Finite Group Databasebeta

← #10692536

Subgroups — #10692536

Order 768.

446 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 89 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#78518C211yesmaximal, characteristic
2#73427C211yesmaximal, characteristic
3#70826C211yesmaximal, characteristic
4#70812C211yesmaximal, characteristic, complemented
5#69863C211yesmaximal, characteristic, complemented
6C3×#1300C211yesmaximal, characteristic, complemented
7#67615C211yesmaximal, characteristic, complemented
8#9390—31nomaximal, Sylow
9C2×#872C2211yescharacteristic
10C2×#870C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×C4×M16C2211yescharacteristic
13C3×#376C2211yescharacteristic
14#5126—21no—
15#5118—21no—
16#4940—21no—
17#4928—21no—
18#4926C2211yescharacteristic
19C4×#208C2211yescharacteristic
20#4709—22no—
21#4690—22no—
22#4677C411yescharacteristic
23#4674C411yescharacteristic
24#4668—21no—
25#4661C411yescharacteristic
26#4659C411yescharacteristic
27#4653—21no—
28#4469—21no—
29#4468—21no—
30#4465—22no—
31#4465—22no—
32#2921—31no—
33#2611—31no—
34#1528—31no—
35#1528—31no—
36#1464—31no—
37#1300—31no—
38#1300D311yescharacteristic, complemented
39C22×C4×D3D411yescharacteristic
40C2×C3×M16—21no—
41C2×C3×C42—21no—
42C2×C3×C42C2311yescharacteristic
43C2×(C2×D3)⋊C4—21no—
44C2×(C2×D3)⋊C4—21no—
45C2×(C2×D3)⋊C4C2×C411yescharacteristic
46C2×(C2×D3)⋊C4D411yescharacteristic
47C2×C4⋊Dic3—21no—
48C2×C4⋊Dic3—21no—
49C2×C4⋊Dic3C2×C411yescharacteristic
50C2×C4⋊Dic3C2×C411yescharacteristic
51C2×Dic3⋊C4—21no—
52C2×Dic3⋊C4—21no—
53C2×Dic3⋊C4C2×C411yescharacteristic
54C2×Dic3⋊C4D411yescharacteristic
55C2×C4×Dic3D411yescharacteristic
56C2×#208—21no—
57#872—21no—
58#872—21no—
59#872—21no—
60#872—21no—
61#870—21no—
62#870—21no—
63#870—21no—
64#870—21no—
65C3×C4⋊C8—21no—
66C3×C22⋊C8—21no—
67C3×#97—21no—
68C3×#97D412yes—
69C3×C4×C8—21no—
70C3×#95—21no—
71C3×#95C2×C411yescharacteristic
72C3×#95C2×C411yescharacteristic
73C3⋊#95—21no—
74C3⋊#95—21no—
75C3⋊#95—21no—
76C3⋊#95—21no—
77#830—21no—
78(C2×D3)⋊C8—24no—
79Dic3⋊C8—24no—
80C4⋊(C3⋊C8)—21no—
81#803—21no—
82#803D412yes—
83C4×C3⋊C8—21no—
84C2×#125—31no—
85C2×#117—31no—
86#433—62no—
87#433—62no—
88#432—31no—
89#432C2×D311yescharacteristic
90#423—61no—
91#407—61no—
92C4×M16—31no—
93C4×M16C2×D311yescharacteristic
94#400—31no—
95#399—31no—
96#398—31no—
97#397—31no—
98#389—61no—
99#388—61no—
100#387—61no—
101#385—61no—
102#376C2×D311yescharacteristic
103#330—62no—
104#330—62no—
105#324—61no—
106#323—61no—
107C23×D3C22⋊C411yescharacteristic
108C22×C3×C4—21no—
109C22×C3×C4—21no—
110C22×C3×C4C22×C411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×Dic3—21no—
114C22×Dic3—21no—
115C22×Dic3C22⋊C411yescharacteristic
116C22×Dic3C22⋊C411yescharacteristic
117C22×Dic3C22⋊C411yescharacteristic
118C2×C4×D3—21no—
119C2×C4×D3—21no—
120C2×C4×D3—41no—
121C2×C4×D3—41no—
122C3×M16—41no—
123C3×M16—41no—
124C2×C3×C8—22no—
125C2×C3×C8—41no—
126C3×C42—42no—
127C3×C42C2×D411yescharacteristic
128C3×C42C2×D411yescharacteristic
129C3×C42#4111yescharacteristic
130C3×C42#4111yescharacteristic
131(C2×D3)⋊C4—21no—
132(C2×D3)⋊C4—21no—
133(C2×D3)⋊C4—21no—
134(C2×D3)⋊C4—21no—
135(C2×D3)⋊C4—42no—
136(C2×D3)⋊C4—42no—
137C4⋊Dic3—21no—
138C4⋊Dic3—21no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141C4⋊Dic3—42no—
142C4⋊Dic3—42no—
143Dic3⋊C4—21no—
144Dic3⋊C4—21no—
145Dic3⋊C4—21no—
146Dic3⋊C4—21no—
147Dic3⋊C4—41no—
148Dic3⋊C4—41no—
149Dic3⋊C4—41no—
150Dic3⋊C4—41no—
151C4×Dic3—21no—
152C4×Dic3—21no—
153#208—41no—
154#208—41no—
155C2×C3⋊C8—22no—
156C2×C3⋊C8—41no—
157C23×C4—31no—
158C2×M16—21no—
159C2×M16—61no—
160#125—61no—
161#125—61no—
162#125—61no—
163#125—61no—
164#117—61no—
165#117—61no—
166#117—61no—
167#117—61no—
168C2×C4⋊C4—31no—
169C2×C4⋊C4—31no—
170C2×C4⋊C4—31no—
171C2×C4⋊C4—31no—
172C2×C4⋊C4—61no—
173C2×C4⋊C4—61no—
174C2×C4⋊C4—61no—
175C2×C4⋊C4—61no—
176C2×C22⋊C4—31no—
177C2×C22⋊C4—31no—
178C2×C22⋊C4—61no—
179C2×C22⋊C4—61no—
180C2×C42—21no—
181C2×C42—31no—
182C2×C42C22×D311yescharacteristic
183C4⋊C8—21no—
184C4⋊C8—64no—
185C4⋊C8—61no—
186C22⋊C8—21no—
187C22⋊C8—64no—
188C22⋊C8—61no—
189#97—21no—
190#97—32no—
191#97—61no—
192#97D1212yes—
193C4×C8—21no—
194C4×C8—61no—
195#95—21no—
196#95—61no—
197#95—61no—
198#95—61no—
199#95—61no—
200#95C4×D311yescharacteristic
201#95C4×D311yescharacteristic
202C23×C3C2×C22⋊C411yescharacteristic
203C22×D3—21no—
204C22×D3—21no—
205C22×D3—21no—
206C22×D3—41no—
207C22×D3—41no—
208C2×C3×C4—21no—
209C2×C3×C4—21no—
210C2×C3×C4—22no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4C4×D411yescharacteristic
219C2×C3×C4C4×D411yescharacteristic
220C2×C3×C4#12011yescharacteristic
221C2×C3×C4#12111yescharacteristic
222C2×C3×C4#12111yescharacteristic
223C2×C3×C4#12311yescharacteristic
224C2×Dic3—21no—
225C2×Dic3—21no—
226C2×Dic3—21no—
227C2×Dic3—21no—
228C2×Dic3—21no—
229C2×Dic3—41no—
230C2×Dic3—41no—
231C2×Dic3—41no—
232C2×Dic3—41no—
233C2×Dic3—41no—
234C2×Dic3—41no—
235C2×Dic3—41no—
236C2×Dic3—41no—
237C4×D3—42no—
238C4×D3—42no—
239C3×C8—42no—
240C3×C8—81no—
241C3⋊C8—42no—
242C3⋊C8—81no—
243C24—31no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—31no—
247C22×C4—31no—
248C22×C4—31no—
249C22×C4—61no—
250C22×C4—61no—
251C22×C4—61no—
252C22×C4—61no—
253C22×C4—121no—
254C22×C4—121no—
255C22×C4C2×C4×D311yescharacteristic
256C22×C4D3×D411yescharacteristic
257C22×C4D3×D411yescharacteristic
258M16—41no—
259M16—41no—
260M16—121no—
261M16—121no—
262C2×C8—22no—
263C2×C8—41no—
264C2×C8—62no—
265C2×C8—121no—
266C4⋊C4—61no—
267C4⋊C4—61no—
268C4⋊C4—61no—
269C4⋊C4—61no—
270C4⋊C4—61no—
271C4⋊C4—61no—
272C4⋊C4—61no—
273C4⋊C4—61no—
274C4⋊C4—122no—
275C4⋊C4—122no—
276C4⋊C4—121no—
277C4⋊C4—121no—
278C4⋊C4—121no—
279C4⋊C4—121no—
280C22⋊C4—61no—
281C22⋊C4—61no—
282C22⋊C4—61no—
283C22⋊C4—61no—
284C22⋊C4—122no—
285C22⋊C4—122no—
286C42—42no—
287C42—61no—
288C42—61no—
289C42C2×D1211yescharacteristic
290C42#23511yescharacteristic
291C42D3×D411yescharacteristic
292C42#23911yescharacteristic
293C22×C3—21no—
294C22×C3—21no—
295C22×C3#38611yescharacteristic
296C22×C3#42111yescharacteristic
297C22×C3#42111yescharacteristic
298C2×D3—42no—
299C2×D3—41no—
300C2×D3—41no—
301C2×D3—41no—
302C2×D3—81no—
303C3×C4—21no—
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—21no—
308C3×C4—21no—
309C3×C4—42no—
310C3×C4—81no—
311Dic3—41no—
312Dic3—41no—
313Dic3—41no—
314Dic3—81no—
315Dic3—81no—
316C23—61no—
317C23—61no—
318C23—61no—
319C23—121no—
320C23—121no—
321C23D3×C22⋊C411yescharacteristic
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—22no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—61no—
333C2×C4—61no—
334C2×C4—61no—
335C2×C4—61no—
336C2×C4—61no—
337C2×C4—122no—
338C2×C4—122no—
339C2×C4—121no—
340C2×C4—121no—
341C2×C4—121no—
342C2×C4—121no—
343C2×C4—121no—
344C2×C4—121no—
345C2×C4—121no—
346C2×C4—121no—
347C2×C4C4×D1211yescharacteristic
348C2×C4#88211yescharacteristic
349C2×C4#88411yescharacteristic
350C2×C4#89311yescharacteristic
351C2×C4#89411yescharacteristic
352C2×C4#89511yescharacteristic
353C8—42no—
354C8—81no—
355C8—122no—
356C8—241no—
357C2×C3—21no—
358C2×C3—21no—
359C2×C3#188411yescharacteristic
360C2×C3#189711yescharacteristic
361C2×C3#191511yescharacteristic
362D3—81no—
363C22—21no—
364C22—21no—
365C22—122no—
366C22—121no—
367C22—121no—
368C22—121no—
369C22—241no—
370C22#466511yescharacteristic
371C22#482111yescharacteristic
372C22#482111yescharacteristic
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—42no—
380C4—81no—
381C4—121no—
382C4—121no—
383C4—121no—
384C4—241no—
385C4—241no—
386C3#939011yescharacteristic, complemented, Sylow
387C2—21no—
388C2—21no—
389C2—241no—
390C2#6875711yescharacteristic
391C2#6877711yescharacteristic
392C2#6904111yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1300C2
socleC22×C3#386
prime radicalC23×D3C22⋊C4
cosocleC2×C3×C42C23
hypercenter#1300D3
hypocenterC3#9390
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1300

Complements

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

NH ≅ G/Ntype
4. #70812C2semidirect
5. #69863C2semidirect
6. C3×#1300C2semidirect
7. #67615C2semidirect
38. #1300D3semidirect
386. C3#9390semidirect

Maximal subgroups

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

subgroupindex
#785182
#734272
#708262
#708122
#698632
C3×#13002
#676152
#93903