Finite Group Databasebeta

← #10697100

Subgroups — #10697100

Order 768.

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

normal subgroups only

This group has 574 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×#487C211yesmaximal, characteristic, complemented
2#83213C211yesmaximal, characteristic
3#83203C211yesmaximal, characteristic
4#71790C211yesmaximal, characteristic, complemented
5#71773C211yesmaximal, characteristic, complemented
6#69735C211yesmaximal, characteristic
7#67656C211yesmaximal, characteristic, complemented
8#10170—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×#942—22no—
12#5477C2211yescharacteristic
13C3×#487—22no—
14C3×#487—22no—
15C2×C3×#103—22no—
16C2×C3×C4×C8C2211yescharacteristic
17#5227—22no—
18#5092—21no—
19#5086—22no—
20#5085—21no—
21#5085—21no—
22#5083—22no—
23#5079—21no—
24#4925C2211yescharacteristic
25#4924C2211yescharacteristic
26C4×#208C2211yescharacteristic
27#4547—22no—
28#4530—22no—
29#4459—22no—
30#3206—31no—
31#3152—31no—
32C2×#487D311yescharacteristic, complemented
33#1997—31no—
34#1546—31no—
35#1492—31no—
36#1278—31no—
37C22×C3×Q8—21no—
38C22×C3×Q8D412yes—
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⋊C4—21no—
53C2×C3×C42C2311yescharacteristic
54C2×#216—24no—
55#942—44no—
56#942—44no—
57C2×#215—24no—
58C2×#215—24no—
59#930—21no—
60#930—21no—
61C2×#208—21no—
62C2×#208D412yes—
63#875—21no—
64#874D411yescharacteristic
65C4×D12—21no—
66C4×D12—21no—
67#869D411yescharacteristic
68C3×#103—28no—
69C3×C4×C8—21no—
70C3×C4×C8D412yes—
71#835—24no—
72#830—21no—
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#497—62no—
84#494—61no—
85#492—61no—
86#492—61no—
87#487—22no—
88#487—22no—
89#487—61no—
90#461—62no—
91#448—62no—
92#431—31no—
93#423—31no—
94C2×#103—22no—
95C4×M16—31no—
96C2×C4×C8C2×D311yescharacteristic
97#343—62no—
98#332—62no—
99D4⋊C8—62no—
100C2×C3×Q8—22no—
101C2×C3×Q8—22no—
102C2×C3×Q8—22no—
103C2×C3×Q8—44no—
104C2×C3×Q8—41no—
105C2×C3×Q8—42no—
106C2×C3×Q8—42no—
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—22no—
121C2×C3×C8—42no—
122C3×C4⋊C4—22no—
123C3×C4⋊C4—22no—
124C3×C4⋊C4—41no—
125C3×C4⋊C4—41no—
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#215—84no—
137(C2×D3)⋊C4—41no—
138(C2×D3)⋊C4—41no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141#208—22no—
142#208—42no—
143#208SD1614yescomplemented
144C2×C3⋊C8—22no—
145C2×C3⋊C8—22no—
146C2×C3⋊C8—41no—
147C2×#41—61no—
148C22×Q8—21no—
149C22×Q8#6712yes—
150#137—124no—
151#137—124no—
152C2×Q16—64no—
153C2×SD16—64no—
154C2×SD16—64no—
155C2×M16—32no—
156C2×M16—61no—
157C22×C8—21no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—31no—
166C4⋊Q8#6712yes—
167#127—31no—
168#124—61no—
169#121—61no—
170#121—61no—
171C4×D4—61no—
172C4×D4—61no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C42C22×D311yescharacteristic
178C4⋊C8—32no—
179C4⋊C8—61no—
180C4⋊C8—61no—
181#103—28no—
182#103—64no—
183C22⋊C8—61no—
184#97—61no—
185C4×C8—21no—
186C4×C8—61no—
187C4×C8—61no—
188C4×C8#6712yes—
189C23×C3#12111yescharacteristic
190C22×D3—41no—
191C3×Q8—44no—
192C3×Q8—44no—
193C3×Q8—44no—
194C3×Q8—81no—
195C3×Q8—81no—
196C2×C3×C4—22no—
197C2×C3×C4—22no—
198C2×C3×C4—42no—
199C2×C3×C4—42no—
200C2×C3×C4—82no—
201C2×C3×C4—82no—
202C2×C3×C4#12011yescharacteristic
203C2×C3×C4#12011yescharacteristic
204C2×C3×C4#12011yescharacteristic
205C2×C3×C4#12111yescharacteristic
206C2×C3×C4#12111yescharacteristic
207C2×C3×C4#12711yescharacteristic
208C2×C3×C4C2×SD1612yes—
209C2×C3×C4AGL(1,ℤ/8)12yes—
210#67—82no—
211C2×Dic3—41no—
212D12—42no—
213D12—42no—
214D12—42no—
215C4×D3—81no—
216C4×D3—81no—
217Dic6—42no—
218C3×C8—42no—
219C3⋊C8—24no—
220C3⋊C8—42no—
221C3⋊C8—81no—
222#41—122no—
223#41—122no—
224C2×Q8—22no—
225C2×Q8—22no—
226C2×Q8—22no—
227C2×Q8—44no—
228C2×Q8—41no—
229C2×Q8—42no—
230C2×Q8—42no—
231C2×Q8—61no—
232C2×D4—61no—
233C2×D4—61no—
234C2×D4—61no—
235C2×D4—121no—
236C22×C4—22no—
237C22×C4—41no—
238C22×C4—121no—
239C22×C4D3×D411yescharacteristic
240C22×C4#23711yescharacteristic
241C22×C4C2×#6711yescharacteristic
242Q16—128no—
243SD16—128no—
244SD16—244no—
245M16—34no—
246M16—62no—
247M16—122no—
248C2×C8—22no—
249C2×C8—42no—
250C2×C8—62no—
251C2×C8—62no—
252C2×C8—121no—
253C4⋊C4—22no—
254C4⋊C4—22no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—61no—
262C4⋊C4—61no—
263C22⋊C4—121no—
264C22⋊C4—121no—
265C42D3×D411yescharacteristic
266C42D3×D411yescharacteristic
267C42C2×#6711yescharacteristic
268C42C2×#6711yescharacteristic
269C22×C3—21no—
270C22×C3—21no—
271C22×C3#39211yescharacteristic
272C22×C3#46012yes—
273C2×D3—81no—
274C2×D3—81no—
275C2×D3—81no—
276C3×C4—21no—
277C3×C4—21no—
278C3×C4—82no—
279C3×C4—82no—
280C3×C4#45012yes—
281C3×C4#45512yes—
282C3×C4#47312yes—
283C3×C4#49211yescharacteristic
284C3×C4#49611yescharacteristic
285Dic3—81no—
286C23—121no—
287C23#93911yescharacteristic
288Q8—44no—
289Q8—44no—
290Q8—44no—
291Q8—81no—
292Q8—81no—
293Q8—122no—
294D4—122no—
295D4—122no—
296D4—122no—
297D4—242no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—121no—
305C2×C4—241no—
306C2×C4—241no—
307C2×C4D3×SD1612yes—
308C2×C4#91412yes—
309C2×C4#93711yescharacteristic
310C2×C4#93711yescharacteristic
311C2×C4#93811yescharacteristic
312C2×C4#93911yescharacteristic
313C2×C4#94011yescharacteristic
314C2×C4#95311yescharacteristic
315C8—42no—
316C8—64no—
317C8—122no—
318C8—241no—
319C2×C3—22no—
320C2×C3#200411yescharacteristic
321C2×C3#201411yescharacteristic
322C2×C3#202011yescharacteristic
323D3—161no—
324C22—21no—
325C22—21no—
326C22—241no—
327C22—241no—
328C22—241no—
329C22#516312yes—
330C22#522311yescharacteristic
331C4—21no—
332C4—21no—
333C4—82no—
334C4—82no—
335C4—241no—
336C4#506112yes—
337C4#516912yes—
338C4#517511yescharacteristic
339C4#517611yescharacteristic
340C4#519212yes—
341C3#1017011yescharacteristic, complemented, Sylow
342C2—22no—
343C2—481no—
344C2#7173111yescharacteristic
345C2#7205611yescharacteristic
346C2#7207411yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#487C2
socleC22×C3#392
prime radical#5477C22
hypercenterC2×#487D3
hypocenterC3#10170
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#487

Complements

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

NH ≅ G/Ntype
1. C2×C3×#487C2semidirect
4. #71790C2semidirect
5. #71773C2semidirect
7. #67656C2semidirect
32. C2×#487D3semidirect
143. #208SD16semidirect
341. C3#10170semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4872
#832132
#832032
#717902
#717732
#697352
#676562
#101703