Finite Group Databasebeta

← #10690199

Subgroups — #10690199

Order 768.

928 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 928 conjugacy classes of subgroups — too many to draw. Only its 127 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×#4991C211yesmaximal, characteristic
2#82155C211yesmaximal, characteristic, complemented
3C2×C3×#324C211yesmaximal, characteristic, complemented
4#69959C212yesmaximal
5#67675C212yesmaximal, complemented
6#9018—31nomaximal, Sylow
7C23×D12C2211yescharacteristic
8C22×C3×C4⋊C4C2211yescharacteristic
9C22×(C2×D3)⋊C4—21no—
10C22×(C2×D3)⋊C4—21no—
11C2×C3×C22⋊C8C2212yes—
12#5212C2212yes—
13#5211—21no—
14C4×#217—21no—
15C2×C3⋊#95C2211yescharacteristic
16C2×#821—24no—
17#4991—22no—
18#4991C414yescomplemented
19#4980—24no—
20C3×#324—22no—
21C2×#394—31no—
22#2313—31no—
23#1786—32no—
24C2×#324D311yescharacteristic, complemented
25#1294—32no—
26C24×D3—21no—
27C23×C3×C4D411yescharacteristic
28C23×C3×C4D411yescharacteristic
29C23×C3×C4C2311yescharacteristic
30C23×Dic3—21no—
31C22×D12—22no—
32C22×D12—22no—
33C22×D12—21no—
34C22×D12—44no—
35C22×D12C2×C412yes—
36C22×C3×C8—22no—
37C2×C3×C4⋊C4—22no—
38C2×C3×C4⋊C4—22no—
39C2×C3×C4⋊C4C2×C412yes—
40C2×C3×C4⋊C4D412yes—
41C2×#217—22no—
42C2×(C2×D3)⋊C4—24no—
43C2×(C2×D3)⋊C4—24no—
44C2×(C2×D3)⋊C4—44no—
45C2×(C2×D3)⋊C4—44no—
46C2×C4⋊Dic3D414yes—
47C2×C4×Dic3—22no—
48C3×C22⋊C8—22no—
49C3×C22⋊C8—22no—
50C3⋊#95—22no—
51C3⋊#95—24no—
52C3⋊#95—21no—
53C3⋊#95—22no—
54C3⋊#95C2×C412yes—
55#821—48no—
56C23×D4—31no—
57C22×C4⋊C4C2×D311yescharacteristic
58C22×C22⋊C4—61no—
59C22×C22⋊C4—61no—
60C2×#102—64no—
61C2×C22⋊C8C2×D312yes—
62#394—34no—
63#394—62no—
64#387—64no—
65#381—61no—
66#380—32no—
67C4×C22⋊C4—61no—
68C2×#95—31no—
69#324—22no—
70C24×C3C2×D411yescharacteristic
71C23×D3—24no—
72C23×D3—22no—
73C23×D3—42no—
74C23×D3—42no—
75C23×D3—42no—
76C23×D3—41no—
77C23×D3—41no—
78C23×D3—42no—
79C23×D3—42no—
80C22×C3×C4—21no—
81C22×C3×C4—21no—
82C22×C3×C4—21no—
83C22×C3×C4—22no—
84C22×C3×C4—22no—
85C22×C3×C4—21no—
86C22×C3×C4—42no—
87C22×C3×C4—42no—
88C22×C3×C4C22⋊C412yes—
89C22×C3×C4C22⋊C412yes—
90C22×C3×C4C22×C411yescharacteristic
91C22×C3×C4C2×D412yes—
92C22×C3×C4C2×D411yescharacteristic
93C22×C3×C4#4112yes—
94C22×Dic3—22no—
95C22×Dic3—24no—
96C22×Dic3—42no—
97C22×Dic3—42no—
98C22×Dic3—42no—
99C2×D12—28no—
100C2×D12—24no—
101C2×D12—44no—
102C2×D12—48no—
103C2×D12—44no—
104C2×D12—44no—
105C2×D12—42no—
106C2×C3×C8—44no—
107C2×C3×C8—42no—
108C3×C4⋊C4—24no—
109C3×C4⋊C4—24no—
110#217—42no—
111#217—42no—
112(C2×D3)⋊C4—48no—
113(C2×D3)⋊C4—48no—
114C4⋊Dic3—44no—
115C4×Dic3—82no—
116C25—61no—
117C22×D4—32no—
118C22×D4—62no—
119C22×D4—62no—
120C22×D4—61no—
121C22×D4—124no—
122C23×C4—61no—
123C23×C4D1211yescharacteristic
124C23×C4D1211yescharacteristic
125C23×C4C22×D311yescharacteristic
126C22×C8—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—34no—
130C2×C4⋊C4C4×D312yes—
131C2×C4⋊C4#6712yes—
132C2×C22⋊C4—62no—
133C2×C22⋊C4—64no—
134C2×C22⋊C4—64no—
135C2×C22⋊C4—124no—
136C2×C22⋊C4—124no—
137C2×C42—62no—
138#102—128no—
139C22⋊C8—22no—
140C22⋊C8—22no—
141#95—32no—
142#95—62no—
143#95—64no—
144#95—62no—
145#95—61no—
146C23×C3—22no—
147C23×C3—22no—
148C23×C3C2×C22⋊C411yescharacteristic
149C23×C3#12011yescharacteristic
150C23×C3#12311yescharacteristic
151C23×C3#13512yes—
152C23×C3AGL(1,ℤ/8)12yes—
153C22×D3—24no—
154C22×D3—42no—
155C22×D3—44no—
156C22×D3—48no—
157C22×D3—48no—
158C22×D3—42no—
159C22×D3—42no—
160C22×D3—42no—
161C22×D3—42no—
162C22×D3—42no—
163C22×D3—44no—
164C22×D3—84no—
165C22×D3—84no—
166C22×D3—82no—
167C22×D3—82no—
168C22×D3—82no—
169C22×D3—82no—
170C2×C3×C4—24no—
171C2×C3×C4—22no—
172C2×C3×C4—22no—
173C2×C3×C4—22no—
174C2×C3×C4—22no—
175C2×C3×C4—41no—
176C2×C3×C4—41no—
177C2×C3×C4—42no—
178C2×C3×C4—42no—
179C2×C3×C4—42no—
180C2×C3×C4—42no—
181C2×C3×C4—42no—
182C2×C3×C4—81no—
183C2×C3×C4C4×D412yes—
184C2×C3×C4#12112yes—
185C2×Dic3—42no—
186C2×Dic3—42no—
187C2×Dic3—42no—
188C2×Dic3—44no—
189C2×Dic3—44no—
190C2×Dic3—44no—
191C2×Dic3—84no—
192D12—28no—
193D12—48no—
194D12—44no—
195C3×C8—82no—
196C24—64no—
197C24—62no—
198C24—122no—
199C24—122no—
200C24—122no—
201C24—121no—
202C24—121no—
203C24—122no—
204C24—122no—
205C24C2×D1211yescharacteristic
206C2×D4—68no—
207C2×D4—64no—
208C2×D4—124no—
209C2×D4—124no—
210C2×D4—128no—
211C2×D4—124no—
212C2×D4—122no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—21no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—64no—
222C22×C4—62no—
223C22×C4—122no—
224C22×C4—122no—
225C22×C4—122no—
226C22×C4(C2×D3)⋊C412yes—
227C22×C4(C2×D3)⋊C412yes—
228C22×C4C2×C4×D311yescharacteristic
229C22×C4D3×D412yes—
230C22×C4#23712yes—
231C22×C4C2×#6711yescharacteristic
232C2×C8—44no—
233C2×C8—42no—
234C4⋊C4—24no—
235C4⋊C4—24no—
236C4⋊C4—124no—
237C22⋊C4—128no—
238C22⋊C4—128no—
239C22⋊C4—122no—
240C22⋊C4—122no—
241C42—242no—
242C22×C3—24no—
243C22×C3—24no—
244C22×C3—21no—
245C22×C3—21no—
246C22×C3—21no—
247C22×C3—21no—
248C22×C3#38611yescharacteristic
249C22×C3#41611yescharacteristic
250C22×C3#41811yescharacteristic
251C22×C3#44912yes—
252C22×C3#45412yes—
253C22×C3B(4,𝔽2)12yes—
254C22×C3#48212yes—
255C2×D3—44no—
256C2×D3—44no—
257C2×D3—44no—
258C2×D3—82no—
259C2×D3—82no—
260C2×D3—88no—
261C2×D3—88no—
262C2×D3—82no—
263C2×D3—82no—
264C3×C4—22no—
265C3×C4—22no—
266C3×C4—42no—
267C3×C4—42no—
268Dic3—82no—
269Dic3—84no—
270C23—22no—
271C23—22no—
272C23—64no—
273C23—122no—
274C23—124no—
275C23—128no—
276C23—128no—
277C23—122no—
278C23—122no—
279C23—122no—
280C23—122no—
281C23—124no—
282C23—122no—
283C23—244no—
284C23—244no—
285C23—242no—
286C23—242no—
287C23—242no—
288C23—242no—
289C23#88211yescharacteristic
290C23#88611yescharacteristic
291C23#90412yes—
292C23#90812yes—
293C23C2×(C2×D3)⋊C411yescharacteristic
294D4—68no—
295D4—128no—
296D4—124no—
297C2×C4—24no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—81no—
310C2×C4—122no—
311C2×C4—122no—
312C2×C4—122no—
313C2×C4—124no—
314C2×C4—124no—
315C2×C4—124no—
316C2×C4—244no—
317C2×C4#67⋊C412yes—
318C2×C4#93912yes—
319C8—82no—
320C2×C3—21no—
321C2×C3—21no—
322C2×C3—22no—
323C2×C3#159714yes—
324C2×C3#187511yescharacteristic
325C2×C3#188811yescharacteristic
326C2×C3#188911yescharacteristic
327D3—84no—
328C22—24no—
329C22—24no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—124no—
335C22—124no—
336C22—124no—
337C22—242no—
338C22—242no—
339C22—248no—
340C22—248no—
341C22—242no—
342C22—242no—
343C22#473412yes—
344C22#473712yes—
345C22#474112yes—
346C22#475312yes—
347C22#495811yescharacteristic
348C22#511311yescharacteristic
349C22#513111yescharacteristic
350C4—22no—
351C4—22no—
352C4—42no—
353C4—42no—
354C4—242no—
355C4—244no—
356C3#901811yescharacteristic, complemented, Sylow
357C2—21no—
358C2—21no—
359C2—22no—
360C2—244no—
361C2#6813014yes—
362C2#6993311yescharacteristic
363C2#7002211yescharacteristic
364C2#7010411yescharacteristic
subgroupG/H
centerC23#882
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#324C2
socleC23×C3#120
prime radicalC23×D12C22
cosocleC23×C3×C4C23
hypercenterC2×#324D3
hypocenterC3#9018
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
2. #82155C2semidirect
3. C2×C3×#324C2semidirect
5. #67675C2semidirect
18. #4991C4semidirect
24. C2×#324D3semidirect
356. C3#9018semidirect

Maximal subgroups

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

subgroupindex
C2×#49912
#821552
C2×C3×#3242
#699592
#676752
#90183