Finite Group Databasebeta

← #10692087

Subgroups — #10692087

Order 768.

536 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 107 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#82450C211yesmaximal, characteristic
2#82448C211yesmaximal, characteristic
3C3×#1491C211yesmaximal, characteristic
4#70836C211yesmaximal, characteristic
5#69868C211yesmaximal, characteristic
6#67749C211yesmaximal, characteristic
7#67665C211yesmaximal, characteristic
8#9324—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10C2×Q8×Dic3—21no—
11#5470C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×C4×M16C2211yescharacteristic
14#5228—21no—
15#5031C412yescomplemented
16#5030—22no—
17#5020C412yescomplemented
18Dic3×C4⋊C4—21no—
19Dic3×C4⋊C4—22no—
20#4971—21no—
21C4×C4⋊Dic3—21no—
22C4×C4⋊Dic3C2211yescharacteristic
23#4932—21no—
24#4932C2211yescharacteristic
25#4926C2211yescharacteristic
26C3×Q8⋊C8—22no—
27#4555—22no—
28#4467—21no—
29#4467—21no—
30#3113—31no—
31#2276—31no—
32#1986—31no—
33#1506—31no—
34#1491D311yescharacteristic
35#1300—31no—
36#1285—31no—
37C22×C3×Q8D411yescharacteristic
38C2×C3×M16—21no—
39C3×C4×Q8—21no—
40C3×C4×Q8—21no—
41C3×C4×Q8C2×C412yes—
42C3×C4×Q8D412yes—
43C2×C3×C4⋊C4—21no—
44C2×C3×C4⋊C4D411yescharacteristic
45C2×C3×C42—21no—
46C2×C3×C42C2311yescharacteristic
47Q8×Dic3—28no—
48C2×#216—22no—
49#926—21no—
50C2×C4⋊Dic3—22no—
51C2×C4⋊Dic3—21no—
52#924—21no—
53#924—21no—
54C2×C4×Dic3—22no—
55C2×C4×Dic3—21no—
56C2×C4×Dic3—22no—
57C2×C4×Dic3D412yes—
58#870—21no—
59#869C2×C411yescharacteristic
60#869C2×C411yescharacteristic
61C4×Dic6—21no—
62C4×Dic6—21no—
63C3×C4⋊C8—21no—
64C3×C22⋊C8—21no—
65C3×#97—21no—
66C3×#97—21no—
67C3×C4×C8—21no—
68#835—22no—
69C3⋊#95—22no—
70C3⋊#95—21no—
71C3⋊#95—21no—
72C3⋊#95D411yescharacteristic
73C3⋊#95D411yescharacteristic
74#830—21no—
75#810—22no—
76#808—22no—
77C4⋊(C3⋊C8)C2×C412yes—
78#803—21no—
79#533—31no—
80C2×C4×Q8—61no—
81C2×C4×Q8C2×D311yescharacteristic
82#483—61no—
83#475—32no—
84#462—32no—
85#451—61no—
86#441—62no—
87#432—31no—
88#432C2×D311yescharacteristic
89C4×M16C2×D311yescharacteristic
90C4×C4⋊C4—31no—
91C4×C4⋊C4—61no—
92C4×C4⋊C4—62no—
93C4×C4⋊C4—61no—
94#376—31no—
95#376—61no—
96#339—62no—
97#328—61no—
98#328—61no—
99Q8⋊C8—22no—
100C2×C3×Q8—24no—
101C2×C3×Q8—21no—
102C2×C3×Q8C22⋊C412yes—
103C22×C3×C4—21no—
104C22×C3×C4—21no—
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×Dic3—21no—
109C22×Dic3—22no—
110C22×Dic3—21no—
111C2×Dic6—21no—
112C2×Dic6—21no—
113C3×M16—22no—
114C3×M16—41no—
115C2×C3×C8—41no—
116C2×C3×C8—41no—
117C3×C4⋊C4—22no—
118C3×C4⋊C4—22no—
119C3×C4⋊C4—21no—
120C3×C4⋊C4C22⋊C412yes—
121C3×C42—22no—
122C3×C42—22no—
123C3×C42C22×C411yescharacteristic
124C3×C42C2×D411yescharacteristic
125C3×C42#4111yescharacteristic
126C3×C42#4111yescharacteristic
127#217—41no—
128#216—44no—
129C4⋊Dic3—28no—
130C4⋊Dic3—24no—
131C4⋊Dic3—21no—
132C4⋊Dic3—21no—
133C4⋊Dic3—21no—
134C4⋊Dic3—21no—
135Dic3⋊C4—41no—
136Dic3⋊C4—41no—
137C4×Dic3—28no—
138C4×Dic3—24no—
139C4×Dic3—41no—
140C4×Dic3—42no—
141C4×Dic3—82no—
142C2×C3⋊C8—22no—
143C22×Q8D1211yescharacteristic
144C2×Q16—62no—
145C2×M16—21no—
146C4⋊Q8—31no—
147C4⋊Q8—31no—
148#125—61no—
149C22⋊Q8—61no—
150C22⋊Q8—61no—
151C4×Q8—21no—
152C4×Q8—21no—
153C4×Q8—68no—
154C4×Q8—61no—
155C4×Q8—61no—
156C4×Q8C4×D312yes—
157C4×Q8#6712yes—
158#117—61no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—62no—
161C2×C4⋊C4—61no—
162C2×C4⋊C4D1211yescharacteristic
163C2×C42—21no—
164C2×C42—32no—
165C2×C42—62no—
166C2×C42—61no—
167C2×C42—62no—
168C2×C42C22×D311yescharacteristic
169#106—62no—
170C4⋊C8—21no—
171C4⋊C8—32no—
172#103—62no—
173#103—62no—
174C22⋊C8—21no—
175C22⋊C8—61no—
176#97—21no—
177#97—21no—
178#97—61no—
179C4×C8—21no—
180#95—31no—
181#95—31no—
182#95—62no—
183#95—61no—
184#95—61no—
185C23×C3#12011yescharacteristic
186C3×Q8—24no—
187C3×Q8—22no—
188C3×Q8#10414yes—
189C2×C3×C4—24no—
190C2×C3×C4—22no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—22no—
194C2×C3×C4—22no—
195C2×C3×C4—41no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4C2×C22⋊C411yescharacteristic
199C2×C3×C4C4×D411yescharacteristic
200C2×C3×C4C4×D411yescharacteristic
201C2×C3×C4#12111yescharacteristic
202C2×C3×C4#12111yescharacteristic
203C2×C3×C4#12311yescharacteristic
204C2×Dic3—22no—
205C2×Dic3—24no—
206C2×Dic3—22no—
207C2×Dic3—41no—
208C2×Dic3—42no—
209C2×Dic3—41no—
210C2×Dic3—42no—
211C2×Dic3—41no—
212C2×Dic3—41no—
213C2×Dic3—41no—
214C2×Dic3—41no—
215Dic6—42no—
216Dic6—81no—
217C3×C8—42no—
218C3×C8—81no—
219C3⋊C8—42no—
220C2×Q8—24no—
221C2×Q8—21no—
222C2×Q8—61no—
223C2×Q8—61no—
224C2×Q8(C2×D3)⋊C412yes—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—61no—
228C22×C4—61no—
229C22×C4—62no—
230C22×C4C2×D1211yescharacteristic
231C22×C4D3×D411yescharacteristic
232C22×C4D3×D411yescharacteristic
233Q16—124no—
234M16—22no—
235M16—41no—
236C2×C8—41no—
237C2×C8—41no—
238C2×C8—62no—
239C4⋊C4—22no—
240C4⋊C4—21no—
241C4⋊C4—22no—
242C4⋊C4—68no—
243C4⋊C4—64no—
244C4⋊C4—61no—
245C4⋊C4—61no—
246C4⋊C4—61no—
247C4⋊C4—61no—
248C4⋊C4—121no—
249C4⋊C4—121no—
250C4⋊C4(C2×D3)⋊C412yes—
251C22⋊C4—121no—
252C42—22no—
253C42—22no—
254C42—68no—
255C42—64no—
256C42—122no—
257C42—121no—
258C42—242no—
259C42C2×C4×D311yescharacteristic
260C42#23711yescharacteristic
261C42#23711yescharacteristic
262C42C2×#6711yescharacteristic
263C22×C3—21no—
264C22×C3—21no—
265C22×C3#38611yescharacteristic
266C22×C3#45511yescharacteristic
267C22×C3#45611yescharacteristic
268C3×C4—21no—
269C3×C4—21no—
270C3×C4—24no—
271C3×C4—21no—
272C3×C4—21no—
273C3×C4—21no—
274C3×C4—42no—
275C3×C4C2×#10412yes—
276Dic3—44no—
277Dic3—42no—
278Dic3—42no—
279Dic3—81no—
280Dic3—81no—
281C23#88211yescharacteristic
282Q8—24no—
283Q8—22no—
284Q8—122no—
285Q8—241no—
286Q8#82514yes—
287C2×C4—24no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—62no—
297C2×C4—62no—
298C2×C4—64no—
299C2×C4—121no—
300C2×C4—121no—
301C2×C4—122no—
302C2×C4—122no—
303C2×C4—121no—
304C2×C4—121no—
305C2×C4—121no—
306C2×C4—121no—
307C2×C4#67⋊C411yescharacteristic
308C2×C4#67⋊C411yescharacteristic
309C2×C4#88611yescharacteristic
310C2×C4C2×(C2×D3)⋊C411yescharacteristic
311C2×C4#93911yescharacteristic
312C2×C4#93911yescharacteristic
313C8—42no—
314C8—81no—
315C8—122no—
316C2×C3—21no—
317C2×C3—21no—
318C2×C3#188211yescharacteristic
319C2×C3#189611yescharacteristic
320C2×C3#190511yescharacteristic
321C22—21no—
322C22—21no—
323C22#482511yescharacteristic
324C22#482611yescharacteristic
325C22#495811yescharacteristic
326C4—24no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—21no—
332C4—42no—
333C4—124no—
334C4—122no—
335C4—122no—
336C4—241no—
337C4—241no—
338C4C2×#82512yes—
339C3#932411yescharacteristic, complemented, Sylow
340C2—21no—
341C2—21no—
342C2#6996511yescharacteristic
343C2#7033411yescharacteristic
344C2#7036511yescharacteristic
subgroupG/H
centerC22#4958
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1491C2
socleC22×C3#386
prime radicalC23×C3#120
cosocleC2×C3×C42C23
hypercenter#1491D3
hypocenterC3#9324
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1491

Complements

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

NH ≅ G/Ntype
15. #5031C4semidirect
17. #5020C4semidirect
339. C3#9324semidirect

Maximal subgroups

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

subgroupindex
#824502
#824482
C3×#14912
#708362
#698682
#677492
#676652
#93243