Finite Group Databasebeta

← #10704828

Subgroups — #10704828

Order 768.

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

normal subgroups only

This group has 394 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#76535C211yesmaximal, characteristic
2#68127C211yesmaximal, characteristic
3#68125C211yesmaximal, characteristic
4C3×#1319C211yesmaximal, characteristic
5#67643C211yesmaximal, characteristic
6#67455C211yesmaximal, characteristic
7#67455C211yesmaximal, characteristic
8#11568—31nomaximal, Sylow
9C2×#944C411yescharacteristic
10C2×#944C411yescharacteristic
11C2×#887C2211yescharacteristic
12C2×#869C411yescharacteristic
13C3×#391C2211yescharacteristic
14#4973C2211yescharacteristic
15C4×Dic3⋊C4C411yescharacteristic
16#4719—21no—
17#4718—21no—
18#4657—21no—
19#4657—21no—
20#4647—21no—
21#4647—21no—
22C3×#324C2211yescharacteristic
23C3×#324C2211yescharacteristic
24#4528C2211yescharacteristic
25#4528C2211yescharacteristic
26#4469—21no—
27#4469—21no—
28#2546—31no—
29#1467—31no—
30#1461—31no—
31#1319—31no—
32#1319—31no—
33#1319—31no—
34#1319D311yescharacteristic
35C22×C3×Q8C2×C411yescharacteristic
36C22×Dic6C2×C411yescharacteristic
37C22×Dic6C2×C411yescharacteristic
38C2×C3×C4⋊C4C2311yescharacteristic
39C2×C3×C42C2×C411yescharacteristic
40#944—41no—
41#944—41no—
42#944—42no—
43C2×C4⋊Dic3—21no—
44C2×C4⋊Dic3—21no—
45C2×Dic3⋊C4—21no—
46C2×Dic3⋊C4—21no—
47C2×Dic3⋊C4—21no—
48C2×Dic3⋊C4—21no—
49C2×Dic3⋊C4—21no—
50C2×Dic3⋊C4D411yescharacteristic
51C2×Dic3⋊C4D411yescharacteristic
52C2×C4×Dic3C2×C411yescharacteristic
53C2×C4×Dic3C2×C411yescharacteristic
54C2×C4×Dic3D411yescharacteristic
55C2×C4×Dic3D411yescharacteristic
56#887—41no—
57#887—41no—
58#869—42no—
59C3×C22⋊C8—21no—
60C3×C22⋊C8—21no—
61C3×#95—21no—
62C3×#95—21no—
63C3⋊#95—21no—
64C3⋊#95—21no—
65C3⋊#95—21no—
66#830—21no—
67#830—21no—
68#815—41no—
69Dic3⋊C8—41no—
70Dic3⋊C8—41no—
71C8×Dic3—41no—
72C2×C4⋊Q8—31no—
73C2×C4⋊Q8—31no—
74C2×C4⋊Q8—31no—
75C2×C4×Q8—31no—
76#432—61no—
77#431—61no—
78#398—61no—
79#398—61no—
80#391—31no—
81#391C2×D311yescharacteristic
82#389—61no—
83#389—61no—
84C4×C4⋊C4—31no—
85#324—31no—
86#324—31no—
87#324—61no—
88#324—61no—
89#324C2×D311yescharacteristic
90#324C2×D311yescharacteristic
91C2×C3×Q8—21no—
92C2×C3×Q8—21no—
93C2×C3×Q8—41no—
94C2×C3×Q8—41no—
95C22×C3×C4—21no—
96C22×C3×C4—21no—
97C22×C3×C4C22×C411yescharacteristic
98C22×C3×C4C2×D411yescharacteristic
99C22×C3×C4C2×D411yescharacteristic
100C22×Dic3—21no—
101C22×Dic3—21no—
102C22×Dic3C22⋊C411yescharacteristic
103C22×Dic3C22⋊C411yescharacteristic
104C22×Dic3C22⋊C411yescharacteristic
105C22×Dic3C22⋊C411yescharacteristic
106C2×Dic6—21no—
107C2×Dic6—21no—
108C2×Dic6—22no—
109C2×Dic6—21no—
110C2×Dic6—21no—
111C2×Dic6—41no—
112C2×Dic6—41no—
113C2×Dic6—41no—
114C2×C3×C8—41no—
115C2×C3×C8—41no—
116C3×C4⋊C4—41no—
117C3×C42—41no—
118C4⋊Dic3—42no—
119C4⋊Dic3—42no—
120Dic3⋊C4—24no—
121Dic3⋊C4—41no—
122Dic3⋊C4—41no—
123Dic3⋊C4—42no—
124Dic3⋊C4—42no—
125Dic3⋊C4—41no—
126Dic3⋊C4—41no—
127Dic3⋊C4—42no—
128C4×Dic3—21no—
129C4×Dic3—21no—
130C4×Dic3—21no—
131C4×Dic3—21no—
132C4×Dic3—21no—
133C4×Dic3—21no—
134C4×Dic3—41no—
135C2×C3⋊C8—41no—
136C2×C3⋊C8—41no—
137C22×Q8—31no—
138C22×Q8—31no—
139C22×Q8C4×D311yescharacteristic
140C4⋊Q8—122no—
141C4⋊Q8—121no—
142C4⋊Q8—121no—
143C4⋊Q8—122no—
144C4×Q8—121no—
145C4×Q8—121no—
146C2×C4⋊C4—31no—
147C2×C4⋊C4—31no—
148C2×C4⋊C4—61no—
149C2×C4⋊C4—61no—
150C2×C4⋊C4—61no—
151C2×C4⋊C4—61no—
152C2×C4⋊C4—61no—
153C2×C4⋊C4—61no—
154C2×C4⋊C4—61no—
155C2×C4⋊C4C22×D311yescharacteristic
156C2×C42—31no—
157C2×C42—31no—
158C2×C42—31no—
159C2×C42—31no—
160C2×C42C4×D311yescharacteristic
161C4⋊C8—121no—
162C4⋊C8—121no—
163C22⋊C8—21no—
164C22⋊C8—21no—
165C22⋊C8—61no—
166C22⋊C8—61no—
167#97—121no—
168C4×C8—121no—
169#95—21no—
170#95—21no—
171#95—61no—
172#95—61no—
173#95—61no—
174C23×C3C2×C22⋊C411yescharacteristic
175C3×Q8—81no—
176C3×Q8—81no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—22no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—41no—
183C2×C3×C4—41no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4—41no—
189C2×C3×C4#13111yescharacteristic
190C2×C3×C4#13111yescharacteristic
191C2×Dic3—21no—
192C2×Dic3—21no—
193C2×Dic3—22no—
194C2×Dic3—22no—
195C2×Dic3—21no—
196C2×Dic3—21no—
197C2×Dic3—21no—
198C2×Dic3—21no—
199C2×Dic3—41no—
200C2×Dic3—41no—
201C2×Dic3—41no—
202C2×Dic3—41no—
203C2×Dic3—41no—
204C2×Dic3—41no—
205C2×Dic3#9911yescharacteristic
206C2×Dic3#9911yescharacteristic
207C2×Dic3#10012yes—
208Dic6—42no—
209Dic6—42no—
210Dic6—81no—
211Dic6—81no—
212C3×C8—81no—
213C3×C8—81no—
214C3⋊C8—81no—
215C3⋊C8—81no—
216C2×Q8—21no—
217C2×Q8—21no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8—62no—
221C2×Q8—61no—
222C2×Q8—61no—
223C2×Q8—61no—
224C2×Q8—61no—
225C2×Q8—121no—
226C2×Q8—121no—
227C2×Q8—121no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—31no—
231C22×C4—31no—
232C22×C4—31no—
233C22×C4—31no—
234C22×C4—61no—
235C22×C4—61no—
236C22×C4C2×C4×D311yescharacteristic
237C22×C4D3×D411yescharacteristic
238C22×C4D3×D411yescharacteristic
239C2×C8—41no—
240C2×C8—41no—
241C2×C8—121no—
242C2×C8—121no—
243C4⋊C4—41no—
244C4⋊C4—64no—
245C4⋊C4—121no—
246C4⋊C4—121no—
247C4⋊C4—121no—
248C4⋊C4—121no—
249C4⋊C4—122no—
250C4⋊C4—122no—
251C4⋊C4—122no—
252C4⋊C4—122no—
253C4⋊C4—122no—
254C42—41no—
255C42—61no—
256C42—61no—
257C42—61no—
258C42—61no—
259C42—61no—
260C42—61no—
261C42—121no—
262C22×C3—21no—
263C22×C3—21no—
264C22×C3#40811yescharacteristic
265C22×C3C2×#9911yescharacteristic
266C22×C3C2×#10011yescharacteristic
267C3×C4—21no—
268C3×C4—21no—
269C3×C4—42no—
270C3×C4—41no—
271C3×C4—41no—
272C3×C4—81no—
273Dic3—21no—
274Dic3—22no—
275Dic3—42no—
276Dic3—41no—
277Dic3—41no—
278Dic3—81no—
279Dic3#35612yes—
280C23D3×C22⋊C411yescharacteristic
281Q8—81no—
282Q8—81no—
283Q8—122no—
284Q8—122no—
285Q8—241no—
286Q8—241no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—31no—
293C2×C4—31no—
294C2×C4—32no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—62no—
303C2×C4—61no—
304C2×C4—61no—
305C2×C4—62no—
306C2×C4—61no—
307C2×C4—61no—
308C2×C4—61no—
309C2×C4—61no—
310C2×C4—121no—
311C2×C4—121no—
312C2×C4—121no—
313C2×C4—121no—
314C2×C4—121no—
315C2×C4—121no—
316C2×C4#90111yescharacteristic
317C2×C4#90711yescharacteristic
318C8—81no—
319C8—81no—
320C8—241no—
321C8—241no—
322C2×C3—21no—
323C2×C3—21no—
324C2×C3#147111yescharacteristic
325C2×C3#212911yescharacteristic
326C2×C3C2×#35611yescharacteristic
327C22—21no—
328C22—21no—
329C22#472711yescharacteristic
330C22D3×#9911yescharacteristic
331C22D3×#10011yescharacteristic
332C4—21no—
333C4—21no—
334C4—32no—
335C4—42no—
336C4—41no—
337C4—41no—
338C4—62no—
339C4—61no—
340C4—81no—
341C4—122no—
342C4—121no—
343C4—121no—
344C4—241no—
345C3#1156811yescharacteristic, complemented, Sylow
346C2—21no—
347C2—21no—
348C2#6812311yescharacteristic
349C2#6915911yescharacteristic
350C2D3×#35611yescharacteristic
subgroupG/H
centerC22D3×#99
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1319C2
socleC22×C3C2×#99
prime radicalC23×C3C2×C22⋊C4
cosocleC2×C3×C4⋊C4C23
hypercenter#1319D3
hypocenterC3#11568
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1319

Complements

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

NH ≅ G/Ntype
345. C3#11568semidirect

Maximal subgroups

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

subgroupindex
#765352
#681272
#681252
C3×#13192
#676432
#674552
#115683