Finite Group Databasebeta

← #10764372

Subgroups — #10764372

Order 768.

618 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D3×#432C211yesmaximal, characteristic
2#79038C211yesmaximal, characteristic
3#78517C211yesmaximal, characteristic
4C3×#1569C211yesmaximal, characteristic, complemented
5#70795C211yesmaximal, characteristic, complemented
6#70492C211yesmaximal, characteristic, complemented
7#69851C211yesmaximal, characteristic, complemented
8#68574C212yesmaximal, complemented
9#68573C212yesmaximal, complemented
10#68534C212yesmaximal, complemented
11#68533C212yesmaximal, complemented
12#17821—31nomaximal, Sylow
13C2×C42×D3C2211yescharacteristic
14C3×#432C2211yescharacteristic
15C3×#423C2211yescharacteristic
16C3×C4×M16C2211yescharacteristic
17#5126—21no—
18#5126C412yes—
19#5118C412yes—
20Dic3×M16—21no—
21#4926C2211yescharacteristic
22#4924C2211yescharacteristic
23C4×#208C2211yescharacteristic
24#4835—21no—
25#4835C2211yescharacteristic
26#4835C2212yes—
27#4835C2211yescharacteristic
28#4833—21no—
29#4833C2212yes—
30D3×C4⋊C8C2211yescharacteristic
31D3×C4⋊C8C2211yescharacteristic
32D3×C22⋊C8—21no—
33#4719—21no—
34#4705—21no—
35#4705C412yes—
36#4705C412yes—
37D3×#97—21no—
38#4685C2212yes—
39#4685C2212yes—
40C3×#335C2212yescomplemented
41C3×#334C2212yescomplemented
42#4480C2212yescomplemented
43#4480C2212yescomplemented
44#4479C2212yescomplemented
45#4479C2212yescomplemented
46#4458C2212yescomplemented
47#4457C2212yescomplemented
48#2920—31no—
49#2920—31no—
50C2×#432—31no—
51#1573—32no—
52#1573—32no—
53#1572—32no—
54#1572—32no—
55#1569—31no—
56#1569—31no—
57#1569—31no—
58#1569D311yescharacteristic, complemented
59C22×C4×D3C2×C411yescharacteristic
60C22×C4×D3D411yescharacteristic
61C22×C4×D3Q811yescharacteristic
62C2×C3×M16—21no—
63C2×C3×M16C2×C412yes—
64C2×C3×C42C2311yescharacteristic
65C2×C4×Dic3C2×C411yescharacteristic
66C2×C4×Dic3D411yescharacteristic
67C2×C4×Dic3Q811yescharacteristic
68C2×#208—21no—
69C2×#208C2×C412yes—
70C2×D3×C8—21no—
71C2×D3×C8—21no—
72C42×D3C2×C411yescharacteristic
73C42×D3C2×C411yescharacteristic
74C42×D3D411yescharacteristic
75C42×D3D411yescharacteristic
76C42×D3Q811yescharacteristic
77C42×D3Q811yescharacteristic
78C42×D3C2311yescharacteristic
79C42×D3C2311yescharacteristic
80C3×C4⋊C8—21no—
81C3×C4⋊C8C2311yescharacteristic
82C3×C4⋊C8C2312yes—
83C3×C4⋊C8C2311yescharacteristic
84C3×C22⋊C8—21no—
85C3×#97—21no—
86C3×#97C2×C412yes—
87C3×C4×C8C2312yes—
88#830—21no—
89(C2×D3)⋊C8—22no—
90(C2×D3)⋊C8—22no—
91(C2×D3)⋊C8—21no—
92(C2×D3)⋊C8—21no—
93(C2×D3)⋊C8C2×C414yes—
94(C2×D3)⋊C8C2×C414yes—
95#815—21no—
96#815—21no—
97#815—22no—
98Dic3⋊C8—21no—
99Dic3⋊C8—21no—
100Dic3⋊C8C2×C414yes—
101Dic3⋊C8C2×C414yes—
102C8×Dic3—22no—
103C4⋊(C3⋊C8)—21no—
104C4⋊(C3⋊C8)C2312yes—
105C4⋊(C3⋊C8)C2311yescharacteristic
106C4⋊(C3⋊C8)C2311yescharacteristic
107#803—21no—
108#803C2×C412yes—
109C4×C3⋊C8C2312yes—
110C22×C42—31no—
111#432—32no—
112#432—32no—
113#432—31no—
114#432—31no—
115#432—31no—
116#432—32no—
117#432—61no—
118#432—61no—
119#432—61no—
120#432C2×D311yescharacteristic
121#431—32no—
122#431—32no—
123#431—32no—
124#431—61no—
125#423—32no—
126#423—31no—
127#423C2×D311yescharacteristic
128C2×C4⋊C8—31no—
129C2×C4⋊C8—31no—
130#407—32no—
131#407—61no—
132C2×C22⋊C8—61no—
133C4×M16—31no—
134C4×M16—61no—
135C4×M16C2×D311yescharacteristic
136C2×#97—61no—
137#335—32no—
138#335—32no—
139#335—32no—
140#335C2×D312yescomplemented
141#334—32no—
142#334—32no—
143#334—32no—
144#334C2×D312yescomplemented
145C23×D3C4⋊C411yescharacteristic
146C22×C3×C4C22×C411yescharacteristic
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2×Q811yescharacteristic
149C22×Dic3C4⋊C411yescharacteristic
150C22×Dic3C4⋊C411yescharacteristic
151C22×Dic3C4⋊C411yescharacteristic
152C2×C4×D3—42no—
153C2×C4×D3—41no—
154C2×C4×D3—41no—
155C2×C4×D3—42no—
156C2×C4×D3C4⋊C411yescharacteristic
157C2×C4×D3C4⋊C411yescharacteristic
158C2×C4×D3C4⋊C411yescharacteristic
159C2×C4×D3C4⋊C411yescharacteristic
160C2×C4×D3C4⋊C411yescharacteristic
161C2×C4×D3C4⋊C411yescharacteristic
162C2×C4×D3C22×C411yescharacteristic
163C2×C4×D3C22×C411yescharacteristic
164C2×C4×D3C2×D411yescharacteristic
165C2×C4×D3C2×D411yescharacteristic
166C2×C4×D3C2×Q811yescharacteristic
167C2×C4×D3C2×Q811yescharacteristic
168C3×M16—42no—
169C3×M16—42no—
170C2×C3×C8—22no—
171C2×C3×C8—21no—
172C2×C3×C8—21no—
173C2×C3×C8C22×C414yes—
174C3×C42C22×C411yescharacteristic
175C3×C42C2×D411yescharacteristic
176C3×C42C2×Q811yescharacteristic
177C3×C42C2411yescharacteristic
178C4×Dic3C4⋊C411yescharacteristic
179C4×Dic3C4⋊C411yescharacteristic
180C4×Dic3C4⋊C411yescharacteristic
181C4×Dic3C4⋊C411yescharacteristic
182C4×Dic3C4⋊C411yescharacteristic
183C4×Dic3C4⋊C411yescharacteristic
184C4×Dic3C22×C411yescharacteristic
185C4×Dic3C22×C411yescharacteristic
186C4×Dic3C2×D411yescharacteristic
187C4×Dic3C2×D411yescharacteristic
188C4×Dic3C2×Q811yescharacteristic
189C4×Dic3C2×Q811yescharacteristic
190#208—42no—
191#208—42no—
192C2×C3⋊C8—21no—
193C2×C3⋊C8—21no—
194C2×C3⋊C8—22no—
195C2×C3⋊C8C22×C414yes—
196D3×C8—42no—
197D3×C8—42no—
198C23×C4—31no—
199C23×C4—31no—
200C23×C4—31no—
201C2×M16—21no—
202C2×M16—32no—
203C2×M16—61no—
204C2×M16C4×D312yes—
205C22×C8—61no—
206C22×C8—61no—
207C2×C42—31no—
208C2×C42—31no—
209C2×C42—31no—
210C2×C42—31no—
211C2×C42—31no—
212C2×C42—31no—
213C2×C42—31no—
214C2×C42—31no—
215C2×C42—31no—
216C2×C42—31no—
217C2×C42—31no—
218C2×C42C22×D311yescharacteristic
219C4⋊C8—21no—
220C4⋊C8—34no—
221C4⋊C8—34no—
222C4⋊C8—31no—
223C4⋊C8—31no—
224C4⋊C8—32no—
225C4⋊C8—61no—
226C4⋊C8—61no—
227C4⋊C8—61no—
228C4⋊C8C22×D311yescharacteristic
229C4⋊C8C22×D312yes—
230C4⋊C8C22×D311yescharacteristic
231C22⋊C8—21no—
232C22⋊C8—34no—
233C22⋊C8—34no—
234C22⋊C8—61no—
235C22⋊C8—61no—
236C22⋊C8—62no—
237C22⋊C8—62no—
238C22⋊C8—61no—
239#97—21no—
240#97—32no—
241#97—61no—
242#97—61no—
243#97—62no—
244#97—61no—
245#97C4×D312yes—
246C4×C8—32no—
247C4×C8—62no—
248C4×C8C22×D312yes—
249C23×C3C2×C4⋊C411yescharacteristic
250C22×D3—22no—
251C22×D3—41no—
252C22×D3—41no—
253C22×D3C2×C4⋊C411yescharacteristic
254C22×D3C2×C4⋊C411yescharacteristic
255C2×C3×C4—22no—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4—41no—
259C2×C3×C4C2×C4⋊C411yescharacteristic
260C2×C3×C4C2×C4⋊C411yescharacteristic
261C2×C3×C4C2×C4⋊C411yescharacteristic
262C2×C3×C4C23×C411yescharacteristic
263C2×C3×C4C22×D411yescharacteristic
264C2×C3×C4C22×Q811yescharacteristic
265C2×Dic3—41no—
266C2×Dic3—41no—
267C2×Dic3—41no—
268C2×Dic3C2×C4⋊C411yescharacteristic
269C2×Dic3C2×C4⋊C411yescharacteristic
270C2×Dic3C2×C4⋊C411yescharacteristic
271C2×Dic3C2×C4⋊C411yescharacteristic
272C2×Dic3C2×C4⋊C411yescharacteristic
273C2×Dic3C2×C4⋊C411yescharacteristic
274C4×D3—22no—
275C4×D3—22no—
276C4×D3—22no—
277C4×D3—22no—
278C4×D3—21no—
279C4×D3—21no—
280C4×D3—21no—
281C4×D3—21no—
282C4×D3—41no—
283C4×D3—41no—
284C4×D3—41no—
285C4×D3—41no—
286C4×D3—41no—
287C4×D3—41no—
288C3×C8—24no—
289C3×C8—41no—
290C3×C8—41no—
291C3×C8—42no—
292C3⋊C8—24no—
293C3⋊C8—41no—
294C3⋊C8—41no—
295C3⋊C8—42no—
296C24—31no—
297C22×C4—31no—
298C22×C4—31no—
299C22×C4—31no—
300C22×C4—31no—
301C22×C4—31no—
302C22×C4—31no—
303C22×C4—31no—
304C22×C4—31no—
305C22×C4—31no—
306C22×C4—31no—
307C22×C4—31no—
308C22×C4—31no—
309C22×C4—31no—
310C22×C4—31no—
311C22×C4—31no—
312C22×C4—122no—
313C22×C4—121no—
314C22×C4—121no—
315C22×C4—122no—
316C22×C4C2×C4×D311yescharacteristic
317C22×C4D3×D411yescharacteristic
318C22×C4D3×Q811yescharacteristic
319M16—42no—
320M16—42no—
321M16—122no—
322M16—122no—
323C2×C8—22no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—34no—
327C2×C8—62no—
328C2×C8—61no—
329C2×C8—61no—
330C2×C8—122no—
331C2×C8—122no—
332C2×C8C2×C4×D314yes—
333C42—31no—
334C42—31no—
335C42—31no—
336C42—31no—
337C42—31no—
338C42—31no—
339C42—31no—
340C42—31no—
341C42—31no—
342C42—31no—
343C42—31no—
344C42—31no—
345C42C2×C4×D311yescharacteristic
346C42D3×D411yescharacteristic
347C42D3×Q811yescharacteristic
348C42C23×D311yescharacteristic
349C22×C3—21no—
350C22×C3—21no—
351C22×C3—21no—
352C22×C3C22×C4⋊C411yescharacteristic
353C2×D3—21no—
354C2×D3—21no—
355C2×D3—21no—
356C2×D3—21no—
357C2×D3—21no—
358C2×D3—44no—
359C2×D3#43012yes—
360C3×C4—21no—
361C3×C4—21no—
362C3×C4—21no—
363C3×C4—21no—
364C3×C4#56812yes—
365C3×C4#57511yescharacteristic
366C3×C4#57511yescharacteristic
367Dic3—21no—
368Dic3—21no—
369Dic3—21no—
370Dic3—21no—
371Dic3—21no—
372Dic3—21no—
373C23—31no—
374C23—31no—
375C23—62no—
376C23—121no—
377C23—121no—
378C23D3×C4⋊C411yescharacteristic
379C2×C4—22no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—31no—
383C2×C4—31no—
384C2×C4—31no—
385C2×C4—31no—
386C2×C4—31no—
387C2×C4—31no—
388C2×C4—41no—
389C2×C4—62no—
390C2×C4—62no—
391C2×C4—62no—
392C2×C4—62no—
393C2×C4—61no—
394C2×C4—61no—
395C2×C4—61no—
396C2×C4—61no—
397C2×C4—121no—
398C2×C4—121no—
399C2×C4—121no—
400C2×C4—121no—
401C2×C4—121no—
402C2×C4—121no—
403C2×C4—121no—
404C2×C4—121no—
405C2×C4—121no—
406C2×C4D3×C4⋊C411yescharacteristic
407C2×C4D3×C4⋊C411yescharacteristic
408C2×C4D3×C4⋊C411yescharacteristic
409C2×C4C22×C4×D311yescharacteristic
410C2×C4C2×D3×D411yescharacteristic
411C2×C4C2×D3×Q811yescharacteristic
412C8—24no—
413C8—41no—
414C8—42no—
415C8—41no—
416C8—64no—
417C8—121no—
418C8—121no—
419C8—122no—
420C2×C3—41no—
421C2×C3#290511yescharacteristic
422C2×C3#291111yescharacteristic
423C2×C3C2×#43011yescharacteristic
424D3—22no—
425D3—41no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—32no—
430C22—61no—
431C22—61no—
432C22—61no—
433C22—61no—
434C22—61no—
435C22—124no—
436C22C2×D3×C4⋊C411yescharacteristic
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—61no—
442C4—61no—
443C4—61no—
444C4—61no—
445C4—61no—
446C4—61no—
447C4#574512yes—
448C4#577311yescharacteristic
449C4#577711yescharacteristic
450C3#1782111yescharacteristic, complemented, Sylow
451C2—41no—
452C2—62no—
453C2—121no—
454C2#7904811yescharacteristic
455C2#7920811yescharacteristic
456C2D3×#43011yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1569C2
socleC22×C3C22×C4⋊C4
prime radicalC23×D3C4⋊C4
cosocleC3×C42C24
hypercenter#1569D3
hypocenterC3#17821
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1569

Complements

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

NH ≅ G/Ntype
4. C3×#1569C2semidirect
5. #70795C2semidirect
6. #70492C2semidirect
7. #69851C2semidirect
8. #68574C2semidirect
9. #68573C2semidirect
10. #68534C2semidirect
11. #68533C2semidirect
40. C3×#335C22semidirect
41. C3×#334C22semidirect
42. #4480C22semidirect
43. #4480C22semidirect
44. #4479C22semidirect
45. #4479C22semidirect
46. #4458C22semidirect
47. #4457C22semidirect
58. #1569D3semidirect
140. #335C2×D3semidirect
144. #334C2×D3semidirect
450. C3#17821semidirect

Maximal subgroups

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

subgroupindex
D3×#4322
#790382
#785172
C3×#15692
#707952
#704922
#698512
#685742
#685732
#685342
#685332
#178213