Finite Group Databasebeta

← #10698913

Subgroups — #10698913

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#83367C211yesmaximal, characteristic
2C3×#1493C211yesmaximal, characteristic
3#71830C211yesmaximal, characteristic
4#70528C211yesmaximal, characteristic
5C2×#4456C211yesmaximal, characteristic
6#69731C211yesmaximal, characteristic
7#67711C211yesmaximal, characteristic
8#10463—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C4×Dic6C2211yescharacteristic
11C3×#423C2211yescharacteristic
12C2×C3×C4×C8C2211yescharacteristic
13C2×#824—21no—
14#5094—21no—
15#5091—21no—
16#5063—22no—
17#5004—21no—
18#4995—21no—
19#4924C2211yescharacteristic
20#4924C2211yescharacteristic
21C2×C4⋊(C3⋊C8)C2211yescharacteristic
22C3×Q8⋊C8—21no—
23C3×Q8⋊C8—21no—
24#4554—21no—
25#4547—21no—
26#4546—21no—
27#4458—22no—
28#4456—21no—
29#4456—21no—
30#3078—31no—
31#1801—31no—
32#1567—31no—
33#1550—31no—
34#1493D311yescharacteristic
35C2×Q8⋊C8—31no—
36#1292—31no—
37C22×Dic6D411yescharacteristic
38C2×C3×M16—21no—
39C22×C3×C8—21no—
40C3×C4⋊Q8D411yescharacteristic
41C3×C4⋊Q8D411yescharacteristic
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
49C2×C4⋊Dic3D411yescharacteristic
50C2×Dic3⋊C4—21no—
51C2×C4×Dic3—21no—
52C2×#208—21no—
53C2×#208—21no—
54C22×C3⋊C8—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×C8D411yescharacteristic
63C3×C4×C8D411yescharacteristic
64#824—42no—
65#810—22no—
66#810—22no—
67#808—22no—
68#807—22no—
69C4⋊(C3⋊C8)—21no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)D412yes—
75#533C2×D311yescharacteristic
76C2×C4×Q8—31no—
77#479—62no—
78#477—61no—
79#475—61no—
80#428—61no—
81#423—31no—
82#423—31no—
83#423C2×D311yescharacteristic
84C2×C4⋊C8—31no—
85#419—61no—
86C2×#101—61no—
87C2×C4×C8C2×D311yescharacteristic
88#343—61no—
89#335—62no—
90#333—61no—
91#332—61no—
92Q8⋊C8—21no—
93Q8⋊C8—21no—
94Q8⋊C8—61no—
95Q8⋊C8—61no—
96C2×C3×Q8—21no—
97C2×C3×Q8—21no—
98C22×C3×C4C2×D411yescharacteristic
99C22×C3×C4#4111yescharacteristic
100C22×C3×C4#4111yescharacteristic
101C22×Dic3—21no—
102C22×Dic3—21no—
103C2×Dic6—21no—
104C2×Dic6—21no—
105C2×Dic6—21no—
106C2×Dic6—21no—
107C2×Dic6—21no—
108C2×Dic6Q1612yes—
109C3×M16—41no—
110C3×M16—41no—
111C2×C3×C8—21no—
112C2×C3×C8—21no—
113C2×C3×C8—42no—
114C2×C3×C8—41no—
115C3×C4⋊C4—21no—
116C3×C4⋊C4—21no—
117C3×C4⋊C4—21no—
118C3×C4⋊C4—21no—
119C3×C4⋊C4—41no—
120C3×C4⋊C4—41no—
121C3×C22⋊C4—41no—
122C3×C42—41no—
123C3×C42C2×D411yescharacteristic
124C3×C42C2×D411yescharacteristic
125C3×C42C2×D411yescharacteristic
126C3×C42#4111yescharacteristic
127C4⋊Dic3—21no—
128C4⋊Dic3—21no—
129Dic3⋊C4—41no—
130Dic3⋊C4—41no—
131C4×Dic3—41no—
132C4×Dic3—41no—
133#208—41no—
134#208—41no—
135#208—41no—
136#208—41no—
137C2×C3⋊C8—22no—
138C2×C3⋊C8—41no—
139C2×C3⋊C8—41no—
140C2×C3⋊C8—81no—
141C22×Q8—31no—
142C2×M16—21no—
143C2×M16—61no—
144C2×M16—61no—
145C22×C8—21no—
146C22×C8—61no—
147C4⋊Q8D1211yescharacteristic
148C4⋊Q8D1211yescharacteristic
149#125—21no—
150C22⋊Q8—21no—
151C22⋊Q8—21no—
152C4×Q8—21no—
153C4×Q8—21no—
154C4×Q8—61no—
155C4×Q8—61no—
156C4×Q8—61no—
157C4×Q8—61no—
158#117—21no—
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—32no—
168C4⋊C8—61no—
169C4⋊C8—61no—
170C4⋊C8—61no—
171C4⋊C8—61no—
172C4⋊C8—61no—
173#103—62no—
174#103—62no—
175#101—122no—
176C4×C8—21no—
177C4×C8#6711yescharacteristic
178C4×C8#6711yescharacteristic
179C23×C3#12311yescharacteristic
180C3×Q8—41no—
181C3×Q8—41no—
182C3×Q8—81no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—41no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4#12011yescharacteristic
189C2×C3×C4#12111yescharacteristic
190C2×C3×C4#12111yescharacteristic
191C2×C3×C4#12111yescharacteristic
192C2×C3×C4#12311yescharacteristic
193C2×C3×C4#12411yescharacteristic
194C2×C3×C4C2×Q1611yescharacteristic
195C2×C3×C4#13511yescharacteristic
196C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
197C2×C3×C4#13711yescharacteristic
198C2×Dic3—22no—
199C2×Dic3—41no—
200C2×Dic3—41no—
201C2×Dic3—41no—
202C2×Dic3—81no—
203Dic6—22no—
204Dic6—22no—
205Dic6—41no—
206Dic6—41no—
207C3×C8—22no—
208C3×C8—41no—
209C3×C8—81no—
210C3⋊C8—42no—
211C3⋊C8—81no—
212C3⋊C8—81no—
213C2×Q8—21no—
214C2×Q8—21no—
215C2×Q8—32no—
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#23711yescharacteristic
226M16—41no—
227M16—41no—
228M16—121no—
229M16—121no—
230M16—121no—
231M16—121no—
232C2×C8—21no—
233C2×C8—21no—
234C2×C8—42no—
235C2×C8—41no—
236C2×C8—62no—
237C2×C8—121no—
238C2×C8—121no—
239C2×C8—241no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—61no—
247C4⋊C4—61no—
248C4⋊C4—121no—
249C4⋊C4—121no—
250C22⋊C4—41no—
251C42—41no—
252C42—121no—
253C42—121no—
254C42C2×D1211yescharacteristic
255C42D3×D411yescharacteristic
256C42#23711yescharacteristic
257C42C2×#6711yescharacteristic
258C22×C3—21no—
259C22×C3—21no—
260C22×C3#39411yescharacteristic
261C22×C3#48411yescharacteristic
262C22×C3#48511yescharacteristic
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—21no—
266C3×C4—81no—
267C3×C4—81no—
268C3×C4#44911yescharacteristic
269C3×C4#45111yescharacteristic
270C3×C4#46711yescharacteristic
271C3×C4#46811yescharacteristic
272C3×C4#47111yescharacteristic
273C3×C4#47311yescharacteristic
274Dic3—42no—
275Dic3—81no—
276C23#88311yescharacteristic
277Q8—41no—
278Q8—41no—
279Q8—62no—
280Q8—62no—
281Q8—81no—
282Q8—121no—
283Q8—121no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—62no—
290C2×C4—121no—
291C2×C4—121no—
292C2×C4—121no—
293C2×C4—241no—
294C2×C4#88211yescharacteristic
295C2×C4#88511yescharacteristic
296C2×C4#88611yescharacteristic
297C2×C4#91411yescharacteristic
298C2×C4#91511yescharacteristic
299C2×C4D3×Q1611yescharacteristic
300C2×C4#91911yescharacteristic
301C2×C4#93011yescharacteristic
302C2×C4#93811yescharacteristic
303C2×C4#93911yescharacteristic
304C8—22no—
305C8—41no—
306C8—81no—
307C8—122no—
308C8—241no—
309C8—241no—
310C2×C3—21no—
311C2×C3—21no—
312C2×C3#203611yescharacteristic
313C2×C3#205011yescharacteristic
314C2×C3#205211yescharacteristic
315C22—21no—
316C22—21no—
317C22#481111yescharacteristic
318C22#481411yescharacteristic
319C22#496111yescharacteristic
320C4—21no—
321C4—21no—
322C4—21no—
323C4—81no—
324C4—81no—
325C4—122no—
326C4—241no—
327C4#480911yescharacteristic
328C4#481011yescharacteristic
329C4#514111yescharacteristic
330C4#514311yescharacteristic
331C4#517411yescharacteristic
332C4#518811yescharacteristic
333C3#1046311yescharacteristic, complemented, Sylow
334C2—21no—
335C2—21no—
336C2#7000411yescharacteristic
337C2#7024411yescharacteristic
338C2#7037911yescharacteristic
subgroupG/H
centerC22#4961
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1493C2
socleC22×C3#394
prime radicalC23×C3#123
hypercenter#1493D3
hypocenterC3#10463
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1493

Complements

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

NH ≅ G/Ntype
333. C3#10463semidirect

Maximal subgroups

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

subgroupindex
#833672
C3×#14932
#718302
#705282
C2×#44562
#697312
#677112
#104633