Finite Group Databasebeta

← #10705702

Subgroups — #10705702

Order 768.

470 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 71 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#77199C211yesmaximal, characteristic
2#68989C211yesmaximal, characteristic, complemented
3#68126C211yesmaximal, characteristic, complemented
4C3×#1354C211yesmaximal, characteristic, complemented
5#67648C211yesmaximal, characteristic, complemented
6#67493C211yesmaximal, characteristic, complemented
7#67456C211yesmaximal, characteristic, complemented
8#11765—31nomaximal, Sylow
9C2×#947C411yescharacteristic
10C2×D3×M16—21no—
11C2×#892C2211yescharacteristic
12C3×#395C2211yescharacteristic
13#5231C411yescharacteristic
14#4991C2211yescharacteristic
15#4990—21no—
16#4988—21no—
17#4719—21no—
18#4671C411yescharacteristic
19#4670—21no—
20#4665—21no—
21#4648C411yescharacteristic
22C3×#343C2211yescharacteristic
23C3×#324C2211yescharacteristic
24#4554—21no—
25#4530C2211yescharacteristic
26#4528C2211yescharacteristic
27#4468—21no—
28#2610—31no—
29#1742—31no—
30#1464—31no—
31#1354—31no—
32#1354D311yescharacteristic, complemented
33#1345—31no—
34#1345—31no—
35C22×C3×Q8C2×C411yescharacteristic
36C22×D12—21no—
37C22×D12C2×C411yescharacteristic
38C22×C4×D3D411yescharacteristic
39C2×C3×M16—21no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C4⋊C4C2311yescharacteristic
42#947—41no—
43#947—41no—
44C2×(C2×D3)⋊C4—21no—
45C2×(C2×D3)⋊C4—21no—
46C2×(C2×D3)⋊C4—21no—
47C2×(C2×D3)⋊C4—21no—
48C2×(C2×D3)⋊C4—21no—
49C2×(C2×D3)⋊C4—21no—
50C2×(C2×D3)⋊C4C2×C411yescharacteristic
51C2×(C2×D3)⋊C4D411yescharacteristic
52C2×C4⋊Dic3D411yescharacteristic
53C2×C4×Dic3—21no—
54C2×C4×Dic3C2×C411yescharacteristic
55C2×C4×Dic3D411yescharacteristic
56C2×#208—21no—
57D3×M16—24no—
58D3×M16—42no—
59C2×ΓL(1,𝔽25)—41no—
60C2×D3×C8—41no—
61#892—41no—
62#892—41no—
63C3×C22⋊C8—21no—
64C3×#95—21no—
65C3×#95C2×C411yescharacteristic
66C3⋊#95—21no—
67C3⋊#95—21no—
68C3⋊#95C2×C411yescharacteristic
69#830—21no—
70#815—41no—
71Dic3⋊C8—41no—
72C22×M16—61no—
73C2×#124—31no—
74C2×#117—31no—
75#432—61no—
76#395C2×D311yescharacteristic
77#394—31no—
78#393—31no—
79#392—31no—
80#390—61no—
81#388—61no—
82#388—61no—
83#386—61no—
84#376—31no—
85#343—31no—
86#343—61no—
87#343C2×D311yescharacteristic
88#324—31no—
89#324C2×D311yescharacteristic
90#323—61no—
91C23×D3—21no—
92C23×D3C22⋊C411yescharacteristic
93C2×C3×Q8—41no—
94C2×C3×Q8—81no—
95C22×C3×C4—21no—
96C22×C3×C4—21no—
97C22×C3×C4C22×C411yescharacteristic
98C22×C3×C4C2×D411yescharacteristic
99C22×C3×C4C2×D411yescharacteristic
100C22×Dic3—21no—
101C22×Dic3C22⋊C411yescharacteristic
102C22×Dic3C22⋊C411yescharacteristic
103C22×Dic3C22⋊C411yescharacteristic
104C2×D12—41no—
105C2×D12—42no—
106C2×D12—42no—
107C2×D12—81no—
108C2×D12—81no—
109C2×C4×D3—22no—
110C2×C4×D3—21no—
111C2×C4×D3—21no—
112C2×C4×D3—21no—
113C2×C4×D3—21no—
114C3×M16—22no—
115C3×M16—41no—
116C2×C3×C8—41no—
117C2×C3×C8—41no—
118C3×C4⋊C4—41no—
119C3×C4⋊C4—81no—
120(C2×D3)⋊C4—41no—
121(C2×D3)⋊C4—41no—
122(C2×D3)⋊C4—42no—
123(C2×D3)⋊C4—42no—
124(C2×D3)⋊C4—41no—
125(C2×D3)⋊C4—41no—
126(C2×D3)⋊C4—41no—
127(C2×D3)⋊C4—41no—
128(C2×D3)⋊C4—81no—
129(C2×D3)⋊C4—81no—
130C4⋊Dic3—41no—
131C4×Dic3—21no—
132C4×Dic3—21no—
133C4×Dic3—21no—
134C4×Dic3—21no—
135C4×Dic3—42no—
136#208—22no—
137#208—41no—
138C2×C3⋊C8—41no—
139C2×C3⋊C8—41no—
140ΓL(1,𝔽25)—44no—
141D3×C8—44no—
142C22×Q8C4×D311yescharacteristic
143C22×D4—31no—
144C22×D4—61no—
145C23×C4—31no—
146C2×M16—21no—
147C2×M16—64no—
148C2×M16—61no—
149C2×M16—121no—
150C2×M16—122no—
151C22×C8—121no—
152#124—121no—
153#124—121no—
154#117—121no—
155#117—121no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—31no—
158C2×C4⋊C4C22×D311yescharacteristic
159C2×C22⋊C4—31no—
160C2×C22⋊C4—31no—
161C2×C22⋊C4—61no—
162C2×C22⋊C4—61no—
163C2×C22⋊C4—61no—
164C2×C22⋊C4—61no—
165C2×C22⋊C4—61no—
166C2×C22⋊C4—61no—
167C2×C42—31no—
168C2×C42—31no—
169C2×C42—61no—
170C4⋊C8—121no—
171C22⋊C8—21no—
172C22⋊C8—61no—
173#97—121no—
174#95—21no—
175#95—31no—
176#95—61no—
177#95—61no—
178#95C4×D311yescharacteristic
179C23×C3C2×C22⋊C411yescharacteristic
180C22×D3—21no—
181C22×D3—21no—
182C22×D3—41no—
183C22×D3—41no—
184C22×D3—41no—
185C22×D3—41no—
186C22×D3—41no—
187C22×D3—41no—
188C22×D3—41no—
189C22×D3—41no—
190C22×D3—41no—
191C22×D3#10412yes—
192C3×Q8—82no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—22no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—41no—
199C2×C3×C4—41no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
206C2×C3×C4#13711yescharacteristic
207C2×Dic3—21no—
208C2×Dic3—21no—
209C2×Dic3—21no—
210C2×Dic3—21no—
211C2×Dic3—41no—
212C2×Dic3—41no—
213C2×Dic3—41no—
214C2×Dic3—41no—
215C2×Dic3—41no—
216C2×Dic3#9911yescharacteristic
217C2×Dic3#9911yescharacteristic
218D12—84no—
219D12—82no—
220C4×D3—22no—
221C4×D3—22no—
222C4×D3—22no—
223C4×D3—22no—
224C3×C8—42no—
225C3×C8—81no—
226C3⋊C8—42no—
227C3⋊C8—81no—
228C24—31no—
229C24—61no—
230C2×Q8—41no—
231C2×Q8—81no—
232C2×D4—121no—
233C2×D4—122no—
234C2×D4—122no—
235C2×D4—241no—
236C2×D4—241no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—31no—
240C22×C4—31no—
241C22×C4—31no—
242C22×C4—61no—
243C22×C4—61no—
244C22×C4—62no—
245C22×C4—61no—
246C22×C4—61no—
247C22×C4—61no—
248C22×C4C2×C4×D311yescharacteristic
249C22×C4D3×D411yescharacteristic
250C22×C4D3×D411yescharacteristic
251M16—22no—
252M16—41no—
253M16—62no—
254M16—124no—
255M16—121no—
256C2×C8—41no—
257C2×C8—41no—
258C2×C8—124no—
259C2×C8—121no—
260C2×C8—121no—
261C4⋊C4—41no—
262C4⋊C4—81no—
263C4⋊C4—121no—
264C22⋊C4—121no—
265C22⋊C4—121no—
266C22⋊C4—122no—
267C22⋊C4—121no—
268C22⋊C4—121no—
269C22⋊C4—122no—
270C22⋊C4—121no—
271C22⋊C4—121no—
272C22⋊C4—241no—
273C22⋊C4—241no—
274C42—61no—
275C42—61no—
276C42—61no—
277C42—61no—
278C42—122no—
279C22×C3—21no—
280C22×C3—21no—
281C22×C3C2×#9911yescharacteristic
282C22×C3#41711yescharacteristic
283C22×C3C2×#10411yescharacteristic
284C2×D3—22no—
285C2×D3—41no—
286C2×D3—41no—
287C2×D3—42no—
288C2×D3—81no—
289C2×D3—81no—
290C2×D3—81no—
291C2×D3—81no—
292C2×D3—81no—
293C2×D3—81no—
294C2×D3—81no—
295C2×D3—81no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—42no—
299C3×C4—41no—
300C3×C4—41no—
301C3×C4—81no—
302Dic3—21no—
303Dic3—21no—
304Dic3—41no—
305Dic3—41no—
306Dic3—81no—
307C23—32no—
308C23—61no—
309C23—61no—
310C23—121no—
311C23—121no—
312C23—121no—
313C23—121no—
314C23—121no—
315C23—121no—
316C23—121no—
317C23—121no—
318C23—121no—
319C23D3×C22⋊C411yescharacteristic
320Q8—82no—
321D4—244no—
322D4—242no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—31no—
329C2×C4—31no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—62no—
338C2×C4—62no—
339C2×C4—62no—
340C2×C4—62no—
341C2×C4—61no—
342C2×C4—61no—
343C2×C4—61no—
344C2×C4—61no—
345C2×C4—121no—
346C2×C4—121no—
347C2×C4—121no—
348C2×C4—121no—
349C2×C4—121no—
350C2×C4#91411yescharacteristic
351C2×C4#91811yescharacteristic
352C8—42no—
353C8—81no—
354C8—122no—
355C8—241no—
356C2×C3—21no—
357C2×C3—21no—
358C2×C3#150411yescharacteristic
359C2×C3#212211yescharacteristic
360C2×C3#213211yescharacteristic
361D3—42no—
362D3—161no—
363C22—21no—
364C22—21no—
365C22—62no—
366C22—122no—
367C22—121no—
368C22—121no—
369C22—241no—
370C22—241no—
371C22—241no—
372C22—241no—
373C22—241no—
374C22—241no—
375C22—241no—
376C22—241no—
377C22D3×#9911yescharacteristic
378C22#480311yescharacteristic
379C22D3×#10411yescharacteristic
380C4—21no—
381C4—21no—
382C4—42no—
383C4—41no—
384C4—41no—
385C4—61no—
386C4—61no—
387C4—81no—
388C4—121no—
389C4—121no—
390C4—241no—
391C3#1176511yescharacteristic, complemented, Sylow
392C2—21no—
393C2—21no—
394C2—122no—
395C2—481no—
396C2#6833411yescharacteristic
397C2#6913711yescharacteristic
398C2#6917211yescharacteristic
subgroupG/H
centerC22D3×#99
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1354C2
socleC22×C3C2×#99
prime radical#4671C4
cosocleC2×C3×C4⋊C4C23
hypercenter#1354D3
hypocenterC3#11765
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1354

Complements

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

NH ≅ G/Ntype
2. #68989C2semidirect
3. #68126C2semidirect
4. C3×#1354C2semidirect
5. #67648C2semidirect
6. #67493C2semidirect
7. #67456C2semidirect
32. #1354D3semidirect
391. C3#11765semidirect

Maximal subgroups

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

subgroupindex
#771992
#689892
#681262
C3×#13542
#676482
#674932
#674562
#117653