Finite Group Databasebeta

← #10698911

Subgroups — #10698911

Order 768.

374 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C3×#3078C211yesmaximal, characteristic
2#71830C211yesmaximal, characteristic
3#71826C211yesmaximal, characteristic
4#71637C211yesmaximal, characteristic
5C2×#4535C211yesmaximal, characteristic
6#70491C211yesmaximal, characteristic
7#67711C211yesmaximal, characteristic
8#10463—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4×Q8C2211yescharacteristic
11C3×#479—22no—
12C3×#477—21no—
13C3×#475—21no—
14C3×#428—21no—
15C3×#423C2211yescharacteristic
16C3×#419—21no—
17C2×#836—21no—
18#4924C2211yescharacteristic
19#4924C2211yescharacteristic
20C2×C4⋊(C3⋊C8)C2211yescharacteristic
21C2×C4×C3⋊C8C2211yescharacteristic
22#4547—21no—
23#4546—21no—
24#4535—21no—
25#4535—21no—
26#4535—21no—
27#4535—21no—
28#4530—21no—
29#4479—22no—
30#3078D311yescharacteristic
31#1801—31no—
32#1567—31no—
33#1550—31no—
34#1493—31no—
35C2×Q8⋊C8—31no—
36#1292—31no—
37C22×C3×Q8D411yescharacteristic
38C2×C3×M16—21no—
39C3×C4⋊Q8D411yescharacteristic
40C3×C4⋊Q8D411yescharacteristic
41C3×#125—21no—
42C3×C22⋊Q8—21no—
43C3×C22⋊Q8—21no—
44C3×C4×Q8—21no—
45C3×C4×Q8—21no—
46C3×C4×Q8—21no—
47C3×C4×Q8—21no—
48C3×C4×Q8—21no—
49C3×C4×Q8—21no—
50C3×#117—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4D411yescharacteristic
53C2×C3×C42—21no—
54C2×C3×C42C2311yescharacteristic
55C2×#208—21no—
56C2×#208—21no—
57C22×C3⋊C8—21no—
58C22×C3⋊C8—21no—
59C3×#107—22no—
60C3×#106—22no—
61C3×C4⋊C8—21no—
62C3×C4⋊C8D412yes—
63C3×#103—22no—
64C3×#103—22no—
65#836—42no—
66C4⋊(C3⋊C8)—21no—
67C4⋊(C3⋊C8)—21no—
68C4⋊(C3⋊C8)—21no—
69C4⋊(C3⋊C8)—21no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)—21no—
72C4×C3⋊C8—21no—
73C4×C3⋊C8D411yescharacteristic
74C4×C3⋊C8D411yescharacteristic
75#533C2×D311yescharacteristic
76C2×C4×Q8C2×D311yescharacteristic
77#479—22no—
78#477—21no—
79#475—21no—
80#428—21no—
81#423—31no—
82#423—31no—
83#423C2×D311yescharacteristic
84C2×C4⋊C8—31no—
85#419—21no—
86C2×#101—61no—
87C2×C4×C8—31no—
88#343—61no—
89#335—62no—
90#333—61no—
91#332—61no—
92Q8⋊C8—61no—
93Q8⋊C8—61no—
94Q8⋊C8—61no—
95Q8⋊C8—61no—
96C2×C3×Q8—21no—
97C2×C3×Q8—21no—
98C2×C3×Q8—21no—
99C2×C3×Q8—21no—
100C2×C3×Q8—21no—
101C2×C3×Q8—21no—
102C2×C3×Q8—21no—
103C2×C3×Q8Q1612yes—
104C22×C3×C4—21no—
105C22×C3×C4—21no—
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4#4111yescharacteristic
108C22×C3×C4#4111yescharacteristic
109C3×M16—41no—
110C3×M16—41no—
111C2×C3×C8—22no—
112C3×C4⋊C4—21no—
113C3×C4⋊C4—21no—
114C3×C4⋊C4—21no—
115C3×C4⋊C4—21no—
116C3×C4⋊C4—21no—
117C3×C4⋊C4—21no—
118C3×C4⋊C4—41no—
119C3×C4⋊C4—41no—
120C3×C4⋊C4—41no—
121C3×C4⋊C4—41no—
122C3×C22⋊C4—41no—
123C3×C42—41no—
124C3×C42—41no—
125C3×C42—41no—
126C3×C42C2×D411yescharacteristic
127C3×C42C2×D411yescharacteristic
128C3×C42C2×D411yescharacteristic
129C3×C42#4111yescharacteristic
130#208—41no—
131#208—41no—
132#208—41no—
133#208—41no—
134C2×C3⋊C8—21no—
135C2×C3⋊C8—21no—
136C2×C3⋊C8—42no—
137C2×C3⋊C8—41no—
138C2×C3⋊C8—41no—
139C2×C3⋊C8—41no—
140C2×C3⋊C8—81no—
141C22×Q8#6711yescharacteristic
142C2×M16—21no—
143C2×M16—61no—
144C2×M16—61no—
145C22×C8—61no—
146C22×C8—61no—
147C4⋊Q8#6711yescharacteristic
148C4⋊Q8#6711yescharacteristic
149#125—21no—
150C22⋊Q8—21no—
151C22⋊Q8—21no—
152C4×Q8—21no—
153C4×Q8—21no—
154C4×Q8—21no—
155C4×Q8—21no—
156C4×Q8—21no—
157C4×Q8—21no—
158#117—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4#6711yescharacteristic
161C2×C42—21no—
162C2×C42C22×D311yescharacteristic
163#107—22no—
164#106—22no—
165C4⋊C8—21no—
166C4⋊C8—61no—
167C4⋊C8—61no—
168C4⋊C8—61no—
169C4⋊C8—61no—
170C4⋊C8—61no—
171C4⋊C8—61no—
172C4⋊C8#6712yes—
173#103—22no—
174#103—22no—
175#101—122no—
176C4×C8—31no—
177C4×C8—31no—
178C4×C8—61no—
179C23×C3#12311yescharacteristic
180C3×Q8—22no—
181C3×Q8—22no—
182C3×Q8—41no—
183C3×Q8—41no—
184C3×Q8—41no—
185C3×Q8—41no—
186C3×Q8—81no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—22no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4—41no—
193C2×C3×C4—41no—
194C2×C3×C4—41no—
195C2×C3×C4—41no—
196C2×C3×C4—81no—
197C2×C3×C4#12011yescharacteristic
198C2×C3×C4#12111yescharacteristic
199C2×C3×C4#12111yescharacteristic
200C2×C3×C4#12111yescharacteristic
201C2×C3×C4#12311yescharacteristic
202C2×C3×C4#12411yescharacteristic
203C2×C3×C4C2×Q1611yescharacteristic
204C2×C3×C4#13511yescharacteristic
205C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
206C2×C3×C4#13711yescharacteristic
207C3×C8—42no—
208C3⋊C8—22no—
209C3⋊C8—41no—
210C3⋊C8—81no—
211C3⋊C8—81no—
212C3⋊C8—81no—
213C2×Q8—21no—
214C2×Q8—21no—
215C2×Q8—21no—
216C2×Q8—21no—
217C2×Q8—21no—
218C2×Q8—21no—
219C2×Q8—21no—
220C2×Q8#21612yes—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4#23711yescharacteristic
224C22×C4#23711yescharacteristic
225C22×C4C2×#6711yescharacteristic
226M16—41no—
227M16—41no—
228M16—121no—
229M16—121no—
230M16—121no—
231M16—121no—
232C2×C8—22no—
233C2×C8—61no—
234C2×C8—61no—
235C2×C8—122no—
236C2×C8—121no—
237C2×C8—121no—
238C2×C8—121no—
239C2×C8—241no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C22⋊C4—41no—
251C42—41no—
252C42—41no—
253C42—41no—
254C42D3×D411yescharacteristic
255C42#23711yescharacteristic
256C42C2×#6711yescharacteristic
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—42no—
267C3×C4—81no—
268C3×C4—81no—
269C3×C4—81no—
270C3×C4#44911yescharacteristic
271C3×C4#45111yescharacteristic
272C3×C4#46711yescharacteristic
273C3×C4#46811yescharacteristic
274C3×C4#47111yescharacteristic
275C3×C4#47311yescharacteristic
276C23#93511yescharacteristic
277Q8—22no—
278Q8—22no—
279Q8—41no—
280Q8—41no—
281Q8—41no—
282Q8—41no—
283Q8—81no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—22no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—81no—
294C2×C4#93511yescharacteristic
295C2×C4#93611yescharacteristic
296C2×C4#93811yescharacteristic
297C2×C4#93911yescharacteristic
298C2×C4#93911yescharacteristic
299C2×C4#94211yescharacteristic
300C2×C4C2×#21611yescharacteristic
301C2×C4#94911yescharacteristic
302C2×C4#95011yescharacteristic
303C2×C4#95311yescharacteristic
304C8—42no—
305C8—62no—
306C8—121no—
307C8—241no—
308C8—241no—
309C8—241no—
310C2×C3—21no—
311C2×C3—21no—
312C2×C3#203611yescharacteristic
313C2×C3#205011yescharacteristic
314C2×C3#205211yescharacteristic
315C22—21no—
316C22—21no—
317C22#504411yescharacteristic
318C22#504511yescharacteristic
319C22#522211yescharacteristic
320C4—21no—
321C4—21no—
322C4—21no—
323C4—42no—
324C4—81no—
325C4—81no—
326C4—81no—
327C4#504911yescharacteristic
328C4#505111yescharacteristic
329C4#520311yescharacteristic
330C4#520411yescharacteristic
331C4#522811yescharacteristic
332C4#523811yescharacteristic
333C3#1046311yescharacteristic, complemented, Sylow
334C2—21no—
335C2—21no—
336C2#7169511yescharacteristic
337C2#7210511yescharacteristic
338C2#7211711yescharacteristic
subgroupG/H
centerC22#5222
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3078C2
socleC22×C3#394
prime radicalC23×C3#123
hypercenter#3078D3
hypocenterC3#10463
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3078

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
C3×#30782
#718302
#718262
#716372
C2×#45352
#704912
#677112
#104633