Finite Group Databasebeta

← #10697114

Subgroups — #10697114

Order 768.

562 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 562 conjugacy classes of subgroups — too many to draw. Only its 109 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
1C2×C3×#498C211yesmaximal, characteristic, complemented
2#83216C211yesmaximal, characteristic
3#83203C211yesmaximal, characteristic
4#71790C211yesmaximal, characteristic, complemented
5#71773C211yesmaximal, characteristic, complemented
6#69735C211yesmaximal, characteristic
7#67656C211yesmaximal, characteristic, complemented
8#10172—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×#942—22no—
12#5477C2211yescharacteristic
13C3×#498—22no—
14C3×#498—22no—
15C2×C3×#106—22no—
16C2×C3×C4×C8C2211yescharacteristic
17#5092—21no—
18#5088—21no—
19#5086—22no—
20#5085—21no—
21#5084—22no—
22#5080—21no—
23#4968—22no—
24#4925C2211yescharacteristic
25#4924C2211yescharacteristic
26C4×#208C2211yescharacteristic
27#4547—22no—
28#4530—22no—
29#4459—22no—
30#3227—31no—
31C2×#498D311yescharacteristic, complemented
32#3152—31no—
33#1997—31no—
34#1546—31no—
35#1492—31no—
36#1278—31no—
37C22×C3×Q8—21no—
38C22×C3×Q8—21no—
39C2×#235—21no—
40C22×C3×C8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—21no—
47C3×C4⋊Q8—21no—
48C3×C4⋊Q8D412yes—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4D412yes—
53C2×C3×C42C2311yescharacteristic
54C2×#216—24no—
55#942—44no—
56#942—44no—
57C2×#215—24no—
58#930—21no—
59#930—21no—
60C2×#208—21no—
61C2×#208D412yes—
62#875—21no—
63#874D411yescharacteristic
64C4×D12—21no—
65C4×D12—21no—
66#869D411yescharacteristic
67C3×#106—28no—
68C3×C4×C8—21no—
69C3×C4×C8D412yes—
70#830—21no—
71#809—24no—
72#808—24no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)D412yes—
76#803—21no—
77C4×C3⋊C8—21no—
78C4×C3⋊C8—21no—
79C2×#137—62no—
80#532—31no—
81C2×C4⋊Q8C2×D311yescharacteristic
82C2×C4⋊Q8C2×D311yescharacteristic
83#498—22no—
84#498—22no—
85#498—61no—
86#497—62no—
87#494—61no—
88#492—61no—
89#486—61no—
90#481—62no—
91#476—62no—
92#431—31no—
93C2×#106—22no—
94#423—31no—
95C4×M16—31no—
96C2×C4×C8C2×D311yescharacteristic
97#343—62no—
98#332—62no—
99D4⋊C8—62no—
100C2×C3×Q8—22no—
101C2×C3×Q8—41no—
102C2×C3×Q8—41no—
103C2×C3×Q8—41no—
104C2×C3×Q8—42no—
105C2×C3×Q8—42no—
106C2×C3×Q8—82no—
107C22×C3×C4—22no—
108C22×C3×C4—41no—
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4#4111yescharacteristic
112C2×#67—41no—
113#235—42no—
114#235—42no—
115C2×D12—21no—
116C2×D12—21no—
117C2×D12—21no—
118C2×C4×D3—41no—
119C2×Dic6—21no—
120C2×C3×C8—24no—
121C2×C3×C8—22no—
122C3×C4⋊C4—22no—
123C3×C4⋊C4—22no—
124C3×C4⋊C4—22no—
125C3×C4⋊C4—22no—
126C3×C4⋊C4—41no—
127C3×C4⋊C4—41no—
128C3×C4⋊C4—41no—
129C3×C4⋊C4—41no—
130C3×C42C2×D411yescharacteristic
131C3×C42C2×D411yescharacteristic
132C3×C42C2×D411yescharacteristic
133C3×C42C2×D411yescharacteristic
134#216—48no—
135#215—48no—
136(C2×D3)⋊C4—41no—
137(C2×D3)⋊C4—41no—
138C4⋊Dic3—21no—
139C4⋊Dic3—21no—
140#208—22no—
141#208—42no—
142#208SD1614yes—
143C2×C3⋊C8—22no—
144C2×C3⋊C8—22no—
145C2×C3⋊C8—41no—
146C2×#41—61no—
147C22×Q8—21no—
148C22×Q8—21no—
149#137—124no—
150#137—124no—
151C2×Q16—64no—
152C2×SD16—64no—
153C2×M16—32no—
154C2×M16—61no—
155C22×C8—21no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—31no—
164C4⋊Q8#6712yes—
165#127—31no—
166#124—61no—
167#121—61no—
168#121—61no—
169C4×D4—61no—
170C4×D4—61no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4#6712yes—
175C2×C42C22×D311yescharacteristic
176#106—28no—
177#106—64no—
178C4⋊C8—32no—
179C4⋊C8—61no—
180C4⋊C8—61no—
181#102—64no—
182C22⋊C8—61no—
183#97—61no—
184C4×C8—21no—
185C4×C8—61no—
186C4×C8—61no—
187C4×C8#6712yes—
188C23×C3#12111yescharacteristic
189C22×D3—41no—
190C3×Q8—44no—
191C3×Q8—82no—
192C3×Q8—82no—
193C3×Q8—81no—
194C3×Q8—81no—
195C2×C3×C4—22no—
196C2×C3×C4—22no—
197C2×C3×C4—42no—
198C2×C3×C4—42no—
199C2×C3×C4—82no—
200C2×C3×C4—82no—
201C2×C3×C4#12011yescharacteristic
202C2×C3×C4#12011yescharacteristic
203C2×C3×C4#12011yescharacteristic
204C2×C3×C4#12111yescharacteristic
205C2×C3×C4#12111yescharacteristic
206C2×C3×C4#12711yescharacteristic
207C2×C3×C4C2×SD1612yes—
208C2×C3×C4AGL(1,ℤ/8)12yes—
209#67—82no—
210C2×Dic3—41no—
211D12—42no—
212D12—42no—
213D12—42no—
214C4×D3—81no—
215C4×D3—81no—
216Dic6—42no—
217C3×C8—24no—
218C3⋊C8—24no—
219C3⋊C8—42no—
220C3⋊C8—81no—
221#41—122no—
222#41—122no—
223C2×Q8—22no—
224C2×Q8—41no—
225C2×Q8—41no—
226C2×Q8—41no—
227C2×Q8—42no—
228C2×Q8—42no—
229C2×Q8—61no—
230C2×Q8—82no—
231C2×D4—61no—
232C2×D4—61no—
233C2×D4—61no—
234C2×D4—121no—
235C22×C4—22no—
236C22×C4—41no—
237C22×C4—121no—
238C22×C4D3×D411yescharacteristic
239C22×C4#23711yescharacteristic
240C22×C4C2×#6711yescharacteristic
241Q16—128no—
242SD16—128no—
243M16—34no—
244M16—62no—
245M16—122no—
246C2×C8—24no—
247C2×C8—22no—
248C2×C8—62no—
249C2×C8—62no—
250C2×C8—121no—
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—22no—
254C4⋊C4—22no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—61no—
260C4⋊C4—61no—
261C22⋊C4—121no—
262C22⋊C4—121no—
263C42D3×D411yescharacteristic
264C42D3×D411yescharacteristic
265C42C2×#6711yescharacteristic
266C42C2×#6711yescharacteristic
267C22×C3—21no—
268C22×C3—21no—
269C22×C3#39211yescharacteristic
270C22×C3#46012yes—
271C2×D3—81no—
272C2×D3—81no—
273C2×D3—81no—
274C3×C4—21no—
275C3×C4—21no—
276C3×C4—82no—
277C3×C4—82no—
278C3×C4#45012yes—
279C3×C4#45612yes—
280C3×C4#47112yes—
281C3×C4#49211yescharacteristic
282C3×C4#49611yescharacteristic
283Dic3—81no—
284C23—121no—
285C23#93911yescharacteristic
286Q8—44no—
287Q8—82no—
288Q8—81no—
289Q8—81no—
290Q8—82no—
291Q8—122no—
292D4—122no—
293D4—122no—
294D4—122no—
295D4—242no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—121no—
303C2×C4—241no—
304C2×C4—241no—
305C2×C4D3×SD1612yes—
306C2×C4#91412yes—
307C2×C4#93711yescharacteristic
308C2×C4#93711yescharacteristic
309C2×C4#93811yescharacteristic
310C2×C4#93911yescharacteristic
311C2×C4#94011yescharacteristic
312C2×C4#95311yescharacteristic
313C8—24no—
314C8—64no—
315C8—122no—
316C8—241no—
317C2×C3—22no—
318C2×C3#200511yescharacteristic
319C2×C3#201411yescharacteristic
320C2×C3#201911yescharacteristic
321D3—161no—
322C22—21no—
323C22—21no—
324C22—241no—
325C22—241no—
326C22—241no—
327C22#516312yes—
328C22#522311yescharacteristic
329C4—21no—
330C4—21no—
331C4—82no—
332C4—82no—
333C4—241no—
334C4#509512yes—
335C4#516912yes—
336C4#517511yescharacteristic
337C4#517611yescharacteristic
338C4#517712yes—
339C3#1017211yescharacteristic, complemented, Sylow
340C2—22no—
341C2—481no—
342C2#7179511yescharacteristic
343C2#7205611yescharacteristic
344C2#7205911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#498C2
socleC22×C3#392
prime radical#5477C22
hypercenterC2×#498D3
hypocenterC3#10172
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#498

Complements

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

NH ≅ G/Ntype
1. C2×C3×#498C2semidirect
4. #71790C2semidirect
5. #71773C2semidirect
7. #67656C2semidirect
31. C2×#498D3semidirect
339. C3#10172semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4982
#832162
#832032
#717902
#717732
#697352
#676562
#101723