Finite Group Databasebeta

← #10699219

Subgroups — #10699219

Order 768.

390 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 99 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×#3229C211yesmaximal, characteristic
2#71841C211yesmaximal, characteristic
3#71830C211yesmaximal, characteristic
4#71789C211yesmaximal, characteristic
5#71780C211yesmaximal, characteristic
6#70746C211yesmaximal, characteristic
7#67746C211yesmaximal, characteristic
8#10507—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11C3×#498—21no—
12C3×#487—21no—
13C3×#483—22no—
14C3×#475—21no—
15C3×#475—21no—
16C3×#431C2211yescharacteristic
17C2×#836—21no—
18C2×#836—21no—
19C4×C4⋊Dic3C2211yescharacteristic
20#4924C2211yescharacteristic
21C2×C4⋊(C3⋊C8)C2211yescharacteristic
22C4×#208C2211yescharacteristic
23#4547—21no—
24#4546—21no—
25#4532—21no—
26#4532—21no—
27#4529—22no—
28#4480—21no—
29#4479—21no—
30#3229D311yescharacteristic
31#1988—31no—
32#1986—31no—
33#1966—31no—
34#1565—31no—
35#1550—31no—
36#1303—31no—
37C22×C3×Q8—21no—
38C3×C4⋊Q8—21no—
39C3×C4⋊Q8—21no—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8D411yescharacteristic
43C3×C4⋊Q8D411yescharacteristic
44C3×#125—21no—
45C3×C22⋊Q8—21no—
46C3×C22⋊Q8—21no—
47C3×C4×Q8—21no—
48C3×C4×Q8—21no—
49C3×#117—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4D412yes—
52C2×C3×C42C2311yescharacteristic
53C2×C4⋊Dic3—21no—
54C2×C4×Dic3—21no—
55C2×#208—21no—
56C2×#208D411yescharacteristic
57C2×#208D411yescharacteristic
58C22×C3⋊C8—21no—
59C3×#106—22no—
60C3×#106—22no—
61C3×C4⋊C8D411yescharacteristic
62C3×C4⋊C8D411yescharacteristic
63C3×#103—22no—
64C3×#103—22no—
65C3×C22⋊C8—21no—
66C3×C4×C8—21no—
67#836—42no—
68#836—42no—
69C3⋊#95—21no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74#803—21no—
75C4×C3⋊C8—21no—
76#533C2×D311yescharacteristic
77C2×C4⋊Q8C2×D311yescharacteristic
78#498—21no—
79#487—21no—
80#483—22no—
81#475—21no—
82#475—21no—
83#431C2×D311yescharacteristic
84#423—31no—
85C2×C4⋊C8—31no—
86C2×#101—61no—
87C2×#101—61no—
88C4×M16—31no—
89C4×C4⋊C4—31no—
90#340—62no—
91#339—61no—
92#339—61no—
93#335—61no—
94#334—61no—
95#333—61no—
96#332—61no—
97C2×C3×Q8—21no—
98C2×C3×Q8—21no—
99C2×C3×Q8—41no—
100C2×C3×Q8—41no—
101C2×C3×Q8—41no—
102C2×C3×Q8—41no—
103C2×C3×Q8—41no—
104C2×C3×Q8—41no—
105C22×C3×C4—22no—
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4#4111yescharacteristic
109C22×Dic3—41no—
110C2×C3×C8—21no—
111C2×C3×C8—21no—
112C3×C4⋊C4—22no—
113C3×C4⋊C4—21no—
114C3×C4⋊C4—21no—
115C3×C4⋊C4—21no—
116C3×C4⋊C4—22no—
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×C42C2×D411yescharacteristic
125C3×C42C2×D411yescharacteristic
126C3×C42#4111yescharacteristic
127C3×C42#4111yescharacteristic
128C4⋊Dic3—22no—
129C4⋊Dic3—41no—
130C4×Dic3—41no—
131C4×Dic3—41no—
132#208—21no—
133#208—21no—
134#208—21no—
135#208—41no—
136#208—41no—
137#208SD1612yes—
138C2×C3⋊C8—21no—
139C2×C3⋊C8—21no—
140C2×C3⋊C8—42no—
141C2×C3⋊C8—41no—
142C2×C3⋊C8—41no—
143C22×Q8—21no—
144C2×M16—31no—
145C2×M16—31no—
146C2×M16—61no—
147C22×C8—61no—
148C4⋊Q8—21no—
149C4⋊Q8—21no—
150C4⋊Q8—21no—
151C4⋊Q8—21no—
152C4⋊Q8#6711yescharacteristic
153C4⋊Q8#6711yescharacteristic
154#125—21no—
155C22⋊Q8—21no—
156C22⋊Q8—21no—
157C4×Q8—21no—
158C4×Q8—21no—
159#117—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—61no—
162C2×C4⋊C4#6712yes—
163C2×C42—61no—
164C2×C42C22×D311yescharacteristic
165#106—22no—
166#106—22no—
167C4⋊C8—61no—
168C4⋊C8—61no—
169C4⋊C8—61no—
170C4⋊C8—61no—
171C4⋊C8#6711yescharacteristic
172C4⋊C8#6711yescharacteristic
173#103—22no—
174#103—22no—
175#101—122no—
176#101—122no—
177C22⋊C8—21no—
178#97—61no—
179C4×C8—21no—
180C4×C8—61no—
181#95—61no—
182C23×C3#12111yescharacteristic
183C3×Q8—41no—
184C3×Q8—41no—
185C3×Q8—81no—
186C3×Q8—81no—
187C3×Q8—81no—
188C3×Q8—81no—
189C3×Q8—81no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—41no—
193C2×C3×C4—41no—
194C2×C3×C4—41no—
195C2×C3×C4—42no—
196C2×C3×C4—42no—
197C2×C3×C4—42no—
198C2×C3×C4#12011yescharacteristic
199C2×C3×C4#12111yescharacteristic
200C2×C3×C4#12111yescharacteristic
201C2×C3×C4#12311yescharacteristic
202C2×C3×C4#12311yescharacteristic
203C2×C3×C4#12411yescharacteristic
204C2×C3×C4C2×SD1611yescharacteristic
205C2×C3×C4#13511yescharacteristic
206C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
207C2×C3×C4#13711yescharacteristic
208C2×Dic3—42no—
209C2×Dic3—81no—
210C2×Dic3—81no—
211C3×C8—41no—
212C3×C8—41no—
213C3⋊C8—22no—
214C3⋊C8—41no—
215C3⋊C8—81no—
216C3⋊C8—81no—
217C2×Q8—21no—
218C2×Q8—21no—
219C2×Q8—41no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×Q8—41no—
223C2×Q8—41no—
224C2×Q8—41no—
225C22×C4—22no—
226C22×C4—121no—
227C22×C4D3×D411yescharacteristic
228C22×C4#23711yescharacteristic
229C22×C4C2×#6711yescharacteristic
230M16—32no—
231M16—61no—
232M16—61no—
233M16—61no—
234M16—121no—
235M16—121no—
236C2×C8—21no—
237C2×C8—21no—
238C2×C8—61no—
239C2×C8—61no—
240C2×C8—122no—
241C2×C8—121no—
242C2×C8—121no—
243C4⋊C4—22no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—22no—
248C4⋊C4—21no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—62no—
254C4⋊C4—121no—
255C22⋊C4—41no—
256C42—41no—
257C42—121no—
258C42—121no—
259C42#23711yescharacteristic
260C42#23711yescharacteristic
261C42C2×#6711yescharacteristic
262C42C2×#6711yescharacteristic
263C22×C3—21no—
264C22×C3—21no—
265C22×C3#39411yescharacteristic
266C22×C3#46111yescharacteristic
267C22×C3#46311yescharacteristic
268C3×C4—21no—
269C3×C4—21no—
270C3×C4—21no—
271C3×C4—81no—
272C3×C4—82no—
273C3×C4—81no—
274C3×C4#45411yescharacteristic
275C3×C4#45611yescharacteristic
276C3×C4#46011yescharacteristic
277C3×C4#46411yescharacteristic
278C3×C4#46511yescharacteristic
279C3×C4#46811yescharacteristic
280Dic3—82no—
281C23#93911yescharacteristic
282Q8—41no—
283Q8—41no—
284Q8—81no—
285Q8—81no—
286Q8—81no—
287Q8—81no—
288Q8—81no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—122no—
298C2×C4—241no—
299C2×C4—241no—
300C2×C4D3×SD1611yescharacteristic
301C2×C4#91411yescharacteristic
302C2×C4#91511yescharacteristic
303C2×C4#91611yescharacteristic
304C2×C4#93511yescharacteristic
305C2×C4#93511yescharacteristic
306C2×C4#93611yescharacteristic
307C2×C4#93811yescharacteristic
308C2×C4#93911yescharacteristic
309C2×C4#95311yescharacteristic
310C8—41no—
311C8—41no—
312C8—62no—
313C8—121no—
314C8—241no—
315C8—241no—
316C2×C3—21no—
317C2×C3—21no—
318C2×C3#203811yescharacteristic
319C2×C3#204911yescharacteristic
320C2×C3#205511yescharacteristic
321C22—21no—
322C22—21no—
323C22#516211yescharacteristic
324C22#516511yescharacteristic
325C22#522211yescharacteristic
326C4—21no—
327C4—21no—
328C4—21no—
329C4—82no—
330C4—81no—
331C4—81no—
332C4—242no—
333C4#516311yescharacteristic
334C4#516611yescharacteristic
335C4#517111yescharacteristic
336C4#517411yescharacteristic
337C4#524511yescharacteristic
338C4#524611yescharacteristic
339C3#1050711yescharacteristic, complemented, Sylow
340C2—21no—
341C2—21no—
342C2#7186311yescharacteristic
343C2#7205111yescharacteristic
344C2#7210411yescharacteristic
subgroupG/H
centerC22#5222
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3229C2
socleC22×C3#394
prime radicalC23×C3#121
hypercenter#3229D3
hypocenterC3#10507
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3229

Complements

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

NH ≅ G/Ntype
339. C3#10507semidirect

Maximal subgroups

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

subgroupindex
C3×#32292
#718412
#718302
#717892
#717802
#707462
#677462
#105073