋊C8)—21no—
69C4⋊(C3⋊C8)—21no—
70C4⋊(C3⋊C8)D412yes—
71C4×C3⋊C8—21no—
72C2×C4⋊Q8—31no—
73C2×#124—61no—
74C2×C4×D4—31no—
75#436—62no—
76#436—62no—
77#431—31no—
78#431C2×D311yescharacteristic
79#423—31no—
80#423C2×D311yescharacteristic
81C2×#100—61no—
82#407—62no—
83#397—34no—
84#391—61no—
85#380—62no—
86C4×C4⋊C4C2×D311yescharacteristic
87#343—61no—
88#332—62no—
89#332—62no—
90C23×D3C22⋊C412yes—
91C22×C3×C4—22no—
92C22×C3×C4C22×C411yescharacteristic
93C22×C3×C4C2×D411yescharacteristic
94C22×C3×C4C2×D411yescharacteristic
95C22×Dic3—22no—
96C22×Dic3C22⋊C412yes—
97C2×D12—21no—
98C2×D12—21no—
99C2×D12—21no—
100C2×D12—21no—
101C2×D12—21no—
102C2×D12—21no—
103C2×C4×D3—22no—
104C2×C4×D3—22no—
105C2×C4×D3—44no—
106C2×Dic6—41no—
107C2×Dic6—41no—
108C2×Dic6—41no—
109C2×Dic6—41no—
110C2×Dic6—41no—
111C2×Dic6—41no—
112C3×M16—41no—
113C3×M16—41no—
114C2×C3×C8—22no—
115C2×C3×C8—41no—
116C3×C4⋊C4—42no—
117C3×C42—41no—
118C3×C42—41no—
119C3×C42C2×D411yescharacteristic
120C3×C42C2×D411yescharacteristic
121C3×C42#4111yescharacteristic
122C3×C42#4111yescharacteristic
123(C2×D3)⋊C4—22no—
124(C2×D3)⋊C4—22no—
125(C2×D3)⋊C4—48no—
126C4⋊Dic3—21no—
127C4⋊Dic3—21no—
128C4⋊Dic3—22no—
129C4⋊Dic3—22no—
130C4⋊Dic3—41no—
131C4⋊Dic3—41no—
132#208—41no—
133#208—41no—
134C2×C3⋊C8—22no—
135C2×C3⋊C8—41no—
136ΓL(1,𝔽25)—84no—
137C22×Q8—61no—
138C22×D4—31no—
139C23×C4—32no—
140C2×M16—21no—
141C2×M16—64no—
142C2×M16—61no—
143C4⋊Q8—61no—
144C4⋊Q8—61no—
145C4⋊Q8—61no—
146C4⋊Q8—61no—
147#124—122no—
148#124—122no—
149C4×D4—61no—
150C4×D4—61no—
151C4×D4—61no—
152C4×D4—61no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—31no—
155C2×C4⋊C4—32no—
156C2×C4⋊C4—61no—
157C2×C22⋊C4—32no—
158C2×C22⋊C4—64no—
159C2×C42—21no—
160C2×C42C22×D311yescharacteristic
161C4⋊C8—21no—
162C4⋊C8—21no—
163C4⋊C8—32no—
164C4⋊C8—61no—
165C4⋊C8—61no—
166C4⋊C8D1212yes—
167#100—122no—
168C22⋊C8—21no—
169C22⋊C8—64no—
170C22⋊C8—61no—
171C4×C8—21no—
172C4×C8—61no—
173#95—64no—
174#95C4×D312yes—
175C23×C3C2×C22⋊C411yescharacteristic
176C22×D3—22no—
177C22×D3—22no—
178C22×D3—22no—
179C22×D3—42no—
180C22×D3—42no—
181C2×C3×C4—22no—
182C2×C3×C4—41no—
183C2×C3×C4—42no—
184C2×C3×C4—42no—
185C2×C3×C4—42no—
186C2×C3×C4C4×D411yescharacteristic
187C2×C3×C4C4×D411yescharacteristic
188C2×C3×C4#12011yescharacteristic
189C2×C3×C4#12111yescharacteristic
190C2×C3×C4#12111yescharacteristic
191C2×C3×C4#12311yescharacteristic
192C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
193C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
194C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
195C2×C3×C4#13711yescharacteristic
196C2×Dic3—22no—
197C2×Dic3—42no—
198C2×Dic3—42no—
199C2×Dic3—42no—
200C2×Dic3—42no—
201D12—41no—
202D12—41no—
203D12—41no—
204D12—41no—
205C4×D3—44no—
206C4×D3—82no—
207Dic6—81no—
208Dic6—81no—
209Dic6—81no—
210Dic6—81no—
211C3×C8—42no—
212C3×C8—81no—
213C3⋊C8—42no—
214C3⋊C8—81no—
215C24—32no—
216C2×Q8—121no—
217C2×Q8—121no—
218C2×Q8—121no—
219C2×Q8—121no—
220C2×Q8—121no—
221C2×Q8—121no—
222C2×D4—61no—
223C2×D4—61no—
224C2×D4—61no—
225C2×D4—61no—
226C2×D4—61no—
227C2×D4—61no—
228C22×C4—22no—
229C22×C4—32no—
230C22×C4—62no—
231C22×C4—62no—
232C22×C4—62no—
233C22×C4—124no—
234C22×C4C2×C4×D311yescharacteristic
235C22×C4D3×D411yescharacteristic
236C22×C4D3×D411yescharacteristic
237M16—41no—
238M16—41no—
239M16—121no—
240M16—121no—
241M16—244no—
242C2×C8—22no—
243C2×C8—41no—
244C2×C8—62no—
245C2×C8—121no—
246C4⋊C4—42no—
247C4⋊C4—61no—
248C4⋊C4—61no—
249C4⋊C4—62no—
250C4⋊C4—62no—
251C4⋊C4—121no—
252C4⋊C4—121no—
253C22⋊C4—62no—
254C22⋊C4—62no—
255C22⋊C4—128no—
256C42—41no—
257C42—41no—
258C42C2×D1211yescharacteristic
259C42#23511yescharacteristic
260C42D3×D411yescharacteristic
261C42#23911yescharacteristic
262C22×C3—21no—
263C22×C3—21no—
264C22×C3#38611yescharacteristic
265C22×C3#41711yescharacteristic
266C22×C3#41811yescharacteristic
267C2×D3—44no—
268C2×D3—42no—
269C2×D3—42no—
270C2×D3—42no—
271C2×D3—82no—
272C3×C4—21no—
273C3×C4—21no—
274C3×C4—21no—
275C3×C4—82no—
276C3×C4#45512yes—
277C3×C4#46811yescharacteristic
278C3×C4#46811yescharacteristic
279C3×C4#46911yescharacteristic
280C3×C4#46911yescharacteristic
281Dic3—42no—
282Dic3—82no—
283C23—62no—
284C23—62no—
285C23—62no—
286C23—122no—
287C23—122no—
288C23D3×C22⋊C411yescharacteristic
289Q8—241no—
290Q8—241no—
291Q8—241no—
292Q8—241no—
293D4—121no—
294D4—121no—
295D4—121no—
296D4—121no—
297C2×C4—22no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—62no—
303C2×C4—124no—
304C2×C4—122no—
305C2×C4—122no—
306C2×C4—122no—
307C2×C4—122no—
308C2×C4—242no—
309C2×C4C4×D1211yescharacteristic
310C2×C4#88211yescharacteristic
311C2×C4#88411yescharacteristic
312C2×C4#89311yescharacteristic
313C2×C4#89411yescharacteristic
314C2×C4#89511yescharacteristic
315C2×C4#91111yescharacteristic
316C2×C4#91411yescharacteristic
317C2×C4#91411yescharacteristic
318C2×C4#91811yescharacteristic
319C8—42no—
320C8—81no—
321C8—122no—
322C8—241no—
323C2×C3—21no—
324C2×C3—21no—
325C2×C3#188511yescharacteristic
326C2×C3#189511yescharacteristic
327C2×C3#191511yescharacteristic
328D3—82no—
329C22—21no—
330C22—21no—
331C22—124no—
332C22—122no—
333C22—122no—
334C22—122no—
335C22—242no—
336C22#466511yescharacteristic
337C22#477011yescharacteristic
338C22#480311yescharacteristic
339C4—21no—
340C4—21no—
341C4—21no—
342C4—82no—
343C4—122no—
344C4—242no—
345C4#478211yescharacteristic
346C4#481211yescharacteristic
347C4#482512yes—
348C4#486511yescharacteristic
349C4#488611yescharacteristic
350C3#940111yescharacteristic, complemented, Sylow
351C2—21no—
352C2—21no—
353C2—242no—
354C2#6875911yescharacteristic
355C2#6888511yescharacteristic
356C2#6904111yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1303C2
socleC22×C3#386
prime radicalC22×D12D4
cosocleC2×C3×C42C23
hypercenter#1303D3
hypocenterC3#9401
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1303

Complements

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

NH ≅ G/Ntype
3. #70778C2semidirect
5. #70506C2semidirect
6. C3×#1303C2semidirect
7. #67620C2semidirect
33. #1303D3semidirect
350. C3#9401semidirect

Maximal subgroups

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

subgroupindex
#792882
#738022
#707782
#705272
#705062
C3×#13032
#676202
#94013