Finite Group Databasebeta

← #10706239

Subgroups — #10706239

Order 768.

476 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 79 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#76946C211yesmaximal, characteristic
2#68997C211yesmaximal, characteristic, complemented
3#68125C211yesmaximal, characteristic, complemented
4C3×#1345C211yesmaximal, characteristic, complemented
5#67647C211yesmaximal, characteristic, complemented
6#67493C211yesmaximal, characteristic, complemented
7#67456C211yesmaximal, characteristic, complemented
8#11882—31nomaximal, Sylow
9C2×#940C411yescharacteristic
10C2×#887C2211yescharacteristic
11C2×#885—21no—
12C3×#394C2211yescharacteristic
13#5222C411yescharacteristic
14C2×#824—21no—
15#5118—21no—
16#4991C2211yescharacteristic
17#4957—21no—
18#4957—21no—
19#4957—21no—
20#4718—21no—
21#4672C411yescharacteristic
22#4649C411yescharacteristic
23C3×#343C2211yescharacteristic
24C3×#324C2211yescharacteristic
25#4530C2211yescharacteristic
26#4528C2211yescharacteristic
27#4469—21no—
28#2548—31no—
29#1801—31no—
30#1461—31no—
31#1345—31no—
32#1345—31no—
33#1345—31no—
34#1345D311yescharacteristic, complemented
35C22×C3×D4C2×C411yescharacteristic
36C22×#67—21no—
37C22×#67—21no—
38C22×D12C2×C411yescharacteristic
39C22×Dic6D411yescharacteristic
40C2×C3×M16—21no—
41C2×C3×C4⋊C4C2311yescharacteristic
42C2×C3×C22⋊C4—21no—
43C2×C3×C22⋊C4—21no—
44C2×#217—21no—
45C2×#217—21no—
46#940—41no—
47#940—41no—
48C2×(C2×D3)⋊C4—21no—
49C2×(C2×D3)⋊C4—21no—
50C2×(C2×D3)⋊C4—21no—
51C2×(C2×D3)⋊C4—21no—
52C2×Dic3⋊C4—21no—
53C2×Dic3⋊C4C2×C411yescharacteristic
54C2×Dic3⋊C4D411yescharacteristic
55C2×C4×Dic3—21no—
56C2×C4×Dic3C2×C411yescharacteristic
57C2×C4×Dic3D411yescharacteristic
58C2×C4×Dic3D411yescharacteristic
59C2×#208—21no—
60#887—41no—
61#887—41no—
62#885—42no—
63#885—42no—
64C3×C22⋊C8—21no—
65C3×#95C2×C411yescharacteristic
66C3⋊#95C2×C411yescharacteristic
67#830—21no—
68#824—42no—
69Dic3⋊C8—41no—
70Dic3⋊C8—41no—
71Dic3⋊C8—41no—
72C8×Dic3—41no—
73C2×#127—31no—
74C2×#124—61no—
75C2×C4×Q8—31no—
76#431—61no—
77#423—61no—
78C2×#101—61no—
79#394—31no—
80#394—31no—
81#394—31no—
82#394C2×D311yescharacteristic
83#390—61no—
84#390—61no—
85#390—61no—
86C4×C4⋊C4—31no—
87#343—31no—
88#343C2×D311yescharacteristic
89#324—31no—
90#324—61no—
91#324C2×D311yescharacteristic
92C24×C3—21no—
93C23×D3—21no—
94C2×C3×D4—41no—
95C2×C3×D4—81no—
96C22×C3×C4—21no—
97C22×C3×C4C22×C411yescharacteristic
98C22×C3×C4C2×D411yescharacteristic
99C22×C3×C4C2×D411yescharacteristic
100C2×#67—42no—
101C2×#67—42no—
102C2×#67—42no—
103C2×#67—42no—
104C2×#67—41no—
105C2×#67—41no—
106C2×#67—42no—
107C22×Dic3—21no—
108C22×Dic3C22⋊C411yescharacteristic
109C22×Dic3C22⋊C411yescharacteristic
110C22×Dic3C22⋊C411yescharacteristic
111C22×Dic3C22⋊C411yescharacteristic
112C2×D12—41no—
113C2×D12—81no—
114C2×Dic6—41no—
115C2×Dic6—41no—
116C2×Dic6D812yes—
117C2×Dic6SD1612yes—
118C3×M16—41no—
119C3×M16—41no—
120C2×C3×C8—41no—
121C2×C3×C8—41no—
122C3×C4⋊C4—41no—
123C3×C22⋊C4—41no—
124C3×C22⋊C4—41no—
125C3×C22⋊C4—81no—
126#217—42no—
127#217—81no—
128(C2×D3)⋊C4—41no—
129(C2×D3)⋊C4—41no—
130(C2×D3)⋊C4—42no—
131(C2×D3)⋊C4—81no—
132(C2×D3)⋊C4—81no—
133Dic3⋊C4—41no—
134Dic3⋊C4—41no—
135Dic3⋊C4—42no—
136C4×Dic3—21no—
137C4×Dic3—21no—
138C4×Dic3—21no—
139C4×Dic3—21no—
140C4×Dic3—21no—
141C4×Dic3—21no—
142C4×Dic3—42no—
143#208—41no—
144#208—41no—
145C2×C3⋊C8—41no—
146C2×C3⋊C8—41no—
147C22×Q8—31no—
148C22×D4—31no—
149C22×D4—61no—
150C22×D4—61no—
151C22×D4C4×D311yescharacteristic
152C2×M16—21no—
153C2×M16—61no—
154#127—121no—
155#127—121no—
156#124—122no—
157#124—122no—
158C4×Q8—121no—
159C4×Q8—121no—
160C2×C4⋊C4—31no—
161C2×C4⋊C4—31no—
162C2×C4⋊C4—61no—
163C2×C4⋊C4C22×D311yescharacteristic
164C2×C22⋊C4—21no—
165C2×C22⋊C4—21no—
166C2×C22⋊C4—61no—
167C2×C22⋊C4—61no—
168C2×C22⋊C4—61no—
169C2×C22⋊C4—61no—
170C2×C22⋊C4—61no—
171C2×C22⋊C4—61no—
172C2×C42—31no—
173C2×C42—31no—
174C2×C42—31no—
175C2×C42—61no—
176C4⋊C8—121no—
177C4⋊C8—121no—
178C4⋊C8—121no—
179#101—122no—
180C22⋊C8—21no—
181C22⋊C8—61no—
182C4×C8—121no—
183#95—31no—
184#95C4×D311yescharacteristic
185C23×C3—41no—
186C23×C3—41no—
187C23×C3—41no—
188C23×C3—41no—
189C23×C3—41no—
190C23×C3—41no—
191C23×C3—41no—
192C23×C3C2×C22⋊C411yescharacteristic
193C22×D3—41no—
194C22×D3—41no—
195C22×D3—41no—
196C22×D3—41no—
197C22×D3—41no—
198C22×D3—41no—
199C22×D3—41no—
200C3×D4—82no—
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4C2×D811yescharacteristic
211C2×C3×C4C2×SD1611yescharacteristic
212#67—82no—
213#67—84no—
214#67—82no—
215C2×Dic3—21no—
216C2×Dic3—21no—
217C2×Dic3—21no—
218C2×Dic3—21no—
219C2×Dic3—22no—
220C2×Dic3—21no—
221C2×Dic3—41no—
222C2×Dic3—41no—
223C2×Dic3—41no—
224C2×Dic3—41no—
225C2×Dic3#9911yescharacteristic
226C2×Dic3#9911yescharacteristic
227C2×Dic3#10212yes—
228C2×Dic3#10212yes—
229D12—82no—
230Dic6—42no—
231Dic6—42no—
232C3×C8—81no—
233C3×C8—81no—
234C3⋊C8—81no—
235C3⋊C8—81no—
236C24—21no—
237C24—61no—
238C2×Q8—32no—
239C2×Q8—32no—
240C2×Q8—121no—
241C2×Q8—121no—
242C2×D4—41no—
243C2×D4—81no—
244C2×D4—121no—
245C2×D4—122no—
246C2×D4—121no—
247C2×D4—121no—
248C2×D4—122no—
249C2×D4—122no—
250C2×D4—122no—
251C2×D4—122no—
252C2×D4—241no—
253C22×C4—21no—
254C22×C4—31no—
255C22×C4—31no—
256C22×C4—31no—
257C22×C4—31no—
258C22×C4—61no—
259C22×C4C2×C4×D311yescharacteristic
260C22×C4D3×D411yescharacteristic
261C22×C4D3×D411yescharacteristic
262M16—41no—
263M16—41no—
264M16—121no—
265M16—121no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8—121no—
269C2×C8—121no—
270C4⋊C4—41no—
271C4⋊C4—121no—
272C4⋊C4—121no—
273C4⋊C4—122no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—81no—
277C22⋊C4—122no—
278C22⋊C4—122no—
279C22⋊C4—121no—
280C22⋊C4—121no—
281C22⋊C4—241no—
282C22⋊C4—241no—
283C22⋊C4—241no—
284C42—61no—
285C42—61no—
286C42—61no—
287C42—61no—
288C42—61no—
289C42—61no—
290C42—122no—
291C22×C3—21no—
292C22×C3—21no—
293C22×C3—81no—
294C22×C3—81no—
295C22×C3—81no—
296C22×C3—81no—
297C22×C3—81no—
298C22×C3—81no—
299C22×C3—81no—
300C22×C3C2×#9911yescharacteristic
301C22×C3C2×#10211yescharacteristic
302C22×C3#42111yescharacteristic
303C2×D3—81no—
304C2×D3—81no—
305C2×D3—81no—
306C2×D3—81no—
307C2×D3—81no—
308C2×D3—81no—
309C2×D3—81no—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—41no—
313C3×C4—41no—
314C3×C4—81no—
315Dic3—21no—
316Dic3—22no—
317Dic3—22no—
318Dic3—41no—
319Dic3—81no—
320Dic3#35512yes—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—121no—
329C23—121no—
330C23—121no—
331C23—121no—
332C23—121no—
333C23—121no—
334C23—121no—
335C23D3×C22⋊C411yescharacteristic
336Q8—122no—
337Q8—122no—
338D4—82no—
339D4—242no—
340D4—242no—
341D4—242no—
342D4—244no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—31no—
348C2×C4—31no—
349C2×C4—32no—
350C2×C4—32no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—62no—
357C2×C4—61no—
358C2×C4—61no—
359C2×C4—61no—
360C2×C4—61no—
361C2×C4—61no—
362C2×C4—121no—
363C2×C4—121no—
364C2×C4—121no—
365C2×C4—121no—
366C2×C4D3×D811yescharacteristic
367C2×C4D3×SD1611yescharacteristic
368C8—81no—
369C8—81no—
370C8—241no—
371C8—241no—
372C2×C3—21no—
373C2×C3—21no—
374C2×C3—161no—
375C2×C3#151911yescharacteristic
376C2×C3#212211yescharacteristic
377C2×C3C2×#35511yescharacteristic
378D3—161no—
379C22—21no—
380C22—21no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22—81no—
387C22—81no—
388C22—241no—
389C22—241no—
390C22—241no—
391C22—241no—
392C22—241no—
393C22—241no—
394C22—241no—
395C22D3×#9911yescharacteristic
396C22D3×#10211yescharacteristic
397C22#482111yescharacteristic
398C4—21no—
399C4—21no—
400C4—32no—
401C4—41no—
402C4—41no—
403C4—61no—
404C4—62no—
405C4—62no—
406C4—81no—
407C4—121no—
408C4—241no—
409C3#1188211yescharacteristic, complemented, Sylow
410C2—21no—
411C2—21no—
412C2—161no—
413C2—481no—
414C2#6830711yescharacteristic
415C2#6913711yescharacteristic
416C2D3×#35511yescharacteristic
subgroupG/H
centerC22D3×#99
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1345C2
socleC22×C3C2×#99
prime radicalC2×#940C4
cosocleC2×C3×C4⋊C4C23
hypercenter#1345D3
hypocenterC3#11882
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1345

Complements

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

NH ≅ G/Ntype
2. #68997C2semidirect
3. #68125C2semidirect
4. C3×#1345C2semidirect
5. #67647C2semidirect
6. #67493C2semidirect
7. #67456C2semidirect
34. #1345D3semidirect
409. C3#11882semidirect

Maximal subgroups

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

subgroupindex
#769462
#689972
#681252
C3×#13452
#676472
#674932
#674562
#118823