Finite Group Databasebeta

← #10698329

Subgroups — #10698329

Order 768.

364 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 89 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#73671C211yesmaximal, characteristic
2#70792C211yesmaximal, characteristic
3#70515C211yesmaximal, characteristic
4#69471C211yesmaximal, characteristic
5C3×#1303C211yesmaximal, characteristic
6#67737C211yesmaximal, characteristic
7#67715C211yesmaximal, characteristic
8#10373—31nomaximal, Sylow
9#5470C2211yescharacteristic
10C3×#431C2211yescharacteristic
11C3×#423C2211yescharacteristic
12C3×C4×C4⋊C4C2211yescharacteristic
13#4986—21no—
14#4982—21no—
15C4×C4⋊Dic3—21no—
16C4×C4⋊Dic3C2211yescharacteristic
17#4932C2211yescharacteristic
18C4×Dic3⋊C4—21no—
19#4925C2211yescharacteristic
20#4661—21no—
21#4660—21no—
22#4555—21no—
23#4555—21no—
24#4467—21no—
25#4467—21no—
26#4452—21no—
27#4452—21no—
28#2402—31no—
29#1995—31no—
30#1503—31no—
31#1503—31no—
32#1303—31no—
33#1303D311yescharacteristic
34#1274—31no—
35C2×C3×M16—21no—
36C2×C3×C4⋊C4—21no—
37C2×C3×C42—21no—
38C2×C3×C42C2311yescharacteristic
39#926—21no—
40C2×C4⋊Dic3—21no—
41C2×C4⋊Dic3—21no—
42C2×C4⋊Dic3—21no—
43#924—21no—
44#924—21no—
45C2×Dic3⋊C4—21no—
46C2×Dic3⋊C4—21no—
47C2×C4×Dic3—21no—
48C2×C4×Dic3D411yescharacteristic
49C2×C4×Dic3D411yescharacteristic
50#870—21no—
51#869Q811yescharacteristic
52#869Q811yescharacteristic
53C4×Dic6—21no—
54C4×Dic6—21no—
55C3×C4⋊C8—21no—
56C3×C4⋊C8—21no—
57C3×C4⋊C8—21no—
58C3×C22⋊C8—21no—
59C3×C4×C8—21no—
60C3×#95D411yescharacteristic
61C3×#95D411yescharacteristic
62C3⋊#95—21no—
63C3⋊#95—21no—
64C3⋊#95—21no—
65C3⋊#95—21no—
66C3⋊#95—21no—
67C3⋊#95—21no—
68C3⋊#95D411yescharacteristic
69C3⋊#95D411yescharacteristic
70#830—21no—
71C4⋊(C3⋊C8)—21no—
72C4×C3⋊C8—21no—
73#533—31no—
74#431—31no—
75#431C2×D311yescharacteristic
76#423C2×D311yescharacteristic
77#400—61no—
78#400—61no—
79#398—61no—
80#398—61no—
81C4×C4⋊C4—31no—
82C4×C4⋊C4—61no—
83C4×C4⋊C4—61no—
84C4×C4⋊C4C2×D311yescharacteristic
85#376—31no—
86#339—61no—
87#339—61no—
88#328—61no—
89#328—61no—
90#328—61no—
91#328—61no—
92C22×C3×C4—21no—
93C22×C3×C4—21no—
94C22×C3×C4C2×D411yescharacteristic
95C22×C3×C4C2×D411yescharacteristic
96C22×C3×C4C2×D411yescharacteristic
97C22×Dic3—21no—
98C22×Dic3—21no—
99C22×Dic3—21no—
100C22×Dic3—21no—
101C2×Dic6—21no—
102C2×Dic6—21no—
103C3×M16—41no—
104C3×M16—41no—
105C2×C3×C8—41no—
106C2×C3×C8—41no—
107C3×C4⋊C4—42no—
108C3×C42—41no—
109C3×C42—41no—
110C3×C42C2×Q811yescharacteristic
111C3×C42#4111yescharacteristic
112C3×C42#4111yescharacteristic
113C3×C42#4111yescharacteristic
114#217—41no—
115C4⋊Dic3—21no—
116C4⋊Dic3—21no—
117C4⋊Dic3—21no—
118C4⋊Dic3—21no—
119C4⋊Dic3—42no—
120C4⋊Dic3—42no—
121C4⋊Dic3—42no—
122Dic3⋊C4—42no—
123Dic3⋊C4—42no—
124Dic3⋊C4—41no—
125Dic3⋊C4—41no—
126C4×Dic3—21no—
127C4×Dic3—21no—
128C4×Dic3—21no—
129C4×Dic3—21no—
130C4×Dic3—41no—
131C4×Dic3—41no—
132C4×Dic3—41no—
133C2×C3⋊C8—41no—
134C2×M16—21no—
135C4⋊Q8—31no—
136C4⋊Q8—31no—
137#125—61no—
138C22⋊Q8—61no—
139C22⋊Q8—61no—
140C4×Q8—61no—
141C4×Q8—61no—
142#117—61no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—61no—
145C2×C4⋊C4—61no—
146C2×C4⋊C4—61no—
147C2×C4⋊C4—61no—
148C2×C4⋊C4—61no—
149C2×C42—21no—
150C2×C42—31no—
151C2×C42—31no—
152C2×C42—61no—
153C2×C42C22×D311yescharacteristic
154C4⋊C8—21no—
155C4⋊C8—21no—
156C4⋊C8—21no—
157C4⋊C8—61no—
158C22⋊C8—21no—
159C22⋊C8—61no—
160C4×C8—21no—
161C4×C8—61no—
162#95—31no—
163#95—31no—
164#95—61no—
165#95—61no—
166#95—61no—
167#95—61no—
168#95—61no—
169#95—61no—
170#95D1211yescharacteristic
171#95D1211yescharacteristic
172C23×C3#12011yescharacteristic
173C2×C3×C4—22no—
174C2×C3×C4—41no—
175C2×C3×C4—41no—
176C2×C3×C4—41no—
177C2×C3×C4—41no—
178C2×C3×C4—41no—
179C2×C3×C4—41no—
180C2×C3×C4—41no—
181C2×C3×C4C22⋊Q811yescharacteristic
182C2×C3×C4C22⋊Q811yescharacteristic
183C2×C3×C4C22⋊Q811yescharacteristic
184C2×C3×C4#12411yescharacteristic
185C2×C3×C4#12411yescharacteristic
186C2×C3×C4#12411yescharacteristic
187C2×C3×C4#13511yescharacteristic
188C2×C3×C4#13511yescharacteristic
189C2×C3×C4#13711yescharacteristic
190C2×C3×C4#13711yescharacteristic
191C2×Dic3—22no—
192C2×Dic3—41no—
193C2×Dic3—41no—
194C2×Dic3—41no—
195C2×Dic3—41no—
196C2×Dic3—41no—
197C2×Dic3—41no—
198C2×Dic3—41no—
199C2×Dic3—41no—
200C2×Dic3—41no—
201C2×Dic3—41no—
202C2×Dic3—41no—
203C2×Dic3—41no—
204C2×Dic3—41no—
205Dic6—81no—
206Dic6—81no—
207C3×C8—81no—
208C3×C8—81no—
209C3⋊C8—81no—
210C2×Q8—61no—
211C2×Q8—61no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—61no—
215C22×C4—61no—
216C22×C4—61no—
217C22×C4—61no—
218C22×C4C2×D1211yescharacteristic
219C22×C4D3×D411yescharacteristic
220C22×C4D3×D411yescharacteristic
221M16—41no—
222M16—41no—
223C2×C8—41no—
224C2×C8—41no—
225C2×C8—121no—
226C4⋊C4—42no—
227C4⋊C4—61no—
228C4⋊C4—61no—
229C4⋊C4—61no—
230C4⋊C4—61no—
231C4⋊C4—122no—
232C4⋊C4—122no—
233C4⋊C4—122no—
234C4⋊C4—122no—
235C4⋊C4—122no—
236C4⋊C4—121no—
237C4⋊C4—121no—
238C22⋊C4—121no—
239C42—41no—
240C42—41no—
241C42—61no—
242C42—61no—
243C42—61no—
244C42—61no—
245C42—121no—
246C42—121no—
247C42—121no—
248C42#23511yescharacteristic
249C42#23511yescharacteristic
250C42#23711yescharacteristic
251C42D3×Q811yescharacteristic
252C22×C3—21no—
253C22×C3—21no—
254C22×C3#39311yescharacteristic
255C22×C3#45211yescharacteristic
256C22×C3#45211yescharacteristic
257C3×C4—21no—
258C3×C4—21no—
259C3×C4—21no—
260C3×C4—21no—
261C3×C4—42no—
262C3×C4—81no—
263C3×C4#47911yescharacteristic
264C3×C4#47911yescharacteristic
265C3×C4#48811yescharacteristic
266C3×C4#49111yescharacteristic
267Dic3—42no—
268Dic3—81no—
269Dic3—81no—
270Dic3—81no—
271Dic3—81no—
272Dic3—81no—
273C23#88211yescharacteristic
274Q8—241no—
275Q8—241no—
276C2×C4—22no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—62no—
285C2×C4—121no—
286C2×C4—121no—
287C2×C4—121no—
288C2×C4—121no—
289C2×C4—121no—
290C2×C4—121no—
291C2×C4—121no—
292C2×C4—121no—
293C2×C4—121no—
294C2×C4—121no—
295C2×C4—121no—
296C2×C4—121no—
297C2×C4—121no—
298C2×C4#87511yescharacteristic
299C2×C4#88511yescharacteristic
300C2×C4#88511yescharacteristic
301C2×C4#89611yescharacteristic
302C2×C4#89611yescharacteristic
303C2×C4#89711yescharacteristic
304C2×C4#91211yescharacteristic
305C2×C4#91511yescharacteristic
306C2×C4#91611yescharacteristic
307C2×C4#91811yescharacteristic
308C8—81no—
309C8—81no—
310C8—241no—
311C2×C3—21no—
312C2×C3—21no—
313C2×C3#202611yescharacteristic
314C2×C3#203111yescharacteristic
315C2×C3#203111yescharacteristic
316C22—21no—
317C22—21no—
318C22#467311yescharacteristic
319C22#478311yescharacteristic
320C22#480811yescharacteristic
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—42no—
326C4—81no—
327C4—122no—
328C4—241no—
329C4—241no—
330C4—241no—
331C4—241no—
332C4—241no—
333C4#476811yescharacteristic
334C4#479711yescharacteristic
335C4#485711yescharacteristic
336C4#487711yescharacteristic
337C3#1037311yescharacteristic, complemented, Sylow
338C2—21no—
339C2—21no—
340C2#6877911yescharacteristic
341C2#6891111yescharacteristic
342C2#6904211yescharacteristic
subgroupG/H
centerC2×C4#885
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1303C2
socleC22×C3#393
prime radicalC23×C3#120
hypercenter#1303D3
hypocenterC3#10373
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1303

Complements

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

NH ≅ G/Ntype
337. C3#10373semidirect

Maximal subgroups

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

subgroupindex
#736712
#707922
#705152
#694712
C3×#13032
#677372
#677152
#103733