Finite Group Databasebeta

← #10691896

Subgroups — #10691896

Order 768.

470 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 109 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#83323C211yesmaximal, characteristic
2#73388C211yesmaximal, characteristic
3C3×#1995C211yesmaximal, characteristic
4#71838C211yesmaximal, characteristic
5#70506C211yesmaximal, characteristic
6#67713C211yesmaximal, characteristic
7#67614C211yesmaximal, characteristic
8#9296—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×#869C2211yescharacteristic
11C2×C4×Dic6—21no—
12C3×#423C2211yescharacteristic
13C3×#376C2211yescharacteristic
14#5080C412yescomplemented
15#5079C412yescomplemented
16#5043—22no—
17#5020—22no—
18#4936C2211yescharacteristic
19#4929—21no—
20#4925C2211yescharacteristic
21#4925C2211yescharacteristic
22#4653—22no—
23#4653—22no—
24C3×#339—21no—
25C3×#339—21no—
26#4538—22no—
27#4482—22no—
28#3229—31no—
29#2327—31no—
30#1995D311yescharacteristic
31#1546—31no—
32#1518—31no—
33#1286—31no—
34#1274—31no—
35C22×Dic6—21no—
36C2×C3×M16—21no—
37C3×C4⋊Q8C2×C411yescharacteristic
38C3×C4⋊Q8C2×C411yescharacteristic
39C3×#125—21no—
40C3×C22⋊Q8—21no—
41C3×C22⋊Q8—21no—
42C3×C4×Q8—21no—
43C3×C4×Q8—21no—
44C3×#117—21no—
45C2×C3×C42D411yescharacteristic
46C2×C3×C42D411yescharacteristic
47C2×C3×C42C2311yescharacteristic
48C2×C4⋊Dic3—21no—
49C2×C4⋊Dic3—21no—
50C2×C4⋊Dic3D412yes—
51C2×Dic3⋊C4—22no—
52C2×C4×Dic3—22no—
53#869—21no—
54#869—21no—
55#869C2×C412yes—
56#869D412yes—
57C4×Dic6—42no—
58C4×Dic6—42no—
59C3×C4⋊C8—21no—
60C3×C4⋊C8—21no—
61C3×#95—21no—
62C3×#95—21no—
63#835—24no—
64C3⋊#95—24no—
65C3⋊#95—21no—
66C3⋊#95D412yes—
67#830—21no—
68#830—21no—
69#808—24no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)—21no—
72C4×C3⋊C8—21no—
73C4×C3⋊C8C2×C412yes—
74#533C2×D311yescharacteristic
75C2×C4⋊Q8—31no—
76C2×C4×Q8—61no—
77#498—32no—
78#487—32no—
79#483—62no—
80#475—62no—
81#431—31no—
82#431—31no—
83#423C2×D311yescharacteristic
84#391—61no—
85#387—62no—
86#387—62no—
87#382—31no—
88#376C2×D311yescharacteristic
89#339—21no—
90#339—21no—
91#332—62no—
92#328—62no—
93C2×C3×Q8—21no—
94C2×C3×Q8—21no—
95C22×C3×C4—21no—
96C22×C3×C4—21no—
97C22×C3×C4C2×D411yescharacteristic
98C22×C3×C4C2×D411yescharacteristic
99C22×C3×C4C2×D411yescharacteristic
100C22×Dic3—22no—
101C22×Dic3—22no—
102C2×Dic6—22no—
103C2×Dic6—22no—
104C2×Dic6—41no—
105C2×Dic6—41no—
106C2×Dic6—41no—
107C2×Dic6—41no—
108C3×M16—41no—
109C3×M16—41no—
110C2×C3×C8—41no—
111C3×C4⋊C4—21no—
112C3×C4⋊C4—21no—
113C3×C4⋊C4—21no—
114C3×C4⋊C4—21no—
115C3×C4⋊C4—41no—
116C3×C4⋊C4—41no—
117C3×C22⋊C4—41no—
118C3×C42—21no—
119C3×C42—21no—
120C3×C42—21no—
121C3×C42—41no—
122C3×C42SD1612yes—
123C3×C42C22×C411yescharacteristic
124C3×C42C2×D411yescharacteristic
125C3×C42#4111yescharacteristic
126C3×C42#4111yescharacteristic
127C4⋊Dic3—22no—
128C4⋊Dic3—22no—
129C4⋊Dic3—22no—
130C4⋊Dic3—41no—
131C4⋊Dic3—41no—
132C4⋊Dic3C22⋊C414yes—
133Dic3⋊C4—44no—
134C4×Dic3—44no—
135C2×C3⋊C8—22no—
136C2×C3⋊C8—41no—
137C22×Q8—61no—
138C2×M16—21no—
139C4⋊Q8—32no—
140C4⋊Q8—32no—
141C4⋊Q8—61no—
142C4⋊Q8—61no—
143C4⋊Q8C4×D311yescharacteristic
144C4⋊Q8C4×D311yescharacteristic
145#125—21no—
146C22⋊Q8—21no—
147C22⋊Q8—21no—
148C4×Q8—21no—
149C4×Q8—21no—
150C4×Q8—122no—
151C4×Q8—122no—
152#117—21no—
153C2×C4⋊C4—32no—
154C2×C4⋊C4—61no—
155C2×C4⋊C4—62no—
156C2×C4⋊C4—61no—
157C2×C42—62no—
158C2×C42#6711yescharacteristic
159C2×C42#6711yescharacteristic
160C2×C42C22×D311yescharacteristic
161#106—64no—
162C4⋊C8—21no—
163C4⋊C8—21no—
164C4⋊C8—61no—
165C4⋊C8—61no—
166#103—64no—
167C22⋊C8—61no—
168C22⋊C8—61no—
169C4×C8—32no—
170C4×C8—61no—
171#95—21no—
172#95—21no—
173#95—32no—
174#95—64no—
175#95—61no—
176C23×C3#12011yescharacteristic
177C3×Q8—42no—
178C3×Q8—81no—
179C2×C3×C4—22no—
180C2×C3×C4—41no—
181C2×C3×C4—41no—
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×C4C2×C22⋊C411yescharacteristic
190C2×C3×C4C4×D411yescharacteristic
191C2×C3×C4C4×D411yescharacteristic
192C2×C3×C4#12111yescharacteristic
193C2×C3×C4#12111yescharacteristic
194C2×C3×C4#12311yescharacteristic
195C2×C3×C4C2×SD1611yescharacteristic
196C2×C3×C4#13511yescharacteristic
197C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
198C2×C3×C4#13711yescharacteristic
199C2×Dic3—24no—
200C2×Dic3—42no—
201C2×Dic3—42no—
202C2×Dic3—42no—
203C2×Dic3—42no—
204C2×Dic3—42no—
205Dic6—42no—
206Dic6—42no—
207Dic6—42no—
208Dic6—42no—
209C3×C8—81no—
210C3⋊C8—42no—
211C3⋊C8—81no—
212C2×Q8—21no—
213C2×Q8—21no—
214C2×Q8—62no—
215C2×Q8—62no—
216C2×Q8—121no—
217C2×Q8—121no—
218C2×Q8—121no—
219C2×Q8—121no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—62no—
223C22×C4—62no—
224C22×C4D3×D411yescharacteristic
225C22×C4D3×D411yescharacteristic
226C22×C4C2×#6711yescharacteristic
227M16—41no—
228M16—41no—
229C2×C8—41no—
230C2×C8—62no—
231C2×C8—121no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—34no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—62no—
240C4⋊C4—62no—
241C4⋊C4—62no—
242C4⋊C4—124no—
243C4⋊C4—121no—
244C4⋊C4—121no—
245C22⋊C4—41no—
246C42—21no—
247C42—21no—
248C42—21no—
249C42—41no—
250C42—124no—
251C42#21512yes—
252C42C2×C4×D311yescharacteristic
253C42#23511yescharacteristic
254C42D3×D411yescharacteristic
255C42#23711yescharacteristic
256C22×C3—21no—
257C22×C3—21no—
258C22×C3#38611yescharacteristic
259C22×C3#45011yescharacteristic
260C22×C3#45211yescharacteristic
261C3×C4—21no—
262C3×C4—21no—
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—42no—
266C3×C4—81no—
267C3×C4—81no—
268C3×C4—81no—
269C3×C4C4×SD1611yescharacteristic
270C3×C4#44011yescharacteristic
271C3×C4#46111yescharacteristic
272C3×C4#46311yescharacteristic
273Dic3—44no—
274Dic3—82no—
275C23#93711yescharacteristic
276Q8—42no—
277Q8—81no—
278Q8—122no—
279Q8—122no—
280Q8—122no—
281Q8—122no—
282C2×C4—22no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—64no—
293C2×C4—122no—
294C2×C4—122no—
295C2×C4—122no—
296C2×C4—122no—
297C2×C4—122no—
298C2×C4D3×C22⋊C411yescharacteristic
299C2×C4#67⋊C411yescharacteristic
300C2×C4#88311yescharacteristic
301C2×C4#88411yescharacteristic
302C2×C4C4×#6711yescharacteristic
303C2×C4#93811yescharacteristic
304C2×C4C2×#21511yescharacteristic
305C2×C4#94211yescharacteristic
306C2×C4#94911yescharacteristic
307C2×C4#95011yescharacteristic
308C8—81no—
309C8—122no—
310C8—241no—
311C2×C3—21no—
312C2×C3—21no—
313C2×C3#187711yescharacteristic
314C2×C3#190511yescharacteristic
315C2×C3#191211yescharacteristic
316C22—21no—
317C22—21no—
318C22#495611yescharacteristic
319C22#504611yescharacteristic
320C22#504711yescharacteristic
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—42no—
326C4—81no—
327C4—81no—
328C4—81no—
329C4—124no—
330C4—242no—
331C4C4×#21511yescharacteristic
332C4#502611yescharacteristic
333C4#505711yescharacteristic
334C4#508311yescharacteristic
335C3#929611yescharacteristic, complemented, Sylow
336C2—21no—
337C2—21no—
338C2#6968711yescharacteristic
339C2#7041011yescharacteristic
340C2#7041111yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1995C2
socleC22×C3#386
prime radicalC23×C3#120
cosocleC2×C3×C42C23
hypercenter#1995D3
hypocenterC3#9296
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1995

Complements

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

NH ≅ G/Ntype
14. #5080C4semidirect
15. #5079C4semidirect
335. C3#9296semidirect

Maximal subgroups

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

subgroupindex
#833232
#733882
C3×#19952
#718382
#705062
#677132
#676142
#92963