Finite Group Databasebeta

← #10691529

Subgroups — #10691529

Order 768.

464 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2984C211yesmaximal, characteristic, complemented
2C2×#5028C211yesmaximal, characteristic
3#71623C211yesmaximal, characteristic, complemented
4#70804C211yesmaximal, characteristic
5#70784C211yesmaximal, characteristic
6#67755C211yesmaximal, characteristic, complemented
7#67744C211yesmaximal, characteristic, complemented
8#9244—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10C22×#216C411yescharacteristic
11C22×#215C411yescharacteristic
12C2×#875C2211yescharacteristic
13C3×#446—21no—
14C3×#446—22no—
15C3×C8×Q8—21no—
16C3×#432—21no—
17C3×#432C2211yescharacteristic
18C3×#431—21no—
19C3×#431C2211yescharacteristic
20#5028—21no—
21#5028—21no—
22#5028—21no—
23#5028—21no—
24C2×#810C411yescharacteristic
25C2×#809C411yescharacteristic
26#4926C2211yescharacteristic
27C2×C4⋊(C3⋊C8)C2211yescharacteristic
28C4×#208C2211yescharacteristic
29#4535—21no—
30#4535—21no—
31#4468—21no—
32#4468—21no—
33#4464—22no—
34C2×#464—31no—
35#2984D311yescharacteristic, complemented
36#1527—31no—
37#1491—31no—
38#1469—31no—
39#1290—31no—
40#1283—31no—
41C22×C3×Q8C2×C411yescharacteristic
42C22×D12C2×C411yescharacteristic
43C22×Dic6C2×C411yescharacteristic
44C3×C4×Q8—21no—
45C3×C4×Q8—21no—
46C3×C4×Q8D411yescharacteristic
47C3×C4×Q8D411yescharacteristic
48C3×C4×Q8D411yescharacteristic
49C3×C4×Q8D411yescharacteristic
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4C2×C411yescharacteristic
52C2×C3×C42—21no—
53C2×C3×C42C2311yescharacteristic
54C2×#216—21no—
55C2×#216—21no—
56C2×#216—42no—
57C2×#215—21no—
58C2×#215—21no—
59C2×#215—42no—
60C2×(C2×D3)⋊C4—21no—
61C2×(C2×D3)⋊C4—21no—
62C2×#208—21no—
63C22×C3⋊C8C2×C411yescharacteristic
64C22×C3⋊C8C2×C411yescharacteristic
65#875—21no—
66#875—21no—
67#875—21no—
68#875—21no—
69C3×C4⋊C8—22no—
70C3×C4⋊C8—22no—
71C3×C4⋊C8—21no—
72C3×C4⋊C8—21no—
73C3×C22⋊C8—21no—
74C3×C22⋊C8D411yescharacteristic
75C3×C22⋊C8D411yescharacteristic
76C3×#97—22no—
77C3×#97D412yes—
78C3×C4×C8—22no—
79C3×C4×C8—21no—
80#830—21no—
81#810—21no—
82#810—21no—
83#809—21no—
84#809—21no—
85C4⋊(C3⋊C8)—21no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)—21no—
88#803—21no—
89#803—21no—
90C4×C3⋊C8—21no—
91C22×Q16—31no—
92C22×SD16—31no—
93C2×#124—31no—
94C2×C4×Q8C2×D311yescharacteristic
95#464—61no—
96#464—61no—
97#464—61no—
98#464—61no—
99#446—21no—
100#446—22no—
101C8×Q8—21no—
102#432—21no—
103#432—31no—
104#432C2×D311yescharacteristic
105#431—21no—
106#431C2×D311yescharacteristic
107C2×C4⋊C8—31no—
108C2×#103—31no—
109C2×#102—31no—
110C4×M16—31no—
111#329—62no—
112Q8⋊C8—61no—
113Q8⋊C8—61no—
114#323—61no—
115#323—61no—
116C23×D3—21no—
117C2×C3×Q8—21no—
118C2×C3×Q8—22no—
119C2×C3×Q8—41no—
120C2×C3×Q8C22⋊C411yescharacteristic
121C2×C3×Q8C22⋊C411yescharacteristic
122C22×C3×C4—21no—
123C22×C3×C4—21no—
124C22×C3×C4C22×C411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4#4111yescharacteristic
127C22×Dic3—21no—
128C2×D12—21no—
129C2×D12—21no—
130C2×D12—41no—
131C2×D12—41no—
132C2×Dic6—21no—
133C2×Dic6—21no—
134C2×Dic6—41no—
135C2×Dic6—41no—
136C2×C3×C8—21no—
137C2×C3×C8—21no—
138C2×C3×C8—22no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4C22⋊C411yescharacteristic
145C3×C4⋊C4C22⋊C411yescharacteristic
146C3×C42—21no—
147C3×C42—21no—
148C3×C42—21no—
149C3×C42—21no—
150C3×C42C2×D411yescharacteristic
151C3×C42C2×D411yescharacteristic
152C3×C42C2×D411yescharacteristic
153C3×C42#4111yescharacteristic
154#216—42no—
155#216—42no—
156#215—42no—
157#215—42no—
158(C2×D3)⋊C4—41no—
159(C2×D3)⋊C4—41no—
160(C2×D3)⋊C4—41no—
161(C2×D3)⋊C4—41no—
162#208—22no—
163#208—41no—
164C2×C3⋊C8—21no—
165C2×C3⋊C8—21no—
166C2×C3⋊C8—41no—
167C2×C3⋊C8—41no—
168C2×C3⋊C8—41no—
169C2×C3⋊C8—41no—
170C22×Q8—31no—
171C22×Q8C4×D311yescharacteristic
172C22×D4—31no—
173C2×Q16—61no—
174C2×Q16—61no—
175C2×Q16—122no—
176C2×SD16—61no—
177C2×SD16—61no—
178C2×SD16—122no—
179C2×M16—61no—
180C22×C8—31no—
181C22×C8—31no—
182#124—61no—
183#124—61no—
184#124—61no—
185#124—61no—
186C4×Q8—21no—
187C4×Q8—21no—
188C4×Q8D1211yescharacteristic
189C4×Q8D1211yescharacteristic
190C4×Q8#6711yescharacteristic
191C4×Q8#6711yescharacteristic
192C2×C4⋊C4—21no—
193C2×C4⋊C4C4×D311yescharacteristic
194C2×C22⋊C4—61no—
195C2×C22⋊C4—61no—
196C2×C42—21no—
197C2×C42C22×D311yescharacteristic
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—21no—
201C4⋊C8—21no—
202C4⋊C8—61no—
203C4⋊C8—61no—
204C4⋊C8—61no—
205#103—61no—
206#103—61no—
207#102—61no—
208#102—61no—
209C22⋊C8—21no—
210C22⋊C8—61no—
211C22⋊C8#6711yescharacteristic
212C22⋊C8#6711yescharacteristic
213#97—22no—
214#97—61no—
215#97—61no—
216#97#6712yes—
217C4×C8—22no—
218C4×C8—21no—
219C4×C8—61no—
220C23×C3C4×D411yescharacteristic
221C22×D3—41no—
222C22×D3—41no—
223C22×D3—41no—
224C22×D3—81no—
225C22×D3—81no—
226C3×Q8—21no—
227C3×Q8—21no—
228C3×Q8—21no—
229C3×Q8—42no—
230C3×Q8#10012yes—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—41no—
240C2×C3×C4—42no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4C2×C22⋊C411yescharacteristic
244C2×C3×C4C4×D411yescharacteristic
245C2×C3×C4#12011yescharacteristic
246C2×C3×C4#12111yescharacteristic
247C2×C3×C4#12111yescharacteristic
248C2×C3×C4#12311yescharacteristic
249C2×Dic3—41no—
250C2×Dic3—81no—
251D12—41no—
252D12—41no—
253D12—81no—
254Dic6—41no—
255Dic6—41no—
256Dic6—81no—
257C3×C8—41no—
258C3×C8—41no—
259C3×C8—42no—
260C3⋊C8—42no—
261C3⋊C8—41no—
262C3⋊C8—41no—
263C3⋊C8—81no—
264C24—61no—
265C2×Q8—21no—
266C2×Q8—22no—
267C2×Q8—41no—
268C2×Q8—61no—
269C2×Q8—61no—
270C2×Q8—121no—
271C2×Q8—121no—
272C2×Q8(C2×D3)⋊C411yescharacteristic
273C2×Q8(C2×D3)⋊C411yescharacteristic
274C2×D4—61no—
275C2×D4—61no—
276C2×D4—121no—
277C2×D4—121no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—61no—
281C22×C4C2×C4×D311yescharacteristic
282C22×C4#23511yescharacteristic
283C22×C4C2×#6711yescharacteristic
284Q16—122no—
285Q16—122no—
286SD16—122no—
287SD16—122no—
288M16—62no—
289M16—121no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—22no—
293C2×C8—61no—
294C2×C8—61no—
295C2×C8—121no—
296C2×C8—121no—
297C2×C8—121no—
298C2×C8—121no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4(C2×D3)⋊C411yescharacteristic
305C4⋊C4(C2×D3)⋊C411yescharacteristic
306C22⋊C4—121no—
307C22⋊C4—121no—
308C22⋊C4—121no—
309C22⋊C4—121no—
310C42—21no—
311C42—21no—
312C42—21no—
313C42—21no—
314C42C2×D1211yescharacteristic
315C42#23511yescharacteristic
316C42C2×#6711yescharacteristic
317C42C2×#6711yescharacteristic
318C22×C3—21no—
319C22×C3—21no—
320C22×C3#38611yescharacteristic
321C22×C3#44311yescharacteristic
322C22×C3#44411yescharacteristic
323C2×D3—81no—
324C2×D3—81no—
325C2×D3—81no—
326C2×D3—161no—
327C2×D3—161no—
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—21no—
331C3×C4—22no—
332C3×C4—21no—
333C3×C4—21no—
334C3×C4—41no—
335C3×C4—41no—
336C3×C4—41no—
337C3×C4C2×#10011yescharacteristic
338C3×C4#41311yescharacteristic
339Dic3—81no—
340C23—121no—
341C23—121no—
342C23—121no—
343C23—241no—
344C23—241no—
345C23C4×#6711yescharacteristic
346Q8—21no—
347Q8—21no—
348Q8—21no—
349Q8—42no—
350Q8—121no—
351Q8—121no—
352Q8—241no—
353Q8#82312yes—
354D4—121no—
355D4—121no—
356D4—241no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—121no—
370C2×C4—241no—
371C2×C4C2×(C2×D3)⋊C411yescharacteristic
372C2×C4C4×#6711yescharacteristic
373C2×C4#92911yescharacteristic
374C2×C4#93011yescharacteristic
375C2×C4#93011yescharacteristic
376C2×C4#95311yescharacteristic
377C8—41no—
378C8—41no—
379C8—42no—
380C8—122no—
381C8—121no—
382C8—121no—
383C8—241no—
384C2×C3—21no—
385C2×C3—21no—
386C2×C3#187611yescharacteristic
387C2×C3#190011yescharacteristic
388C2×C3#190511yescharacteristic
389D3—161no—
390C22—21no—
391C22—21no—
392C22—241no—
393C22—241no—
394C22—241no—
395C22—481no—
396C22—481no—
397C22#514411yescharacteristic
398C22#514511yescharacteristic
399C22#521311yescharacteristic
400C4—21no—
401C4—22no—
402C4—21no—
403C4—21no—
404C4—21no—
405C4—21no—
406C4—41no—
407C4—41no—
408C4—41no—
409C4—241no—
410C4C2×#82311yescharacteristic
411C4#513211yescharacteristic
412C3#924411yescharacteristic, complemented, Sylow
413C2—21no—
414C2—21no—
415C2—481no—
416C2#7166211yescharacteristic
417C2#7200211yescharacteristic
418C2#7201011yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2984C2
socleC22×C3#386
prime radicalC22×D12C2×C4
cosocleC2×C3×C42C23
hypercenter#2984D3
hypocenterC3#9244
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2984

Complements

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

NH ≅ G/Ntype
1. C3×#2984C2semidirect
3. #71623C2semidirect
6. #67755C2semidirect
7. #67744C2semidirect
35. #2984D3semidirect
412. C3#9244semidirect

Maximal subgroups

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

subgroupindex
C3×#29842
C2×#50282
#716232
#708042
#707842
#677552
#677442
#92443