Finite Group Databasebeta

← #10705772

Subgroups — #10705772

Order 768.

464 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 73 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#77764C211yesmaximal, characteristic
2#68972C211yesmaximal, characteristic
3#68145C211yesmaximal, characteristic
4C3×#1346C211yesmaximal, characteristic, complemented
5#67650C211yesmaximal, characteristic, complemented
6#67495C211yesmaximal, characteristic, complemented
7#67458C211yesmaximal, characteristic, complemented
8#11780—31nomaximal, Sylow
9C2×#894—21no—
10C2×#893C2211yescharacteristic
11C2×D3×C4⋊C4C411yescharacteristic
12C3×#398C2211yescharacteristic
13C2×#823—21no—
14Dic3×M16—21no—
15#4992C411yescharacteristic
16#4981C2211yescharacteristic
17#4976—21no—
18#4726—21no—
19#4674C411yescharacteristic
20#4667—21no—
21#4667C411yescharacteristic
22#4664—21no—
23C3×#343C2211yescharacteristic
24C3×#324C2211yescharacteristic
25#4530C2211yescharacteristic
26#4528C2211yescharacteristic
27#4468—21no—
28#2486—31no—
29C4×#100—31no—
30#1465—31no—
31#1346—31no—
32#1346—31no—
33#1346—31no—
34#1346D311yescharacteristic, complemented
35C22×D12D411yescharacteristic
36C22×C4×D3—21no—
37C22×C4×D3C2×C411yescharacteristic
38C22×C4×D3D411yescharacteristic
39C2×C3×M16—21no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C4⋊C4C2×C411yescharacteristic
42C2×C3×C4⋊C4C2311yescharacteristic
43C2×(C2×D3)⋊C4—21no—
44C2×(C2×D3)⋊C4—21no—
45C2×(C2×D3)⋊C4—21no—
46C2×(C2×D3)⋊C4C2×C411yescharacteristic
47C2×(C2×D3)⋊C4D411yescharacteristic
48C2×C4⋊Dic3C2×C411yescharacteristic
49C2×Dic3⋊C4—21no—
50C2×Dic3⋊C4—21no—
51C2×Dic3⋊C4—21no—
52C2×C4×Dic3D411yescharacteristic
53C2×#208—21no—
54C2×ΓL(1,𝔽25)—41no—
55#894—42no—
56#894—42no—
57#893—41no—
58#893—41no—
59D3×C4⋊C4—41no—
60D3×C4⋊C4—41no—
61C3×C22⋊C8—21no—
62C3×#95—21no—
63C3×#95C2×C411yescharacteristic
64C3⋊#95—21no—
65C3⋊#95—21no—
66C3⋊#95—21no—
67C3⋊#95C2×C411yescharacteristic
68#830—21no—
69#823—24no—
70(C2×D3)⋊C8—41no—
71#815—41no—
72C8×Dic3—41no—
73C2×#123—61no—
74C2×C4×D4—31no—
75C22×C4⋊C4—31no—
76C2×#100—61no—
77#407—61no—
78C4×M16—61no—
79#398—31no—
80#398C2×D311yescharacteristic
81#397—31no—
82#397—31no—
83#389—61no—
84#381—61no—
85#380—31no—
86#380—61no—
87#343—31no—
88#343C2×D311yescharacteristic
89#324—31no—
90#324C2×D311yescharacteristic
91#323—61no—
92C23×D3C22⋊C411yescharacteristic
93C23×D3C22⋊C411yescharacteristic
94C22×C3×C4—21no—
95C22×C3×C4—21no—
96C22×C3×C4C22×C411yescharacteristic
97C22×C3×C4C2×D411yescharacteristic
98C22×C3×C4C2×D411yescharacteristic
99C22×Dic3—21no—
100C22×Dic3—21no—
101C22×Dic3C22⋊C411yescharacteristic
102C22×Dic3C22⋊C411yescharacteristic
103C2×D12—22no—
104C2×D12—21no—
105C2×D12—21no—
106C2×D12—41no—
107C2×C4×D3—21no—
108C2×C4×D3—21no—
109C2×C4×D3—21no—
110C2×C4×D3—21no—
111C2×C4×D3—42no—
112C2×C4×D3—42no—
113C2×C4×D3—42no—
114C2×C4×D3—42no—
115C2×C4×D3—42no—
116C3×M16—22no—
117C3×M16—41no—
118C2×C3×C8—41no—
119C2×C3×C8—41no—
120C3×C4⋊C4—41no—
121C3×C4⋊C4—41no—
122C3×C4⋊C4—41no—
123C3×C4⋊C4—41no—
124(C2×D3)⋊C4—41no—
125(C2×D3)⋊C4—41no—
126(C2×D3)⋊C4—42no—
127(C2×D3)⋊C4—42no—
128(C2×D3)⋊C4—42no—
129C4⋊Dic3—41no—
130Dic3⋊C4—42no—
131Dic3⋊C4—41no—
132Dic3⋊C4—41no—
133Dic3⋊C4—41no—
134Dic3⋊C4—41no—
135C4×Dic3—21no—
136C4×Dic3—21no—
137#208—22no—
138#208—41no—
139C2×C3⋊C8—41no—
140C2×C3⋊C8—41no—
141ΓL(1,𝔽25)—82no—
142C22×D4—31no—
143C23×C4—31no—
144C23×C4—31no—
145C23×C4—61no—
146C2×M16—21no—
147C2×M16—61no—
148C2×M16—121no—
149#123—122no—
150#123—122no—
151C4×D4—121no—
152C4×D4—121no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—31no—
155C2×C4⋊C4—61no—
156C2×C4⋊C4—61no—
157C2×C4⋊C4—61no—
158C2×C4⋊C4—121no—
159C2×C4⋊C4—121no—
160C2×C4⋊C4C4×D311yescharacteristic
161C2×C4⋊C4C22×D311yescharacteristic
162C2×C22⋊C4—31no—
163C2×C22⋊C4—31no—
164C2×C22⋊C4—61no—
165C2×C22⋊C4—61no—
166C2×C22⋊C4—61no—
167C2×C42—31no—
168#100—64no—
169C22⋊C8—21no—
170C22⋊C8—61no—
171C22⋊C8—121no—
172#97—121no—
173C4×C8—121no—
174#95—21no—
175#95—31no—
176#95—61no—
177#95—61no—
178#95—61no—
179#95C4×D311yescharacteristic
180C23×C3C2×C22⋊C411yescharacteristic
181C22×D3—21no—
182C22×D3—21no—
183C22×D3—21no—
184C22×D3—21no—
185C22×D3—21no—
186C22×D3—41no—
187C22×D3—41no—
188C22×D3—41no—
189C22×D3—41no—
190C22×D3#10412yes—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—41no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4—41no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
204C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
205C2×Dic3—22no—
206C2×Dic3—21no—
207C2×Dic3—41no—
208C2×Dic3—41no—
209C2×Dic3—41no—
210C2×Dic3—41no—
211C2×Dic3—41no—
212C2×Dic3—41no—
213C2×Dic3—41no—
214C2×Dic3#9911yescharacteristic
215C2×Dic3#9911yescharacteristic
216D12—42no—
217D12—42no—
218C4×D3—42no—
219C4×D3—42no—
220C4×D3—42no—
221C4×D3—81no—
222C4×D3—84no—
223C3×C8—42no—
224C3×C8—81no—
225C3⋊C8—42no—
226C3⋊C8—81no—
227C24—31no—
228C24—31no—
229C2×D4—62no—
230C2×D4—61no—
231C2×D4—61no—
232C2×D4—121no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—31no—
236C22×C4—31no—
237C22×C4—61no—
238C22×C4—61no—
239C22×C4—61no—
240C22×C4—61no—
241C22×C4—61no—
242C22×C4—61no—
243C22×C4—122no—
244C22×C4—122no—
245C22×C4—122no—
246C22×C4—122no—
247C22×C4—122no—
248C22×C4C2×C4×D311yescharacteristic
249C22×C4D3×D411yescharacteristic
250C22×C4D3×D411yescharacteristic
251M16—22no—
252M16—41no—
253M16—62no—
254M16—121no—
255M16—242no—
256C2×C8—41no—
257C2×C8—41no—
258C2×C8—121no—
259C2×C8—121no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—121no—
265C4⋊C4—121no—
266C4⋊C4—121no—
267C4⋊C4—122no—
268C4⋊C4—121no—
269C4⋊C4—121no—
270C22⋊C4—121no—
271C22⋊C4—121no—
272C22⋊C4—122no—
273C22⋊C4—122no—
274C22⋊C4—122no—
275C42—61no—
276C42—61no—
277C22×C3—21no—
278C22×C3—21no—
279C22×C3C2×#9911yescharacteristic
280C22×C3#41811yescharacteristic
281C22×C3C2×#10411yescharacteristic
282C2×D3—22no—
283C2×D3—42no—
284C2×D3—41no—
285C2×D3—41no—
286C2×D3—42no—
287C2×D3—42no—
288C2×D3—41no—
289C2×D3—41no—
290C2×D3—41no—
291C2×D3—81no—
292C3×C4—21no—
293C3×C4—21no—
294C3×C4—41no—
295C3×C4—41no—
296C3×C4—81no—
297C3×C4—81no—
298Dic3—21no—
299Dic3—41no—
300Dic3—81no—
301Dic3—81no—
302Dic3#35212yes—
303C23—32no—
304C23—61no—
305C23—61no—
306C23—61no—
307C23—61no—
308C23—61no—
309C23—121no—
310C23—121no—
311C23—121no—
312C23—121no—
313C23D3×C22⋊C411yescharacteristic
314D4—122no—
315D4—122no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—31no—
321C2×C4—31no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—62no—
331C2×C4—61no—
332C2×C4—122no—
333C2×C4—122no—
334C2×C4—122no—
335C2×C4—121no—
336C2×C4—121no—
337C2×C4—121no—
338C2×C4—121no—
339C2×C4—121no—
340C2×C4—121no—
341C2×C4—121no—
342C2×C4—241no—
343C2×C4—244no—
344C2×C4#91111yescharacteristic
345C2×C4#91411yescharacteristic
346C8—42no—
347C8—81no—
348C8—122no—
349C8—241no—
350C2×C3—21no—
351C2×C3—21no—
352C2×C3#150711yescharacteristic
353C2×C3C2×#35211yescharacteristic
354C2×C3#213511yescharacteristic
355D3—42no—
356D3—81no—
357C22—21no—
358C22—21no—
359C22—62no—
360C22—122no—
361C22—122no—
362C22—122no—
363C22—121no—
364C22—121no—
365C22—121no—
366C22—121no—
367C22—121no—
368C22—241no—
369C22D3×#9911yescharacteristic
370C22#477011yescharacteristic
371C22D3×#10411yescharacteristic
372C4—21no—
373C4—21no—
374C4—32no—
375C4—41no—
376C4—41no—
377C4—61no—
378C4—81no—
379C4—81no—
380C4—121no—
381C4—241no—
382C4—241no—
383C3#1178011yescharacteristic, complemented, Sylow
384C2—21no—
385C2—21no—
386C2—122no—
387C2—241no—
388C2#6830311yescharacteristic
389C2D3×#35211yescharacteristic
390C2#6917311yescharacteristic
subgroupG/H
centerC22D3×#99
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1346C2
socleC22×C3C2×#99
prime radicalC22×D12D4
cosocleC2×C3×C4⋊C4C23
hypercenter#1346D3
hypocenterC3#11780
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1346

Complements

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

NH ≅ G/Ntype
4. C3×#1346C2semidirect
5. #67650C2semidirect
6. #67495C2semidirect
7. #67458C2semidirect
34. #1346D3semidirect
383. C3#11780semidirect

Maximal subgroups

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

subgroupindex
#777642
#689722
#681452
C3×#13462
#676502
#674952
#674582
#117803