Finite Group Databasebeta

← #10686608

Subgroups — #10686608

Order 768.

930 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5516C211yesmaximal, characteristic
2C3×#1458C211yesmaximal, characteristic
3#71571C211yesmaximal, characteristic
4#70767C212yesmaximal, complemented
5#70766C212yesmaximal, complemented
6#8323—31nomaximal, Sylow
7C23×Dic6C411yescharacteristic
8C2×C3×#117C2211yescharacteristic
9C2×#926C411yescharacteristic
10#5516—24no—
11#5516—24no—
12C2×#887—21no—
13C2×#887—21no—
14C2×#877—21no—
15C2×#877—21no—
16C2×C4×Dic6C411yescharacteristic
17C2×C4×Dic6C411yescharacteristic
18C2×C4×Dic6C2211yescharacteristic
19C2×C4×Dic6C2211yescharacteristic
20C3×#431C2212yes—
21#4925C2212yes—
22C3×#324—21no—
23C3×#323—21no—
24#4528—21no—
25#4525—21no—
26#4484—24no—
27#4456—24no—
28C2×#519—31no—
29#1479—32no—
30#1479—32no—
31#1458—31no—
32#1458D311yescharacteristic
33C23×C3×C4C2×C411yescharacteristic
34C23×Dic3—21no—
35C22×Dic6—21no—
36C22×Dic6—22no—
37C22×Dic6—22no—
38C22×Dic6—22no—
39C22×Dic6—22no—
40C22×Dic6—22no—
41C22×Dic6C814yescomplemented
42C22×Dic6C2×C411yescharacteristic
43C22×Dic6C2×C411yescharacteristic
44C3×#117—22no—
45C3×#117—22no—
46C2×C3×C4⋊C4D411yescharacteristic
47C2×C3×C4⋊C4D411yescharacteristic
48C2×C3×C22⋊C4D411yescharacteristic
49C2×C3×C22⋊C4D411yescharacteristic
50C2×C3×C42C2×C411yescharacteristic
51C2×C3×C42C2311yescharacteristic
52C2×#217—21no—
53#926—22no—
54#926—22no—
55C2×C4⋊Dic3C2×C411yescharacteristic
56C2×C4⋊Dic3C2×C411yescharacteristic
57C2×Dic3⋊C4—21no—
58C2×Dic3⋊C4—21no—
59C2×Dic3⋊C4—21no—
60C2×Dic3⋊C4—21no—
61C2×C4×Dic3—21no—
62C2×C4×Dic3—21no—
63C2×C4×Dic3—21no—
64C2×C4×Dic3—21no—
65C2×C4×Dic3—21no—
66#887—28no—
67#887—28no—
68#877—28no—
69#877—28no—
70C4×Dic6—22no—
71C4×Dic6—22no—
72C4×Dic6—22no—
73C4×Dic6—22no—
74C4×Dic6—22no—
75C4×Dic6—22no—
76C4×Dic6D414yes—
77C4×Dic6D414yes—
78C3×C4⋊C8—22no—
79C3×C22⋊C8—22no—
80C3×C4×C8—22no—
81#830—22no—
82C4⋊(C3⋊C8)—22no—
83C4×C3⋊C8—22no—
84C23×Q8—31no—
85#519—64no—
86#519—64no—
87C2×C4×Q8—31no—
88C2×C4×Q8—31no—
89C2×C4×Q8—31no—
90C2×C4×Q8—31no—
91C2×C4×Q8—61no—
92C2×C4×Q8—61no—
93C2×#117—31no—
94C2×#117—61no—
95C2×#117—61no—
96C2×#117C2×D311yescharacteristic
97#431—32no—
98#431C2×D312yes—
99Q8⋊C8—64no—
100Q8⋊C8—64no—
101#324—21no—
102#324—61no—
103#323—21no—
104#323—61no—
105C24×C3C22⋊C411yescharacteristic
106C22×C3×C4—21no—
107C22×C3×C4—21no—
108C22×C3×C4—21no—
109C22×C3×C4—21no—
110C22×C3×C4C22⋊C411yescharacteristic
111C22×C3×C4C22⋊C411yescharacteristic
112C22×C3×C4C22⋊C411yescharacteristic
113C22×C3×C4C2×C812yes—
114C22×C3×C4M1612yes—
115C22×C3×C4C22×C411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×Dic3—24no—
119C22×Dic3—24no—
120C22×Dic3—24no—
121C22×Dic3—21no—
122C22×Dic3—21no—
123C22×Dic3—21no—
124C22×Dic3—21no—
125C2×Dic6—24no—
126C2×Dic6—24no—
127C2×Dic6—28no—
128C2×Dic6—28no—
129C2×Dic6—24no—
130C2×Dic6—21no—
131C2×Dic6—21no—
132C2×Dic6—21no—
133C2×Dic6—21no—
134C2×Dic6—21no—
135C2×Dic6—21no—
136C2×Dic6—28no—
137C2×Dic6—28no—
138C2×Dic6C22⋊C412yes—
139C2×Dic6C22⋊C412yes—
140C2×Dic6C2×C812yes—
141C2×Dic6C2×C812yes—
142C2×Dic6M1612yes—
143C2×Dic6M1612yes—
144C2×C3×C8—42no—
145C3×C4⋊C4—22no—
146C3×C4⋊C4—22no—
147C3×C22⋊C4—22no—
148C3×C22⋊C4—22no—
149C3×C42—21no—
150C3×C42—21no—
151C3×C42C2×D412yes—
152C3×C42C2×D412yes—
153#217—24no—
154C4⋊Dic3—21no—
155C4⋊Dic3—21no—
156C4⋊Dic3C22⋊C412yes—
157C4⋊Dic3C22⋊C412yes—
158Dic3⋊C4—24no—
159Dic3⋊C4—24no—
160Dic3⋊C4—24no—
161Dic3⋊C4—24no—
162C4×Dic3—24no—
163C4×Dic3—24no—
164C4×Dic3—24no—
165C4×Dic3—24no—
166C4×Dic3—24no—
167C2×C3⋊C8—42no—
168C22×Q8—34no—
169C22×Q8—31no—
170C22×Q8—31no—
171C22×Q8—62no—
172C22×Q8—62no—
173C22×Q8—62no—
174C22×Q8—62no—
175C22×Q8—62no—
176C22×Q8—61no—
177C23×C4—61no—
178C23×C4C4×D311yescharacteristic
179C4×Q8—34no—
180C4×Q8—34no—
181C4×Q8—62no—
182C4×Q8—62no—
183C4×Q8—62no—
184C4×Q8—62no—
185C4×Q8—68no—
186C4×Q8—68no—
187C4×Q8—62no—
188C4×Q8—62no—
189#117—22no—
190#117—22no—
191#117—68no—
192#117—68no—
193#117—62no—
194#117—62no—
195C2×C4⋊C4—31no—
196C2×C4⋊C4—31no—
197C2×C4⋊C4—61no—
198C2×C4⋊C4—61no—
199C2×C4⋊C4—61no—
200C2×C4⋊C4—61no—
201C2×C4⋊C4D1211yescharacteristic
202C2×C4⋊C4#6711yescharacteristic
203C2×C22⋊C4—61no—
204C2×C22⋊C4D1211yescharacteristic
205C2×C22⋊C4#6711yescharacteristic
206C2×C42—61no—
207C2×C42—61no—
208C2×C42—61no—
209C2×C42—61no—
210C2×C42—61no—
211C2×C42C4×D311yescharacteristic
212C2×C42C22×D311yescharacteristic
213C4⋊C8—22no—
214C4⋊C8—62no—
215C22⋊C8—22no—
216C22⋊C8—62no—
217C4×C8—22no—
218C4×C8—62no—
219C23×C3—21no—
220C23×C3—21no—
221C23×C3—41no—
222C23×C3—41no—
223C23×C3C22⋊C812yes—
224C23×C3C2×C22⋊C411yescharacteristic
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—22no—
228C2×C3×C4—22no—
229C2×C3×C4—22no—
230C2×C3×C4—22no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4C22⋊C812yes—
238C2×C3×C4C2×C22⋊C411yescharacteristic
239C2×C3×C4C2×C22⋊C411yescharacteristic
240C2×C3×C4#12012yes—
241C2×C3×C4#12012yes—
242C2×C3×C4C22×C811yescharacteristic
243C2×C3×C4C2×M1611yescharacteristic
244C2×C3×C4C2×M1611yescharacteristic
245C2×C3×C4C2×M1611yescharacteristic
246C2×Dic3—216no—
247C2×Dic3—22no—
248C2×Dic3—22no—
249C2×Dic3—22no—
250C2×Dic3—22no—
251C2×Dic3—22no—
252C2×Dic3—22no—
253C2×Dic3—22no—
254C2×Dic3—22no—
255C2×Dic3—42no—
256C2×Dic3—42no—
257Dic6—216no—
258Dic6—22no—
259Dic6—22no—
260Dic6—22no—
261Dic6—22no—
262Dic6—22no—
263Dic6—22no—
264Dic6C22⋊C814yes—
265Dic6C22⋊C814yes—
266C3×C8—82no—
267C3⋊C8—82no—
268C24(C2×D3)⋊C411yescharacteristic
269C2×Q8—32no—
270C2×Q8—32no—
271C2×Q8—32no—
272C2×Q8—32no—
273C2×Q8—32no—
274C2×Q8—32no—
275C2×Q8—68no—
276C2×Q8—68no—
277C2×Q8—64no—
278C2×Q8—64no—
279C2×Q8—68no—
280C2×Q8—68no—
281C2×Q8—64no—
282C2×Q8—61no—
283C2×Q8—61no—
284C2×Q8—61no—
285C2×Q8—61no—
286C2×Q8—61no—
287C2×Q8—61no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—64no—
293C22×C4—64no—
294C22×C4—64no—
295C22×C4—61no—
296C22×C4—61no—
297C22×C4—61no—
298C22×C4—61no—
299C22×C4D3×C812yes—
300C22×C4ΓL(1,𝔽25)12yes—
301C22×C4(C2×D3)⋊C411yescharacteristic
302C22×C4(C2×D3)⋊C411yescharacteristic
303C22×C4(C2×D3)⋊C411yescharacteristic
304C22×C4C2×C4×D311yescharacteristic
305C22×C4C2×D1211yescharacteristic
306C22×C4C2×#6711yescharacteristic
307C2×C8—42no—
308C2×C8—122no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—32no—
312C4⋊C4—32no—
313C4⋊C4—64no—
314C4⋊C4—64no—
315C4⋊C4—61no—
316C4⋊C4—61no—
317C4⋊C4—64no—
318C4⋊C4—64no—
319C22⋊C4—22no—
320C22⋊C4—22no—
321C22⋊C4—64no—
322C42—21no—
323C42—21no—
324C42—64no—
325C42—64no—
326C42—64no—
327C42—64no—
328C42—64no—
329C42D3×D412yes—
330C42D3×D412yes—
331C22×C3—21no—
332C22×C3—21no—
333C22×C3—22no—
334C22×C3—42no—
335C22×C3—42no—
336C22×C3—41no—
337C22×C3—41no—
338C22×C3#37911yescharacteristic
339C22×C3C2×C22⋊C811yescharacteristic
340C22×C3#40711yescharacteristic
341C3×C4—22no—
342C3×C4—22no—
343C3×C4—22no—
344C3×C4—41no—
345C3×C4—41no—
346C3×C4C2×C22⋊C811yescharacteristic
347C3×C4C2×C22⋊C811yescharacteristic
348C3×C4#40711yescharacteristic
349C3×C4#40711yescharacteristic
350Dic3—24no—
351Dic3—24no—
352Dic3—42no—
353Dic3—42no—
354C23—21no—
355C23—21no—
356C23—41no—
357C23—41no—
358C23(C2×D3)⋊C812yes—
359C23C2×(C2×D3)⋊C411yescharacteristic
360Q8—34no—
361Q8—34no—
362Q8—616no—
363Q8—62no—
364Q8—62no—
365Q8—62no—
366Q8—62no—
367Q8—62no—
368Q8—62no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—616no—
382C2×C4—62no—
383C2×C4—62no—
384C2×C4—62no—
385C2×C4—62no—
386C2×C4—62no—
387C2×C4—62no—
388C2×C4—62no—
389C2×C4—62no—
390C2×C4—122no—
391C2×C4—122no—
392C2×C4(C2×D3)⋊C812yes—
393C2×C4D3×C22⋊C411yescharacteristic
394C2×C4D3×C22⋊C411yescharacteristic
395C2×C4#88212yes—
396C2×C4C2×D3×C811yescharacteristic
397C2×C4C2×ΓL(1,𝔽25)11yescharacteristic
398C2×C4D3×M1611yescharacteristic
399C2×C4D3×M1611yescharacteristic
400C2×C4#93712yes—
401C8—82no—
402C8—242no—
403C2×C3—21no—
404C2×C3—21no—
405C2×C3—42no—
406C2×C3#177911yescharacteristic
407C2×C3#179611yescharacteristic
408C2×C3#179611yescharacteristic
409C22—21no—
410C22—21no—
411C22—22no—
412C22—42no—
413C22—42no—
414C22—41no—
415C22—41no—
416C22#495511yescharacteristic
417C22C2×(C2×D3)⋊C811yescharacteristic
418C22#512611yescharacteristic
419C4—22no—
420C4—22no—
421C4—22no—
422C4—41no—
423C4—41no—
424C4—64no—
425C4—64no—
426C4—122no—
427C4—122no—
428C4D3×C22⋊C811yescharacteristic
429C4D3×C22⋊C811yescharacteristic
430C4#472611yescharacteristic
431C4#472611yescharacteristic
432C3#832311yescharacteristic, complemented, Sylow
433C2—21no—
434C2—21no—
435C2—42no—
436C2#6995511yescharacteristic
437C2#7005111yescharacteristic
438C2#7014111yescharacteristic
subgroupG/H
centerC22#4955
commutatorC2×C3×C4C22×C8
FrattiniC2×C42C22×D3
FittingC3×#1458C2
socleC22×C3#379
prime radicalC24×C3C22⋊C4
cosocleC2×C3×C42C23
hypercenter#1458D3
hypocenterC3#8323
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1458

Complements

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

NH ≅ G/Ntype
4. #70767C2semidirect
5. #70766C2semidirect
41. C22×Dic6C8semidirect
432. C3#8323semidirect

Maximal subgroups

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

subgroupindex
C2×#55162
C3×#14582
#715712
#707672
#707662
#83233