Finite Group Databasebeta

← #10691182

Subgroups — #10691182

Order 768.

436 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 91 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#83042C211yesmaximal, characteristic
2#79314C211yesmaximal, characteristic
3C3×#1528C211yesmaximal, characteristic
4#71735C211yesmaximal, characteristic
5C4×#824C211yesmaximal, characteristic
6#67752C211yesmaximal, characteristic
7#67666C211yesmaximal, characteristic
8#9190—31nomaximal, Sylow
9C2×C3×#125C2211yescharacteristic
10C2×C4×Dic6C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×C4×M16C2211yescharacteristic
13C2×#824—21no—
14#4986—21no—
15#4985C411yescharacteristic
16#4985C411yescharacteristic
17#4983C411yescharacteristic
18#4983C411yescharacteristic
19#4979—21no—
20#4978—21no—
21#4978—21no—
22#4937C2211yescharacteristic
23#4926C2211yescharacteristic
24C4×#208C2211yescharacteristic
25#4828—22no—
26#4719—21no—
27#4718—21no—
28#4701—22no—
29C3×#330—22no—
30#4554—21no—
31#4543—22no—
32#2983—31no—
33#2634—31no—
34C4×#101—31no—
35#1528—31no—
36#1528D311yescharacteristic
37#1288—31no—
38#1288—31no—
39C22×Dic6D411yescharacteristic
40C2×C3×M16—21no—
41C3×#125—21no—
42C3×#125—21no—
43C3×#125—21no—
44C3×#125—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4C2×C411yescharacteristic
48C2×C3×C4⋊C4C2×C411yescharacteristic
49C2×C3×C42C2311yescharacteristic
50C2×C4⋊Dic3D411yescharacteristic
51C2×Dic3⋊C4—21no—
52C2×Dic3⋊C4—21no—
53C2×Dic3⋊C4C2×C411yescharacteristic
54C2×Dic3⋊C4C2×C411yescharacteristic
55C2×C4×Dic3D411yescharacteristic
56C2×C4×Dic3D411yescharacteristic
57C2×#208—21no—
58C4×Dic6—21no—
59C4×Dic6—21no—
60C4×Dic6—21no—
61C4×Dic6—21no—
62C3×C4⋊C8—21no—
63C3×C22⋊C8—21no—
64C3×#97—21no—
65C3×#97D412yes—
66C3×C4×C8—21no—
67C3⋊#95—21no—
68C3⋊#95—21no—
69C3⋊#95—21no—
70C3⋊#95—21no—
71C3⋊#95—21no—
72C3⋊#95—21no—
73C3⋊#95—21no—
74C3⋊#95—21no—
75C3⋊#95C2×C411yescharacteristic
76C3⋊#95C2×C411yescharacteristic
77#830—21no—
78#824—24no—
79#815—22no—
80Dic3⋊C8—22no—
81Dic3⋊C8—22no—
82C8×Dic3—22no—
83C4⋊(C3⋊C8)—21no—
84#803—21no—
85#803D412yes—
86C4×C3⋊C8—21no—
87C2×#125C2×D311yescharacteristic
88C2×C4×Q8—31no—
89#446—62no—
90#446—62no—
91#432—31no—
92#432—61no—
93#432C2×D311yescharacteristic
94#431—61no—
95C2×#101—61no—
96C4×M16—31no—
97C4×M16C2×D311yescharacteristic
98#400—31no—
99#400—31no—
100#400—61no—
101#398—31no—
102#398—31no—
103#395—61no—
104#387—61no—
105#387—61no—
106#383—31no—
107#343—61no—
108#330—22no—
109#330—62no—
110C22×C3×C4—21no—
111C22×C3×C4—21no—
112C22×C3×C4C22×C411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×Dic3—21no—
116C22×Dic3—21no—
117C22×Dic3C22⋊C411yescharacteristic
118C22×Dic3C22⋊C411yescharacteristic
119C22×Dic3C22⋊C411yescharacteristic
120C22×Dic3C22⋊C411yescharacteristic
121C2×Dic6—21no—
122C2×Dic6—21no—
123C2×Dic6—22no—
124C2×Dic6—41no—
125C3×M16—22no—
126C3×M16—41no—
127C2×C3×C8—22no—
128C2×C3×C8—41no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—41no—
134C3×C4⋊C4—41no—
135C3×C4⋊C4—41no—
136C3×C4⋊C4—41no—
137C3×C42C2×D411yescharacteristic
138C3×C42C2×D411yescharacteristic
139C3×C42#4111yescharacteristic
140C3×C42#4111yescharacteristic
141C4⋊Dic3—21no—
142C4⋊Dic3—21no—
143Dic3⋊C4—21no—
144Dic3⋊C4—21no—
145Dic3⋊C4—21no—
146Dic3⋊C4—21no—
147Dic3⋊C4—42no—
148Dic3⋊C4—42no—
149C4×Dic3—21no—
150C4×Dic3—21no—
151C4×Dic3—21no—
152C4×Dic3—21no—
153#208—22no—
154#208—41no—
155C2×C3⋊C8—22no—
156C2×C3⋊C8—41no—
157C22×Q8—31no—
158C2×M16—21no—
159C2×M16—61no—
160#125—21no—
161#125—21no—
162#125—21no—
163#125—21no—
164C4×Q8—61no—
165C4×Q8—61no—
166C4×Q8—61no—
167C4×Q8—61no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—31no—
171C2×C4⋊C4—31no—
172C2×C4⋊C4—31no—
173C2×C4⋊C4—61no—
174C2×C4⋊C4—61no—
175C2×C4⋊C4C4×D311yescharacteristic
176C2×C4⋊C4C4×D311yescharacteristic
177C2×C42—31no—
178C2×C42—31no—
179C2×C42C22×D311yescharacteristic
180C4⋊C8—21no—
181C4⋊C8—62no—
182C4⋊C8—62no—
183C4⋊C8—61no—
184#101—64no—
185C22⋊C8—21no—
186C22⋊C8—61no—
187#97—21no—
188#97—32no—
189#97—62no—
190#97—61no—
191#97#6712yes—
192C4×C8—21no—
193C4×C8—62no—
194C4×C8—61no—
195#95—31no—
196#95—31no—
197#95—61no—
198#95—61no—
199#95—61no—
200#95—61no—
201#95—61no—
202#95—61no—
203#95—61no—
204#95—61no—
205C23×C3C2×C22⋊C411yescharacteristic
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—42no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4C4×D411yescharacteristic
216C2×C3×C4C4×D411yescharacteristic
217C2×C3×C4#12011yescharacteristic
218C2×C3×C4#12111yescharacteristic
219C2×C3×C4#12111yescharacteristic
220C2×C3×C4#12311yescharacteristic
221C2×Dic3—21no—
222C2×Dic3—21no—
223C2×Dic3—21no—
224C2×Dic3—21no—
225C2×Dic3—21no—
226C2×Dic3—21no—
227C2×Dic3—21no—
228C2×Dic3—41no—
229C2×Dic3—41no—
230C2×Dic3—41no—
231C2×Dic3—41no—
232C2×Dic3—41no—
233C2×Dic3—41no—
234C2×Dic3—41no—
235C2×Dic3—41no—
236C2×Dic3#10412yes—
237Dic6—42no—
238Dic6—42no—
239C3×C8—42no—
240C3×C8—42no—
241C3⋊C8—42no—
242C3⋊C8—42no—
243C2×Q8—62no—
244C2×Q8—61no—
245C2×Q8—61no—
246C2×Q8—121no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—31no—
250C22×C4—31no—
251C22×C4—31no—
252C22×C4—31no—
253C22×C4—61no—
254C22×C4—61no—
255C22×C4C2×C4×D311yescharacteristic
256C22×C4D3×D411yescharacteristic
257C22×C4D3×D411yescharacteristic
258M16—22no—
259M16—41no—
260M16—62no—
261M16—121no—
262C2×C8—22no—
263C2×C8—41no—
264C2×C8—62no—
265C2×C8—121no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—61no—
275C4⋊C4—61no—
276C4⋊C4—61no—
277C4⋊C4—61no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C4⋊C4—122no—
281C4⋊C4—122no—
282C42—61no—
283C42—61no—
284C42—61no—
285C42—61no—
286C42#23511yescharacteristic
287C42D3×D411yescharacteristic
288C42#23711yescharacteristic
289C42C2×#6711yescharacteristic
290C22×C3—21no—
291C22×C3—21no—
292C22×C3#38611yescharacteristic
293C22×C3C2×#10411yescharacteristic
294C22×C3#42111yescharacteristic
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—81no—
302C3×C4—81no—
303Dic3—22no—
304Dic3—41no—
305Dic3—41no—
306Dic3—41no—
307Dic3—81no—
308Dic3—81no—
309C23D3×C22⋊C411yescharacteristic
310Q8—122no—
311Q8—122no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—32no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—61no—
323C2×C4—61no—
324C2×C4—61no—
325C2×C4—61no—
326C2×C4—61no—
327C2×C4—61no—
328C2×C4—61no—
329C2×C4—121no—
330C2×C4—121no—
331C2×C4—121no—
332C2×C4—121no—
333C2×C4—121no—
334C2×C4—121no—
335C2×C4—121no—
336C2×C4—121no—
337C2×C4#67⋊C411yescharacteristic
338C2×C4#88311yescharacteristic
339C2×C4#88411yescharacteristic
340C2×C4C4×#6711yescharacteristic
341C2×C4#93711yescharacteristic
342C2×C4#93811yescharacteristic
343C8—42no—
344C8—42no—
345C8—122no—
346C8—122no—
347C2×C3—21no—
348C2×C3—21no—
349C2×C3#187111yescharacteristic
350C2×C3#191211yescharacteristic
351C2×C3#191511yescharacteristic
352C22—21no—
353C22—21no—
354C22D3×#10411yescharacteristic
355C22#482111yescharacteristic
356C22#495611yescharacteristic
357C4—21no—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—62no—
364C4—81no—
365C4—81no—
366C4—121no—
367C4—121no—
368C4—121no—
369C4—241no—
370C4—241no—
371C3#919011yescharacteristic, complemented, Sylow
372C2—21no—
373C2—21no—
374C2#7012411yescharacteristic
375C2#7038111yescharacteristic
376C2#7041111yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1528C2
socleC22×C3#386
prime radicalC23×C3C2×C22⋊C4
cosocleC2×C3×C42C23
hypercenter#1528D3
hypocenterC3#9190
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1528

Complements

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

NH ≅ G/Ntype
371. C3#9190semidirect

Maximal subgroups

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

subgroupindex
#830422
#793142
C3×#15282
#717352
C4×#8242
#677522
#676662
#91903