Finite Group Databasebeta

← #10853323

Subgroups — #10853323

Order 768.

734 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#73716C211yesmaximal, characteristic, complemented
2#73638C211yesmaximal, characteristic
3C3×#1939C211yesmaximal, characteristic, complemented
4#70484C212yesmaximal
5#70479C211yesmaximal, characteristic
6#70461C212yesmaximal, complemented
7#70453C211yesmaximal, characteristic, complemented
8#69762C211yesmaximal, characteristic, complemented
9C4×#821C211yesmaximal, characteristic
10#68679C212yesmaximal
11#68624C212yesmaximal, complemented
12#26025—31nomaximal, Sylow
13C2×#892—21no—
14C2×C4×D12C2211yescharacteristic
15C2×#870C2212yes—
16C2×C3×C4⋊C8C2211yescharacteristic
17C2×C3×C4⋊C8C2212yes—
18C2×C3×C4×C8C2211yescharacteristic
19C3×C4×C4⋊C4C2211yescharacteristic
20C2×#821C2211yescharacteristic
21C2×#821C2211yescharacteristic
22C2×#821C2212yes—
23C2×#818C2212yes—
24#4987—21no—
25#4975—21no—
26#4972—21no—
27#4939C2212yes—
28#4935C2211yescharacteristic
29#4935C2211yescharacteristic
30C4×C4⋊Dic3C2212yes—
31C4×C4⋊Dic3C2211yescharacteristic
32#4928C2212yes—
33#4670—22no—
34#4657—22no—
35C3×#336C2212yes—
36#4551C2211yescharacteristic
37#4551C2212yes—
38#4551C2212yes—
39#4551C2212yes—
40#4551C2211yescharacteristic
41#4551C2212yes—
42#2552—31no—
43#2354—31no—
44#2067—32no—
45#2025—32no—
46#1989—32no—
47#1947—31no—
48#1939D311yescharacteristic, complemented
49#1921—32no—
50#1834—31no—
51#1802—31no—
52C4×#102—31no—
53C22×D12C2311yescharacteristic
54C22×C4×D3—21no—
55C22×C3×C8C2311yescharacteristic
56C22×C3×C8C2311yescharacteristic
57C22×C3×C8C2312yes—
58C2×C3×C4⋊C4D411yescharacteristic
59C2×C3×C4⋊C4D411yescharacteristic
60C2×C3×C42C2312yes—
61C2×C3×C42C2311yescharacteristic
62C2×(C2×D3)⋊C4—21no—
63C2×(C2×D3)⋊C4—22no—
64C2×(C2×D3)⋊C4—22no—
65C2×C4⋊Dic3—21no—
66C2×C4⋊Dic3—21no—
67C2×C4⋊Dic3—21no—
68C2×C4⋊Dic3—21no—
69C2×C4⋊Dic3C2312yes—
70C2×C4⋊Dic3C2312yes—
71C2×C4⋊Dic3C2311yescharacteristic
72C2×C4⋊Dic3C2311yescharacteristic
73C2×C4⋊Dic3C2311yescharacteristic
74C2×Dic3⋊C4—22no—
75C2×Dic3⋊C4—22no—
76C2×Dic3⋊C4—22no—
77C2×C4×Dic3—22no—
78C2×C4×Dic3—21no—
79#892—44no—
80C4×D12—22no—
81C4×D12Q814yes—
82#870—24no—
83#870—42no—
84C3×C4⋊C8—22no—
85C3×C4⋊C8—42no—
86C3×C4×C8—22no—
87C3×#95D412yes—
88C3⋊#95—21no—
89C3⋊#95—22no—
90C3⋊#95—22no—
91#821—22no—
92#821—22no—
93#821—42no—
94#818—24no—
95C2×#125—32no—
96C2×C4×D4—31no—
97C2×#117—61no—
98C2×#107—32no—
99C2×C4⋊C8C2×D311yescharacteristic
100C2×C4⋊C8C2×D312yes—
101C2×#102—31no—
102C2×#102—31no—
103C2×#102—32no—
104C2×C4×C8C2×D311yescharacteristic
105#398—62no—
106#390—32no—
107#389—32no—
108#389—61no—
109#388—62no—
110#384—31no—
111#384—31no—
112#384—61no—
113#380—61no—
114C4×C4⋊C4—32no—
115C4×C4⋊C4—31no—
116C4×C4⋊C4C2×D311yescharacteristic
117#340—32no—
118#340—32no—
119#340—32no—
120#340—31no—
121#340—32no—
122#340—31no—
123#336C2×D312yes—
124C23×D3—21no—
125C22×C3×C4C2×D412yes—
126C22×C3×C4C2×D412yes—
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2411yescharacteristic
130C22×Dic3—21no—
131C22×Dic3—21no—
132C22×Dic3—21no—
133C22×Dic3—22no—
134C22×Dic3—22no—
135C2×D12—21no—
136C2×D12—21no—
137C2×D12—21no—
138C2×D12—21no—
139C2×D12—21no—
140C2×D12C2×Q812yes—
141C2×C4×D3—24no—
142C2×C4×D3—42no—
143C2×C4×D3—42no—
144C2×C3×C8—22no—
145C2×C3×C8—21no—
146C2×C3×C8—21no—
147C2×C3×C8—21no—
148C2×C3×C8—21no—
149C2×C3×C8—22no—
150C2×C3×C8—22no—
151C2×C3×C8—21no—
152C2×C3×C8—21no—
153C3×C4⋊C4—22no—
154C3×C4⋊C4—22no—
155C3×C42—22no—
156C3×C42—22no—
157C3×C42C2×Q812yes—
158C3×C42#4112yes—
159(C2×D3)⋊C4—24no—
160(C2×D3)⋊C4—44no—
161(C2×D3)⋊C4—82no—
162C4⋊Dic3—21no—
163C4⋊Dic3—21no—
164C4⋊Dic3—21no—
165C4⋊Dic3—22no—
166C4⋊Dic3—22no—
167C4⋊Dic3—22no—
168C4⋊Dic3—22no—
169C4⋊Dic3—42no—
170C4⋊Dic3—42no—
171C4⋊Dic3—81no—
172C4⋊Dic3—81no—
173C4⋊Dic3C2×Q812yes—
174C4⋊Dic3#4112yes—
175C4⋊Dic3#4112yes—
176Dic3⋊C4—44no—
177Dic3⋊C4—44no—
178Dic3⋊C4—82no—
179C4×Dic3—44no—
180C4×Dic3—42no—
181C22×D4—31no—
182C23×C4—61no—
183C22×C8C22×D311yescharacteristic
184C22×C8C22×D312yes—
185C22×C8C22×D311yescharacteristic
186#125—64no—
187#125—122no—
188C4×D4—34no—
189C4×D4—62no—
190#117—124no—
191C2×C4⋊C4—31no—
192C2×C4⋊C4—31no—
193C2×C4⋊C4—31no—
194C2×C4⋊C4—32no—
195C2×C4⋊C4—32no—
196C2×C4⋊C4—61no—
197C2×C4⋊C4—62no—
198C2×C4⋊C4—62no—
199C2×C4⋊C4—61no—
200C2×C4⋊C4—61no—
201C2×C4⋊C4—61no—
202C2×C4⋊C4—62no—
203C2×C4⋊C4D1211yescharacteristic
204C2×C4⋊C4D1211yescharacteristic
205C2×C22⋊C4—61no—
206C2×C22⋊C4—62no—
207C2×C22⋊C4—62no—
208C2×C42—62no—
209C2×C42—61no—
210C2×C42C22×D312yes—
211C2×C42C22×D311yescharacteristic
212#107—64no—
213C4⋊C8—22no—
214C4⋊C8—42no—
215#102—62no—
216#102—62no—
217#102—122no—
218C4×C8—22no—
219#95—62no—
220#95—61no—
221#95—62no—
222#95D1212yes—
223C23×C3C22×D411yescharacteristic
224C22×D3—22no—
225C22×D3—41no—
226C22×D3—41no—
227C22×D3—41no—
228C22×D3—41no—
229C22×D3—41no—
230C22×D3—41no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—22no—
234C2×C3×C4—22no—
235C2×C3×C4—22no—
236C2×C3×C4—22no—
237C2×C3×C4—22no—
238C2×C3×C4—22no—
239C2×C3×C4C22⋊Q812yes—
240C2×C3×C4C22⋊Q812yes—
241C2×C3×C4#13512yes—
242C2×C3×C4AGL(1,ℤ/8)12yes—
243C2×C3×C4C22×Q811yescharacteristic
244C2×C3×C4C2×#4111yescharacteristic
245C2×C3×C4C2×#4111yescharacteristic
246C2×C3×C4C2×#4111yescharacteristic
247C2×C3×C4H(5,𝔽2)11yescharacteristic
248C2×C3×C42−1+411yescharacteristic
249C2×Dic3—22no—
250C2×Dic3—22no—
251C2×Dic3—22no—
252C2×Dic3—41no—
253C2×Dic3—41no—
254C2×Dic3—41no—
255C2×Dic3—41no—
256C2×Dic3—41no—
257C2×Dic3—41no—
258C2×Dic3—42no—
259C2×Dic3—42no—
260C2×Dic3—42no—
261C2×Dic3—42no—
262C2×Dic3—42no—
263C2×Dic3—42no—
264D12—22no—
265D12—22no—
266D12—41no—
267D12—41no—
268C4×D3—48no—
269C3×C8—42no—
270C3×C8—41no—
271C3×C8—41no—
272C24—61no—
273C2×D4—32no—
274C2×D4—61no—
275C2×D4—61no—
276C2×D4—61no—
277C2×D4—61no—
278C2×D4—61no—
279C22×C4—61no—
280C22×C4—61no—
281C22×C4—61no—
282C22×C4—64no—
283C22×C4—62no—
284C22×C4—62no—
285C22×C4—122no—
286C22×C4—122no—
287C22×C4C2×D1212yes—
288C22×C4C2×D1212yes—
289C22×C4C2×D1211yescharacteristic
290C22×C4C2×D1211yescharacteristic
291C22×C4C23×D311yescharacteristic
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—22no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—32no—
304C4⋊C4—32no—
305C4⋊C4—32no—
306C4⋊C4—61no—
307C4⋊C4—61no—
308C4⋊C4—61no—
309C4⋊C4—62no—
310C4⋊C4—62no—
311C4⋊C4—62no—
312C4⋊C4—62no—
313C4⋊C4—124no—
314C4⋊C4—124no—
315C4⋊C4—122no—
316C4⋊C4—122no—
317C4⋊C4—242no—
318C4⋊C4—241no—
319C4⋊C4—241no—
320C22⋊C4—64no—
321C22⋊C4—124no—
322C22⋊C4—242no—
323C42—22no—
324C42—22no—
325C42—124no—
326C42—122no—
327C42#23712yes—
328C42D3×Q812yes—
329C22×C3C2×C22⋊Q811yescharacteristic
330C22×C3#54611yescharacteristic
331C22×C3#54811yescharacteristic
332C22×C3C2×#13511yescharacteristic
333C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
334C22×C3#57611yescharacteristic
335C22×C3#57811yescharacteristic
336C2×D3—42no—
337C2×D3—42no—
338C2×D3—42no—
339C2×D3—81no—
340C2×D3—81no—
341C2×D3—81no—
342C2×D3—81no—
343C3×C4—22no—
344C3×C4—42no—
345C3×C4—42no—
346C3×C4#47814yes—
347C3×C4#53311yescharacteristic
348C3×C4#54411yescharacteristic
349C3×C4#55511yescharacteristic
350C3×C4#55611yescharacteristic
351Dic3—42no—
352Dic3—42no—
353Dic3—42no—
354Dic3—82no—
355Dic3—82no—
356C23—62no—
357C23—121no—
358C23—121no—
359C23—121no—
360C23—121no—
361C23—121no—
362C23—121no—
363C23C22×D1211yescharacteristic
364D4—62no—
365D4—62no—
366D4—121no—
367D4—121no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—62no—
377C2×C4—62no—
378C2×C4—62no—
379C2×C4—121no—
380C2×C4—121no—
381C2×C4—121no—
382C2×C4—121no—
383C2×C4—121no—
384C2×C4—121no—
385C2×C4—128no—
386C2×C4—122no—
387C2×C4—122no—
388C2×C4—122no—
389C2×C4—122no—
390C2×C4—122no—
391C2×C4—122no—
392C2×C4#89712yes—
393C2×C4#89712yes—
394C2×C4#90412yes—
395C2×C4#90812yes—
396C2×C4C2×#23711yescharacteristic
397C2×C4#100411yescharacteristic
398C2×C4C2×D3×Q811yescharacteristic
399C2×C4#100711yescharacteristic
400C2×C4D3×#4111yescharacteristic
401C2×C4D3×#4111yescharacteristic
402C8—41no—
403C8—41no—
404C8—42no—
405C2×C3#243911yescharacteristic
406C2×C3C2×#47811yescharacteristic
407C2×C3#308311yescharacteristic
408C2×C3#330511yescharacteristic
409C2×C3#333011yescharacteristic
410C2×C3#336411yescharacteristic
411C2×C3#336611yescharacteristic
412D3—82no—
413C22—122no—
414C22—122no—
415C22—122no—
416C22—241no—
417C22—241no—
418C22—241no—
419C22—241no—
420C22C2×#89711yescharacteristic
421C22#555411yescharacteristic
422C22#557611yescharacteristic
423C22C2×#90411yescharacteristic
424C22C2×#90811yescharacteristic
425C22#575211yescharacteristic
426C22#575311yescharacteristic
427C4—22no—
428C4—42no—
429C4—42no—
430C4—122no—
431C4—122no—
432C4—122no—
433C4—242no—
434C4—242no—
435C4#484714yes—
436C4#561611yescharacteristic
437C4#564111yescharacteristic
438C4#568811yescharacteristic
439C4#572911yescharacteristic
440C3#2602511yescharacteristic, complemented, Sylow
441C2—242no—
442C2#7599711yescharacteristic
443C2C2×#484711yescharacteristic
444C2#7909111yescharacteristic
445C2#7945211yescharacteristic
446C2#7955711yescharacteristic
447C2#7993411yescharacteristic
448C2#7996911yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1939C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1939D3
hypocenterC3#26025
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1939

Complements

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

NH ≅ G/Ntype
1. #73716C2semidirect
3. C3×#1939C2semidirect
6. #70461C2semidirect
7. #70453C2semidirect
8. #69762C2semidirect
11. #68624C2semidirect
48. #1939D3semidirect
440. C3#26025semidirect

Maximal subgroups

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

subgroupindex
#737162
#736382
C3×#19392
#704842
#704792
#704612
#704532
#697622
C4×#8212
#686792
#686242
#260253