Finite Group Databasebeta

← #10698909

Subgroups — #10698909

Order 768.

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

normal subgroups only

This group has 374 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#79068C211yesmaximal, characteristic
2C3×#1566C211yesmaximal, characteristic
3#70528C211yesmaximal, characteristic
4#70509C211yesmaximal, characteristic
5#69738C211yesmaximal, characteristic
6C2×#4456C211yesmaximal, characteristic
7#67711C211yesmaximal, characteristic
8#10462—31nomaximal, Sylow
9#5470C2211yescharacteristic
10C2×C4×Dic6C2211yescharacteristic
11C3×#423C2211yescharacteristic
12C3×#423C2211yescharacteristic
13C2×C3×C4×C8C2211yescharacteristic
14C2×#824—21no—
15#5121—21no—
16#5120—21no—
17#4924C2211yescharacteristic
18C2×C4⋊(C3⋊C8)C2211yescharacteristic
19#4851—21no—
20#4850—21no—
21#4829—22no—
22C3×#334—22no—
23#4554—21no—
24#4486—21no—
25#4481—21no—
26#4456—21no—
27#4456—21no—
28#4456—21no—
29#4456—21no—
30#3078—31no—
31#1801—31no—
32#1566D311yescharacteristic
33#1550—31no—
34#1493—31no—
35C2×Q8⋊C8—31no—
36#1292—31no—
37C22×Dic6D411yescharacteristic
38C2×C3×M16—21no—
39C2×C3×M16—21no—
40C22×C3×C8—21no—
41C2×C3×C42C2311yescharacteristic
42#926—21no—
43C2×C4⋊Dic3D411yescharacteristic
44#924—21no—
45#924—21no—
46C2×Dic3⋊C4—21no—
47C2×C4×Dic3—21no—
48C2×#208—21no—
49C22×C3⋊C8—21no—
50#870—21no—
51#869D411yescharacteristic
52#869D411yescharacteristic
53C4×Dic6—21no—
54C4×Dic6—21no—
55C4×Dic6—21no—
56C4×Dic6—21no—
57C4×Dic6—21no—
58C4×Dic6—21no—
59C3×C4⋊C8—21no—
60C3×C4⋊C8—21no—
61C3×C4⋊C8—21no—
62C3×C4⋊C8D412yes—
63C3×C4×C8—21no—
64C3×C4×C8D411yescharacteristic
65C3×C4×C8D411yescharacteristic
66#824—42no—
67#818—22no—
68#817—22no—
69#816—22no—
70#816—22no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)—21no—
75#533—31no—
76C2×C4×Q8—31no—
77#479—62no—
78#477—61no—
79#475—61no—
80#428—61no—
81#423—31no—
82#423C2×D311yescharacteristic
83#423C2×D311yescharacteristic
84C2×C4⋊C8—31no—
85#419—61no—
86C2×#101—61no—
87C2×C4×C8C2×D311yescharacteristic
88#343—61no—
89#334—22no—
90#333—61no—
91#332—61no—
92Q8⋊C8—61no—
93Q8⋊C8—61no—
94Q8⋊C8—61no—
95Q8⋊C8—61no—
96C22×C3×C4C2×D411yescharacteristic
97C22×C3×C4#4111yescharacteristic
98C22×C3×C4#4111yescharacteristic
99C22×Dic3—21no—
100C22×Dic3—21no—
101C2×Dic6—21no—
102C2×Dic6—21no—
103C2×Dic6—21no—
104C2×Dic6—21no—
105C2×Dic6—21no—
106C2×Dic6—21no—
107C2×Dic6—21no—
108C2×Dic6SD1612yes—
109C3×M16—41no—
110C3×M16—41no—
111C3×M16—41no—
112C3×M16—41no—
113C2×C3×C8—21no—
114C2×C3×C8—22no—
115C2×C3×C8—21no—
116C2×C3×C8—42no—
117C2×C3×C8—41no—
118C3×C42C2×D411yescharacteristic
119C3×C42C2×D411yescharacteristic
120C3×C42C2×D411yescharacteristic
121C3×C42#4111yescharacteristic
122#217—41no—
123C4⋊Dic3—21no—
124C4⋊Dic3—21no—
125C4⋊Dic3—21no—
126C4⋊Dic3—21no—
127C4⋊Dic3—21no—
128C4⋊Dic3—21no—
129Dic3⋊C4—41no—
130Dic3⋊C4—41no—
131Dic3⋊C4—41no—
132Dic3⋊C4—41no—
133C4×Dic3—41no—
134C4×Dic3—41no—
135C4×Dic3—41no—
136#208—41no—
137#208—41no—
138C2×C3⋊C8—41no—
139C2×C3⋊C8—41no—
140C2×C3⋊C8—81no—
141C22×Q8—31no—
142C2×M16—21no—
143C2×M16—21no—
144C2×M16—61no—
145C22×C8—21no—
146C22×C8—61no—
147C4⋊Q8—31no—
148C4⋊Q8—31no—
149#125—61no—
150C22⋊Q8—61no—
151C22⋊Q8—61no—
152C4×Q8—61no—
153C4×Q8—61no—
154C4×Q8—61no—
155C4×Q8—61no—
156C4×Q8—61no—
157C4×Q8—61no—
158#117—61no—
159C2×C4⋊C4—31no—
160C2×C4⋊C4—61no—
161C2×C42—61no—
162C2×C42C22×D311yescharacteristic
163#107—62no—
164#106—62no—
165C4⋊C8—21no—
166C4⋊C8—21no—
167C4⋊C8—21no—
168C4⋊C8—61no—
169C4⋊C8—61no—
170C4⋊C8—61no—
171C4⋊C8—61no—
172C4⋊C8#6712yes—
173#103—62no—
174#103—62no—
175#101—122no—
176C4×C8—21no—
177C4×C8D1211yescharacteristic
178C4×C8D1211yescharacteristic
179C23×C3#12311yescharacteristic
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—41no—
183C2×C3×C4#12011yescharacteristic
184C2×C3×C4#12111yescharacteristic
185C2×C3×C4#12111yescharacteristic
186C2×C3×C4#12111yescharacteristic
187C2×C3×C4#12311yescharacteristic
188C2×C3×C4#12411yescharacteristic
189C2×C3×C4C2×SD1611yescharacteristic
190C2×C3×C4#13511yescharacteristic
191C2×C3×C4#13711yescharacteristic
192C2×C3×C4#13711yescharacteristic
193C2×Dic3—22no—
194C2×Dic3—41no—
195C2×Dic3—41no—
196C2×Dic3—41no—
197C2×Dic3—41no—
198C2×Dic3—41no—
199C2×Dic3—81no—
200Dic6—22no—
201Dic6—22no—
202Dic6—41no—
203Dic6—41no—
204Dic6—41no—
205Dic6—41no—
206Dic6—81no—
207C3×C8—22no—
208C3×C8—41no—
209C3×C8—42no—
210C3×C8—81no—
211C3⋊C8—81no—
212C3⋊C8—81no—
213C2×Q8—32no—
214C2×Q8—61no—
215C2×Q8—61no—
216C2×Q8—61no—
217C2×Q8—61no—
218C2×Q8—61no—
219C2×Q8—61no—
220C2×Q8—61no—
221C22×C4—61no—
222C22×C4—61no—
223C22×C4#23511yescharacteristic
224C22×C4D3×D411yescharacteristic
225C22×C4#23911yescharacteristic
226M16—41no—
227M16—41no—
228M16—41no—
229M16—41no—
230M16—121no—
231M16—121no—
232C2×C8—21no—
233C2×C8—21no—
234C2×C8—22no—
235C2×C8—42no—
236C2×C8—41no—
237C2×C8—121no—
238C2×C8—121no—
239C2×C8—241no—
240C4⋊C4—61no—
241C4⋊C4—61no—
242C4⋊C4—61no—
243C4⋊C4—61no—
244C4⋊C4—61no—
245C4⋊C4—61no—
246C4⋊C4—121no—
247C4⋊C4—121no—
248C4⋊C4—121no—
249C4⋊C4—121no—
250C22⋊C4—121no—
251C42—121no—
252C42—121no—
253C42—121no—
254C42C2×D1211yescharacteristic
255C42D3×D411yescharacteristic
256C42#23911yescharacteristic
257C42C2×#6711yescharacteristic
258C22×C3—21no—
259C22×C3—21no—
260C22×C3#39411yescharacteristic
261C22×C3#48311yescharacteristic
262C22×C3#48511yescharacteristic
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—21no—
266C3×C4#44811yescharacteristic
267C3×C4#45211yescharacteristic
268C3×C4#46511yescharacteristic
269C3×C4#47011yescharacteristic
270C3×C4#47111yescharacteristic
271C3×C4#47311yescharacteristic
272Dic3—42no—
273Dic3—81no—
274Dic3—81no—
275Dic3—81no—
276C23#89411yescharacteristic
277Q8—62no—
278Q8—62no—
279Q8—121no—
280Q8—121no—
281Q8—121no—
282Q8—121no—
283Q8—241no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—41no—
287C2×C4—62no—
288C2×C4—121no—
289C2×C4—121no—
290C2×C4—121no—
291C2×C4—121no—
292C2×C4—121no—
293C2×C4—241no—
294C2×C4#89411yescharacteristic
295C2×C4#89511yescharacteristic
296C2×C4#89511yescharacteristic
297C2×C4D3×SD1611yescharacteristic
298C2×C4#91511yescharacteristic
299C2×C4#91511yescharacteristic
300C2×C4#91611yescharacteristic
301C2×C4#93011yescharacteristic
302C2×C4#93711yescharacteristic
303C2×C4#94711yescharacteristic
304C8—22no—
305C8—41no—
306C8—42no—
307C8—81no—
308C8—241no—
309C8—241no—
310C2×C3—21no—
311C2×C3—21no—
312C2×C3#203511yescharacteristic
313C2×C3#205011yescharacteristic
314C2×C3#205211yescharacteristic
315C22—21no—
316C22—21no—
317C22#486211yescharacteristic
318C22#486811yescharacteristic
319C22#499111yescharacteristic
320C4—21no—
321C4—21no—
322C4—21no—
323C4—122no—
324C4—241no—
325C4—241no—
326C4—241no—
327C4#486411yescharacteristic
328C4#486711yescharacteristic
329C4#489611yescharacteristic
330C4#489811yescharacteristic
331C4#517011yescharacteristic
332C4#517311yescharacteristic
333C3#1046211yescharacteristic, complemented, Sylow
334C2—21no—
335C2—21no—
336C2#7053111yescharacteristic
337C2#7058211yescharacteristic
338C2#7071611yescharacteristic
subgroupG/H
centerC22#4991
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1566C2
socleC22×C3#394
prime radicalC23×C3#123
hypercenter#1566D3
hypocenterC3#10462
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1566

Complements

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

NH ≅ G/Ntype
333. C3#10462semidirect

Maximal subgroups

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

subgroupindex
#790682
C3×#15662
#705282
#705092
#697382
C2×#44562
#677112
#104623