Finite Group Databasebeta

← #10764134

Subgroups — #10764134

Order 768.

870 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86204C211yesmaximal, characteristic
2#79340C212yesmaximal
3C3×#1558C211yesmaximal, characteristic, complemented
4C2×#4546C211yesmaximal, characteristic, complemented
5#70820C212yesmaximal, complemented
6#68488C212yesmaximal, complemented
7#68486C212yesmaximal, complemented
8#68484C212yesmaximal, complemented
9#68481C212yesmaximal, complemented
10#17780—31nomaximal, Sylow
11C2×C3×C4⋊Q8C2211yescharacteristic
12#5816C412yes—
13#5812C412yes—
14#5732—22no—
15#5731C414yes—
16#5728C2212yes—
17#5728C2212yes—
18#5721—22no—
19#5513—22no—
20#5513—22no—
21C4×#235C2211yescharacteristic
22#5470C2211yescharacteristic
23C3×#432C2212yes—
24C2×C4⋊(C3⋊C8)C2212yes—
25#4837C2212yes—
26#4835C2212yes—
27#4833C2212yes—
28D3×C4⋊C8C2212yes—
29#4730—22no—
30#67⋊C8—22no—
31#4706—22no—
32#4701—22no—
33C3×#333C2212yes—
34C3×#333C2211yescharacteristic
35C3×#333C2211yescharacteristic
36#4546C2211yescharacteristic
37#4546C2212yes—
38#4546C2211yescharacteristic
39#4486C2212yes—
40#4486C2212yes—
41#4482C2212yes—
42#4482C2212yes—
43#3509—31no—
44#2990—32no—
45#1559—32no—
46#1558D311yescharacteristic, complemented
47#1554—32no—
48#1552—32no—
49#1550—32no—
50#1542—32no—
51C2×#333—31no—
52C22×C3×Q8C2×C412yes—
53C2×#235C2×C411yescharacteristic
54C3×C4⋊Q8—22no—
55C3×C4⋊Q8C2×C412yes—
56C3×C4⋊Q8C2311yescharacteristic
57C3×C4⋊Q8C2311yescharacteristic
58C2×C3×C4⋊C4—22no—
59C2×C3×C42C2311yescharacteristic
60#947—22no—
61#946C2×C414yes—
62#946C2×C414yes—
63Q8×Dic3—24no—
64Q8×Dic3C2×C414yes—
65Q8×Dic3C2×C414yes—
66#944—22no—
67#929—24no—
68C4×#67D412yes—
69C4×#67D412yes—
70#926—22no—
71#926—22no—
72#926C2×C412yes—
73#926C2×C411yescharacteristic
74#924—22no—
75C22×C3⋊C8—22no—
76C2×ΓL(1,𝔽25)—22no—
77C2×D3×C8—22no—
78#898—24no—
79#898—24no—
80#897—22no—
81#897—22no—
82#892—22no—
83#892—22no—
84#890—22no—
85#890—22no—
86#889—24no—
87#889—24no—
88#887—22no—
89#887—22no—
90C4×D12D412yes—
91#872C2×C412yes—
92C42×D3C2312yes—
93#870C2×C412yes—
94#869C2312yes—
95C4×Dic6—22no—
96C4×Dic6D412yes—
97C3×C4⋊C8C2312yes—
98C3×C4⋊C8C2312yes—
99C3×C22⋊C8—22no—
100C3×#97—22no—
101(C2×D3)⋊C8—22no—
102(C2×D3)⋊C8—22no—
103#815—22no—
104Dic3⋊C8—22no—
105Dic3⋊C8—22no—
106C8×Dic3—22no—
107C4⋊(C3⋊C8)—22no—
108C4⋊(C3⋊C8)C2312yes—
109C4⋊(C3⋊C8)C2312yes—
110#557—62no—
111#556—62no—
112#552—62no—
113#552—62no—
114#541—34no—
115#536—32no—
116#533—32no—
117#533—32no—
118#533—31no—
119C2×C4⋊Q8C2×D311yescharacteristic
120#519—32no—
121C4×#41—31no—
122#446—62no—
123#435—62no—
124#435—62no—
125C8×D4—62no—
126#432—32no—
127#432C2×D312yes—
128#431—32no—
129#423—32no—
130C2×C4⋊C8—32no—
131C2×C4⋊C8—32no—
132#333—31no—
133#333—32no—
134#333—31no—
135#333C2×D312yes—
136#333C2×D311yescharacteristic
137#333C2×D311yescharacteristic
138#332—32no—
139#332—32no—
140#332—32no—
141#332—32no—
142C2×C3×Q8—22no—
143C2×C3×Q8—44no—
144C2×C3×Q8C22×C412yes—
145C2×C3×Q8C22×C412yes—
146C22×C3×C4—22no—
147C22×C3×C4C22×C411yescharacteristic
148C22×C3×C4C2×D412yes—
149C2×#67C22⋊C412yes—
150#235—42no—
151C2×D12C22⋊C411yescharacteristic
152C2×C4×D3C22×C412yes—
153C2×C4×D3C2×D412yes—
154C2×C4×D3C2×D412yes—
155C2×Dic6—22no—
156C2×Dic6C22⋊C411yescharacteristic
157C2×C3×C8—22no—
158C2×C3×C8—22no—
159C3×C4⋊C4—24no—
160C3×C4⋊C4—22no—
161C3×C4⋊C4—22no—
162C3×C42C22×C411yescharacteristic
163C3×C42C2×D412yes—
164C3×C42C2411yescharacteristic
165#217—22no—
166#217C22⋊C411yescharacteristic
167#217C22⋊C411yescharacteristic
168(C2×D3)⋊C4—24no—
169(C2×D3)⋊C4—24no—
170(C2×D3)⋊C4C22⋊C412yes—
171(C2×D3)⋊C4C22⋊C412yes—
172C4⋊Dic3—22no—
173C4⋊Dic3—24no—
174C4⋊Dic3—24no—
175C4⋊Dic3C22⋊C412yes—
176C4⋊Dic3C22×C414yes—
177Dic3⋊C4—24no—
178Dic3⋊C4—24no—
179Dic3⋊C4—24no—
180Dic3⋊C4C22⋊C412yes—
181Dic3⋊C4C22⋊C412yes—
182C4×Dic3—22no—
183C4×Dic3—24no—
184C4×Dic3—24no—
185C4×Dic3C22×C412yes—
186C4×Dic3C2×D412yes—
187C4×Dic3C2×D412yes—
188C2×C3⋊C8—22no—
189C2×C3⋊C8—22no—
190C2×C3⋊C8—82no—
191ΓL(1,𝔽25)—82no—
192D3×C8—82no—
193C2×#41—31no—
194C22×Q8C4×D312yes—
195C2×M16—62no—
196C22×C8—62no—
197C22×C8—62no—
198C4⋊Q8—22no—
199C4⋊Q8—32no—
200C4⋊Q8—62no—
201C4⋊Q8C4×D312yes—
202C4⋊Q8C22×D311yescharacteristic
203C4⋊Q8C22×D311yescharacteristic
204#126—64no—
205#126—64no—
206#125—32no—
207#125—62no—
208#125—62no—
209#125—64no—
210#125—64no—
211#124—62no—
212#123—64no—
213C22⋊Q8—34no—
214C22⋊Q8—34no—
215C22⋊Q8—62no—
216C22⋊Q8—62no—
217C22⋊Q8—62no—
218C4×Q8—32no—
219C4×Q8—34no—
220C4×Q8—34no—
221C4×Q8—62no—
222C4×Q8—62no—
223C4×Q8—64no—
224C4×Q8—62no—
225C4×D4—32no—
226C4×D4—32no—
227C4×D4—32no—
228#117—31no—
229#117—32no—
230#117—32no—
231#117—62no—
232#117—62no—
233#117—62no—
234#117—62no—
235C2×C4⋊C4—22no—
236C2×C42—32no—
237C2×C42C22×D311yescharacteristic
238C4⋊C8—32no—
239C4⋊C8—32no—
240C4⋊C8—62no—
241C4⋊C8—62no—
242C4⋊C8—62no—
243C4⋊C8C22×D312yes—
244C4⋊C8C22×D312yes—
245C22⋊C8—22no—
246C22⋊C8—62no—
247C22⋊C8—62no—
248#97—22no—
249#97—62no—
250C4×C8—62no—
251C23×C3C2×C22⋊C411yescharacteristic
252C22×D3C2×C22⋊C412yes—
253C3×Q8—42no—
254C3×Q8—42no—
255C3×Q8—44no—
256C2×C3×C4—22no—
257C2×C3×C4—22no—
258C2×C3×C4—22no—
259C2×C3×C4—22no—
260C2×C3×C4—22no—
261C2×C3×C4—44no—
262C2×C3×C4C2×C22⋊C411yescharacteristic
263C2×C3×C4C2×C22⋊C412yes—
264C2×C3×C4C23×C411yescharacteristic
265C2×C3×C4C22×D412yes—
266#67—24no—
267C2×Dic3—24no—
268C2×Dic3C2×C22⋊C412yes—
269C2×Dic3C2×C22⋊C412yes—
270C2×Dic3C2×C22⋊C412yes—
271D12—41no—
272C4×D3—22no—
273C4×D3—22no—
274C4×D3—22no—
275C4×D3—22no—
276C4×D3—42no—
277Dic6—41no—
278Dic6—42no—
279Dic6—42no—
280C3×C8—42no—
281C3×C8—42no—
282C3⋊C8—42no—
283C3⋊C8—42no—
284#41—122no—
285C2×Q8—22no—
286C2×Q8—31no—
287C2×Q8—44no—
288C2×Q8—62no—
289C2×Q8C2×C4×D312yes—
290C2×Q8C2×C4×D312yes—
291C2×D4—32no—
292C2×D4—31no—
293C22×C4—22no—
294C22×C4—32no—
295C22×C4—32no—
296C22×C4—32no—
297C22×C4C2×C4×D311yescharacteristic
298C22×C4D3×D412yes—
299M16—242no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—62no—
303C2×C8—62no—
304C2×C8—242no—
305C2×C8—242no—
306C4⋊C4—24no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—32no—
310C4⋊C4—32no—
311C4⋊C4—32no—
312C4⋊C4—34no—
313C4⋊C4—64no—
314C4⋊C4—64no—
315C4⋊C4—64no—
316C4⋊C4—64no—
317C4⋊C4—64no—
318C4⋊C4—62no—
319C22⋊C4—31no—
320C22⋊C4—31no—
321C22⋊C4—32no—
322C22⋊C4—32no—
323C22⋊C4—64no—
324C22⋊C4—64no—
325C22⋊C4—62no—
326C42—32no—
327C42—32no—
328C42—32no—
329C42—64no—
330C42—64no—
331C42—62no—
332C42C2×C4×D311yescharacteristic
333C42D3×D412yes—
334C42C23×D311yescharacteristic
335C22×C3—22no—
336C22×C3#41312yes—
337C22×C3C22×C22⋊C411yescharacteristic
338C2×D3—22no—
339C2×D3—22no—
340C2×D3—22no—
341C3×C4—21no—
342C3×C4—21no—
343C3×C4—22no—
344C3×C4—42no—
345C3×C4—42no—
346C3×C4#57712yes—
347C3×C4#57812yes—
348Dic3—22no—
349Dic3—22no—
350Dic3—22no—
351Dic3—44no—
352C23—32no—
353C23D3×C22⋊C411yescharacteristic
354Q8—42no—
355Q8—42no—
356Q8—44no—
357Q8—121no—
358Q8—122no—
359Q8—122no—
360D4—64no—
361D4—121no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—32no—
368C2×C4—32no—
369C2×C4—32no—
370C2×C4—44no—
371C2×C4—62no—
372C2×C4—62no—
373C2×C4—62no—
374C2×C4—62no—
375C2×C4—64no—
376C2×C4—122no—
377C2×C4D3×C22⋊C411yescharacteristic
378C2×C4D3×C22⋊C412yes—
379C2×C4C22×C4×D311yescharacteristic
380C2×C4C2×D3×D412yes—
381C8—42no—
382C8—42no—
383C8—122no—
384C8—122no—
385C2×C3—22no—
386C2×C3C2×#41311yescharacteristic
387C2×C3#289812yes—
388D3—42no—
389C22—22no—
390C22—62no—
391C22—62no—
392C22—62no—
393C22#474512yes—
394C22C2×D3×C22⋊C411yescharacteristic
395C4—21no—
396C4—21no—
397C4—22no—
398C4—42no—
399C4—42no—
400C4—62no—
401C4—62no—
402C4—62no—
403C4—124no—
404C4#577812yes—
405C4#577912yes—
406C3#1778011yescharacteristic, complemented, Sylow
407C2—22no—
408C2—122no—
409C2C2×#474511yescharacteristic
410C2#7895712yes—
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1558C2
socleC22×C3C22×C22⋊C4
prime radicalC2×#235C2×C4
cosocleC3×C42C24
hypercenter#1558D3
hypocenterC3#17780
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1558

Complements

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

NH ≅ G/Ntype
3. C3×#1558C2semidirect
4. C2×#4546C2semidirect
5. #70820C2semidirect
6. #68488C2semidirect
7. #68486C2semidirect
8. #68484C2semidirect
9. #68481C2semidirect
46. #1558D3semidirect
406. C3#17780semidirect

Maximal subgroups

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

subgroupindex
#862042
#793402
C3×#15582
C2×#45462
#708202
#684882
#684862
#684842
#684812
#177803