Finite Group Databasebeta

← #10690500

Subgroups — #10690500

Order 768.

404 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 97 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#83339C211yesmaximal, characteristic
2#79267C211yesmaximal, characteristic
3C3×#1485C211yesmaximal, characteristic
4#71825C211yesmaximal, characteristic
5#69862C211yesmaximal, characteristic
6#67739C211yesmaximal, characteristic
7#67667C211yesmaximal, characteristic
8#9068—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×C4×M16C2211yescharacteristic
13#5080C411yescharacteristic
14#5080C411yescharacteristic
15#5078C411yescharacteristic
16#5078C411yescharacteristic
17#5065—22no—
18#4998—22no—
19#4925C2211yescharacteristic
20C4×#208C2211yescharacteristic
21C2×C4×C3⋊C8C2211yescharacteristic
22#4831—21no—
23#4830—21no—
24#4720—22no—
25#4679—21no—
26C8×Dic6—21no—
27C3×Q8⋊C8—21no—
28C3×Q8⋊C8—21no—
29#4535—21no—
30#4535—21no—
31#4465—22no—
32#3160—31no—
33#2986—31no—
34#1526—31no—
35#1485—31no—
36#1485D311yescharacteristic
37#1278—31no—
38#1278—31no—
39C2×C3×M16—21no—
40C3×C4⋊Q8C2×C411yescharacteristic
41C3×C4⋊Q8C2×C411yescharacteristic
42C3×#125—21no—
43C3×C22⋊Q8—21no—
44C3×C22⋊Q8—21no—
45C3×C4×Q8—21no—
46C3×C4×Q8—21no—
47C3×#117—21no—
48C2×C3×C42C2311yescharacteristic
49#926D412yes—
50#924—21no—
51#924—21no—
52C2×#208—21no—
53C22×C3⋊C8—21no—
54#870D412yes—
55#869C2×C411yescharacteristic
56#869C2×C411yescharacteristic
57C4×Dic6—21no—
58C4×Dic6—21no—
59C3×C4⋊C8D411yescharacteristic
60C3×C4⋊C8D411yescharacteristic
61C3×C22⋊C8—21no—
62C3×#97—21no—
63C3×C4×C8—21no—
64C3×C4×C8—21no—
65#830—21no—
66#815—22no—
67Dic3⋊C8—22no—
68Dic3⋊C8—22no—
69C8×Dic3—22no—
70#808—22no—
71#808—22no—
72#807—22no—
73#807—22no—
74C4⋊(C3⋊C8)D411yescharacteristic
75C4⋊(C3⋊C8)D411yescharacteristic
76#803—21no—
77C4×C3⋊C8—21no—
78C4×C3⋊C8—21no—
79C4×C3⋊C8—21no—
80C4×C3⋊C8C2×C411yescharacteristic
81C4×C3⋊C8C2×C411yescharacteristic
82#533—31no—
83#533C2×D311yescharacteristic
84#500—31no—
85#500—31no—
86#499—62no—
87#498—31no—
88#498—31no—
89#446—61no—
90#446—61no—
91C8×Q8—61no—
92C8×Q8—61no—
93#433—62no—
94#431—31no—
95#431C2×D311yescharacteristic
96#427—62no—
97C4×M16—31no—
98C4×M16C2×D311yescharacteristic
99C2×C4×C8—31no—
100#330—62no—
101Q8⋊C8—21no—
102Q8⋊C8—21no—
103Q8⋊C8—61no—
104Q8⋊C8—61no—
105C2×C3×Q8—21no—
106C2×C3×Q8—21no—
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4#4111yescharacteristic
109C22×C3×C4#4111yescharacteristic
110C2×Dic6—21no—
111C2×Dic6—21no—
112C3×M16—22no—
113C3×M16—41no—
114C2×C3×C8—21no—
115C2×C3×C8—21no—
116C2×C3×C8—41no—
117C3×C4⋊C4—21no—
118C3×C4⋊C4—21no—
119C3×C4⋊C4—21no—
120C3×C4⋊C4—21no—
121C3×C4⋊C4—41no—
122C3×C4⋊C4—41no—
123C3×C22⋊C4—41no—
124C3×C42—41no—
125C3×C42C22×C411yescharacteristic
126C3×C42C2×D411yescharacteristic
127C3×C42C2×D411yescharacteristic
128C3×C42C2×D411yescharacteristic
129#217—22no—
130C4⋊Dic3—21no—
131C4⋊Dic3—21no—
132C4⋊Dic3C22⋊C412yes—
133C4⋊Dic3C22⋊C412yes—
134Dic3⋊C4—22no—
135Dic3⋊C4—22no—
136C4×Dic3—22no—
137#208—22no—
138#208—41no—
139C2×C3⋊C8—22no—
140C2×C3⋊C8—22no—
141C2×C3⋊C8—21no—
142C2×C3⋊C8—21no—
143C2×C3⋊C8—21no—
144C2×C3⋊C8—21no—
145C2×C3⋊C8—41no—
146C2×M16—21no—
147C2×M16—61no—
148C22×C8—61no—
149C4⋊Q8—31no—
150C4⋊Q8—31no—
151C4⋊Q8C4×D311yescharacteristic
152C4⋊Q8C4×D311yescharacteristic
153#125—21no—
154#125—32no—
155C22⋊Q8—21no—
156C22⋊Q8—21no—
157C22⋊Q8—61no—
158C22⋊Q8—61no—
159C4×Q8—21no—
160C4×Q8—21no—
161C4×Q8—61no—
162C4×Q8—61no—
163#117—21no—
164#117—32no—
165C2×C42C22×D311yescharacteristic
166#107—62no—
167#107—62no—
168#106—62no—
169#106—62no—
170C4⋊C8—31no—
171C4⋊C8—31no—
172C4⋊C8—62no—
173C4⋊C8—62no—
174C4⋊C8#6711yescharacteristic
175C4⋊C8#6711yescharacteristic
176C22⋊C8—21no—
177C22⋊C8—61no—
178#97—21no—
179#97—62no—
180#97—61no—
181C4×C8—21no—
182C4×C8—21no—
183C4×C8—31no—
184C4×C8—31no—
185C4×C8—62no—
186C4×C8—61no—
187C4×C8—61no—
188C4×C8—61no—
189C23×C3#12311yescharacteristic
190C3×Q8—41no—
191C3×Q8—41no—
192C3×Q8—41no—
193C3×Q8—41no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4C2×C22⋊C411yescharacteristic
203C2×C3×C4C4×D411yescharacteristic
204C2×C3×C4C4×D411yescharacteristic
205C2×C3×C4#12011yescharacteristic
206C2×C3×C4#12111yescharacteristic
207C2×C3×C4#12111yescharacteristic
208C2×Dic3—22no—
209C2×Dic3—22no—
210Dic6—41no—
211Dic6—41no—
212Dic6—41no—
213Dic6—41no—
214C3×C8—42no—
215C3×C8—41no—
216C3×C8—41no—
217C3⋊C8—22no—
218C3⋊C8—22no—
219C3⋊C8—42no—
220C3⋊C8—41no—
221C3⋊C8—41no—
222C2×Q8—21no—
223C2×Q8—21no—
224C2×Q8—61no—
225C2×Q8—61no—
226C22×C4#23511yescharacteristic
227C22×C4D3×D411yescharacteristic
228C22×C4#23711yescharacteristic
229M16—22no—
230M16—41no—
231M16—62no—
232M16—121no—
233C2×C8—21no—
234C2×C8—21no—
235C2×C8—41no—
236C2×C8—62no—
237C2×C8—62no—
238C2×C8—61no—
239C2×C8—61no—
240C2×C8—61no—
241C2×C8—61no—
242C2×C8—121no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—32no—
248C4⋊C4—32no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—61no—
252C4⋊C4—62no—
253C4⋊C4—61no—
254C4⋊C4—62no—
255C22⋊C4—41no—
256C22⋊C4—62no—
257C42—41no—
258C42—62no—
259C42C2×C4×D311yescharacteristic
260C42D3×D411yescharacteristic
261C42D3×D411yescharacteristic
262C42C2×#6711yescharacteristic
263C22×C3—21no—
264C22×C3—21no—
265C22×C3—21no—
266C22×C3#38611yescharacteristic
267C3×C4—21no—
268C3×C4—21no—
269C3×C4—21no—
270C3×C4—81no—
271C3×C4—81no—
272C3×C4#44311yescharacteristic
273C3×C4#44311yescharacteristic
274C3×C4#44411yescharacteristic
275C3×C4#44411yescharacteristic
276C3×C4#45611yescharacteristic
277C3×C4#45611yescharacteristic
278Dic3—42no—
279Dic3—42no—
280C23#88311yescharacteristic
281Q8—41no—
282Q8—41no—
283Q8—41no—
284Q8—41no—
285Q8—121no—
286Q8—121no—
287Q8—121no—
288Q8—121no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—62no—
298C2×C4—62no—
299C2×C4D3×C22⋊C411yescharacteristic
300C2×C4#67⋊C411yescharacteristic
301C2×C4#88411yescharacteristic
302C2×C4C4×#6711yescharacteristic
303C2×C4#93711yescharacteristic
304C2×C4#93811yescharacteristic
305C8—42no—
306C8—41no—
307C8—41no—
308C8—62no—
309C8—62no—
310C8—122no—
311C8—121no—
312C8—121no—
313C2×C3—41no—
314C2×C3#187611yescharacteristic
315C2×C3#187611yescharacteristic
316C2×C3#188611yescharacteristic
317C22—21no—
318C22—21no—
319C22—21no—
320C22#495611yescharacteristic
321C4—21no—
322C4—21no—
323C4—21no—
324C4—81no—
325C4—81no—
326C4—122no—
327C4—122no—
328C4#482311yescharacteristic
329C4#482411yescharacteristic
330C4#514411yescharacteristic
331C4#514511yescharacteristic
332C4#520511yescharacteristic
333C4#520511yescharacteristic
334C3#906811yescharacteristic, complemented, Sylow
335C2—41no—
336C2#7001111yescharacteristic
337C2#7034511yescharacteristic
338C2#7039011yescharacteristic
subgroupG/H
centerC2×C4#937
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1485C2
socleC22×C3#386
prime radicalC23×C3#123
cosocleC2×C3×C42C23
hypercenter#1485D3
hypocenterC3#9068
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1485

Complements

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

NH ≅ G/Ntype
334. C3#9068semidirect

Maximal subgroups

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

subgroupindex
#833392
#792672
C3×#14852
#718252
#698622
#677392
#676672
#90683