Finite Group Databasebeta

← #10698972

Subgroups — #10698972

Order 768.

366 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 93 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#79068C211yesmaximal, characteristic
2C3×#1571C211yesmaximal, characteristic
3#70793C211yesmaximal, characteristic
4#70786C211yesmaximal, characteristic
5#70486C211yesmaximal, characteristic
6#70437C211yesmaximal, characteristic
7#67713C211yesmaximal, characteristic
8#10471—31nomaximal, Sylow
9#5470C2211yescharacteristic
10C2×C4×Dic6C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×#423C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14#5121—21no—
15#5120—21no—
16C2×#816—21no—
17#4936C2211yescharacteristic
18#4925C2211yescharacteristic
19#4851—21no—
20#4850—21no—
21#4829—22no—
22C3×#335—22no—
23#4551—21no—
24#4486—21no—
25#4481—21no—
26#4469—22no—
27#4467—22no—
28#3078—31no—
29#1805—31no—
30#1571D311yescharacteristic
31#1542—31no—
32#1518—31no—
33#1461—31no—
34#1286—31no—
35C22×Dic6D411yescharacteristic
36C2×C3×M16—21no—
37C22×C3×C8—21no—
38C2×C3×C42C2311yescharacteristic
39#926—21no—
40C2×C4⋊Dic3—21no—
41C2×C4⋊Dic3D411yescharacteristic
42#924—21no—
43#924—21no—
44C2×Dic3⋊C4—21no—
45C2×C4×Dic3—21no—
46#870—21no—
47#869D411yescharacteristic
48#869D411yescharacteristic
49C4×Dic6—21no—
50C4×Dic6—21no—
51C4×Dic6—21no—
52C4×Dic6—21no—
53C4×Dic6—21no—
54C4×Dic6—21no—
55C3×C4⋊C8—21no—
56C3×C4⋊C8—21no—
57C3×C4⋊C8—21no—
58C3×C4⋊C8—21no—
59C3×C4⋊C8D412yes—
60C3×C22⋊C8D412yes—
61C3×C4×C8—21no—
62C3⋊#95—21no—
63C3⋊#95—21no—
64#830—21no—
65#818—22no—
66#817—22no—
67#816—22no—
68#816—22no—
69#816—42no—
70C4⋊(C3⋊C8)—21no—
71C4×C3⋊C8—21no—
72#533—31no—
73C2×C4×Q8—31no—
74#479—62no—
75#477—61no—
76#475—61no—
77#431—31no—
78#431C2×D311yescharacteristic
79#428—61no—
80#423C2×D311yescharacteristic
81C2×C4⋊C8C2×D311yescharacteristic
82#419—61no—
83C2×#103—61no—
84#382—31no—
85#340—61no—
86#335—22no—
87#333—61no—
88#332—61no—
89#328—62no—
90#324—62no—
91C22×C3×C4C2×D411yescharacteristic
92C22×C3×C4C2×D411yescharacteristic
93C22×C3×C4#4111yescharacteristic
94C22×Dic3—21no—
95C22×Dic3—21no—
96C22×Dic3—41no—
97C2×Dic6—21no—
98C2×Dic6—21no—
99C2×Dic6—21no—
100C2×Dic6—21no—
101C2×Dic6—21no—
102C2×Dic6—21no—
103C2×Dic6—21no—
104C2×Dic6Q1612yes—
105C3×M16—41no—
106C3×M16—41no—
107C2×C3×C8—22no—
108C2×C3×C8—22no—
109C2×C3×C8—41no—
110C2×C3×C8—41no—
111C2×C3×C8—41no—
112C3×C42C2×D411yescharacteristic
113C3×C42C2×D411yescharacteristic
114C3×C42#4111yescharacteristic
115C3×C42#4111yescharacteristic
116#217—41no—
117C4⋊Dic3—21no—
118C4⋊Dic3—21no—
119C4⋊Dic3—21no—
120C4⋊Dic3—21no—
121C4⋊Dic3—21no—
122C4⋊Dic3—21no—
123C4⋊Dic3—41no—
124C4⋊Dic3—41no—
125Dic3⋊C4—41no—
126Dic3⋊C4—41no—
127Dic3⋊C4—41no—
128Dic3⋊C4—41no—
129C4×Dic3—41no—
130C4×Dic3—41no—
131C4×Dic3—41no—
132C2×C3⋊C8—41no—
133C22×Q8—31no—
134C2×M16—21no—
135C22×C8—21no—
136C4⋊Q8—31no—
137C4⋊Q8—31no—
138#125—61no—
139C22⋊Q8—61no—
140C22⋊Q8—61no—
141C4×Q8—61no—
142C4×Q8—61no—
143C4×Q8—61no—
144C4×Q8—61no—
145C4×Q8—61no—
146C4×Q8—61no—
147#117—61no—
148C2×C4⋊C4—31no—
149C2×C4⋊C4—61no—
150C2×C4⋊C4—61no—
151C2×C42—61no—
152C2×C42C22×D311yescharacteristic
153#107—62no—
154#106—62no—
155C4⋊C8—21no—
156C4⋊C8—21no—
157C4⋊C8—21no—
158C4⋊C8—21no—
159C4⋊C8—61no—
160C4⋊C8#6712yes—
161#103—62no—
162#103—62no—
163#103—122no—
164C22⋊C8—61no—
165C22⋊C8D1212yes—
166C4×C8—21no—
167C4×C8—61no—
168#95—61no—
169#95—61no—
170C23×C3#12111yescharacteristic
171C2×C3×C4—41no—
172C2×C3×C4—41no—
173C2×C3×C4#12011yescharacteristic
174C2×C3×C4#12111yescharacteristic
175C2×C3×C4#12111yescharacteristic
176C2×C3×C4#12311yescharacteristic
177C2×C3×C4#12311yescharacteristic
178C2×C3×C4#12411yescharacteristic
179C2×C3×C4C2×Q1611yescharacteristic
180C2×C3×C4#13511yescharacteristic
181C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
182C2×C3×C4#13711yescharacteristic
183C2×Dic3—22no—
184C2×Dic3—41no—
185C2×Dic3—41no—
186C2×Dic3—41no—
187C2×Dic3—41no—
188C2×Dic3—41no—
189C2×Dic3—81no—
190C2×Dic3—81no—
191C2×Dic3—81no—
192C2×Dic3—81no—
193Dic6—22no—
194Dic6—22no—
195Dic6—41no—
196Dic6—41no—
197Dic6—81no—
198Dic6—81no—
199C3×C8—42no—
200C3×C8—42no—
201C3×C8—81no—
202C3⋊C8—81no—
203C2×Q8—32no—
204C2×Q8—61no—
205C2×Q8—61no—
206C2×Q8—61no—
207C2×Q8—61no—
208C2×Q8—61no—
209C2×Q8—61no—
210C2×Q8—61no—
211C22×C4—61no—
212C22×C4—61no—
213C22×C4—121no—
214C22×C4C2×D1211yescharacteristic
215C22×C4D3×D411yescharacteristic
216C22×C4#23911yescharacteristic
217M16—41no—
218M16—41no—
219C2×C8—22no—
220C2×C8—22no—
221C2×C8—41no—
222C2×C8—41no—
223C2×C8—41no—
224C2×C8—121no—
225C4⋊C4—61no—
226C4⋊C4—61no—
227C4⋊C4—61no—
228C4⋊C4—61no—
229C4⋊C4—61no—
230C4⋊C4—61no—
231C4⋊C4—121no—
232C4⋊C4—121no—
233C4⋊C4—121no—
234C4⋊C4—121no—
235C4⋊C4—121no—
236C4⋊C4—121no—
237C22⋊C4—121no—
238C42—121no—
239C42—121no—
240C42—121no—
241C42#23511yescharacteristic
242C42D3×D411yescharacteristic
243C42#23911yescharacteristic
244C42C2×#6711yescharacteristic
245C22×C3—21no—
246C22×C3—21no—
247C22×C3#39411yescharacteristic
248C22×C3#46711yescharacteristic
249C22×C3#46811yescharacteristic
250C3×C4—21no—
251C3×C4—21no—
252C3×C4—21no—
253C3×C4—21no—
254C3×C4#44911yescharacteristic
255C3×C4#45111yescharacteristic
256C3×C4#48411yescharacteristic
257C3×C4#48511yescharacteristic
258Dic3—42no—
259Dic3—81no—
260Dic3—81no—
261Dic3—81no—
262Dic3—161no—
263C23#89511yescharacteristic
264Q8—62no—
265Q8—62no—
266Q8—121no—
267Q8—121no—
268Q8—241no—
269Q8—241no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—62no—
273C2×C4—121no—
274C2×C4—121no—
275C2×C4—121no—
276C2×C4—121no—
277C2×C4—121no—
278C2×C4—241no—
279C2×C4—241no—
280C2×C4—241no—
281C2×C4—241no—
282C2×C4#89411yescharacteristic
283C2×C4#89411yescharacteristic
284C2×C4#89511yescharacteristic
285C2×C4#91111yescharacteristic
286C2×C4#91211yescharacteristic
287C2×C4D3×Q1611yescharacteristic
288C2×C4#91811yescharacteristic
289C2×C4#93011yescharacteristic
290C2×C4#93711yescharacteristic
291C2×C4#94711yescharacteristic
292C8—42no—
293C8—42no—
294C8—81no—
295C8—241no—
296C2×C3—21no—
297C2×C3—21no—
298C2×C3#203611yescharacteristic
299C2×C3#204911yescharacteristic
300C2×C3#205311yescharacteristic
301C22—21no—
302C22—21no—
303C22#488611yescharacteristic
304C22#488711yescharacteristic
305C22#499111yescharacteristic
306C4—21no—
307C4—21no—
308C4—21no—
309C4—21no—
310C4—122no—
311C4—241no—
312C4—241no—
313C4—241no—
314C4—481no—
315C4#488411yescharacteristic
316C4#488811yescharacteristic
317C4#515811yescharacteristic
318C4#518611yescharacteristic
319C3#1047111yescharacteristic, complemented, Sylow
320C2—21no—
321C2—21no—
322C2#7054711yescharacteristic
323C2#7063511yescharacteristic
324C2#7073711yescharacteristic
subgroupG/H
centerC22#4991
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1571C2
socleC22×C3#394
prime radicalC23×C3#121
hypercenter#1571D3
hypocenterC3#10471
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1571

Complements

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

NH ≅ G/Ntype
319. C3#10471semidirect

Maximal subgroups

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

subgroupindex
#790682
C3×#15712
#707932
#707862
#704862
#704372
#677132
#104713