Finite Group Databasebeta

← #10876142

Subgroups — #10876142

Order 768.

916 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82070C211yesmaximal, characteristic
2#82062C211yesmaximal, characteristic, complemented
3C3×#1817C211yesmaximal, characteristic, complemented
4#69960C212yesmaximal
5#69929C212yesmaximal, complemented
6#68691C211yesmaximal, characteristic, complemented
7#68682C211yesmaximal, characteristic, complemented
8#68636C211yesmaximal, characteristic, complemented
9#68635C211yesmaximal, characteristic, complemented
10#68627C212yesmaximal, complemented
11#68625C212yesmaximal, complemented
12#27584—31nomaximal, Sylow
13C22×#235C2211yescharacteristic
14C22×C3×C42C2211yescharacteristic
15C2×#926—21no—
16C2×#926C2211yescharacteristic
17C2×C3×C22⋊C8C2212yes—
18#5213—21no—
19#5213—21no—
20#5212—21no—
21#5212C2212yes—
22C2×#821C2212yes—
23C2×#821C2212yes—
24C2×#818C2212yes—
25C2×#817C2212yes—
26C2×#816C2212yes—
27C2×#816C2212yes—
28#4939C2211yescharacteristic
29#4939C2211yescharacteristic
30C4×(C2×D3)⋊C4—21no—
31C4×(C2×D3)⋊C4—21no—
32#4936—21no—
33#4935C2211yescharacteristic
34#4935C2211yescharacteristic
35C4×C4⋊Dic3—21no—
36C4×C4⋊Dic3C2212yes—
37#4929C2211yescharacteristic
38#4929C2211yescharacteristic
39C3×#336C2212yes—
40C3×#336C2211yescharacteristic
41C3×#336C2211yescharacteristic
42#4551C2212yes—
43#4551C2212yes—
44#2297—31no—
45#2285—31no—
46#2095—31no—
47#2094—31no—
48#2072—31no—
49#2071—31no—
50#1927—32no—
51#1925—32no—
52#1854—32no—
53#1817D311yescharacteristic, complemented
54#1788—32no—
55C23×C3×C4D411yescharacteristic
56C23×C3×C4D411yescharacteristic
57C23×C3×C4C2311yescharacteristic
58C22×#67—21no—
59C2×#235—22no—
60C2×#235—24no—
61C2×#235—22no—
62C2×#235—42no—
63C22×D12C2311yescharacteristic
64C22×C4×D3—21no—
65C22×Dic6C2311yescharacteristic
66C22×C3×C8C2312yes—
67C22×C3×C8C2312yes—
68C2×C3×C42—24no—
69C2×C3×C42D411yescharacteristic
70C2×C3×C42D411yescharacteristic
71C2×C3×C42C2311yescharacteristic
72C2×C3×C42C2311yescharacteristic
73C2×#217—22no—
74C2×#217—21no—
75C2×(C2×D3)⋊C4—21no—
76C2×(C2×D3)⋊C4—21no—
77C2×(C2×D3)⋊C4—21no—
78C2×(C2×D3)⋊C4—21no—
79#926—42no—
80#926—44no—
81C2×C4⋊Dic3—21no—
82C2×C4⋊Dic3—21no—
83C2×C4⋊Dic3—21no—
84C2×C4⋊Dic3—21no—
85C2×C4⋊Dic3C2311yescharacteristic
86C2×C4⋊Dic3C2311yescharacteristic
87C2×C4⋊Dic3C2312yes—
88C2×C4⋊Dic3C2312yes—
89C2×C4×Dic3—22no—
90C2×C4×Dic3—21no—
91C3×C22⋊C8—24no—
92C3⋊#95—21no—
93C3⋊#95—21no—
94C3⋊#95—21no—
95C3⋊#95—21no—
96C3⋊#95—22no—
97C3⋊#95—22no—
98#821—24no—
99#821—42no—
100#818—42no—
101#817—42no—
102#816—24no—
103#816—42no—
104C22×#41—31no—
105C2×#117—31no—
106C2×#117—61no—
107C22×C42C2×D311yescharacteristic
108C2×#107—32no—
109C2×#106—32no—
110C2×#103—32no—
111C2×#103—32no—
112C2×#102—32no—
113C2×#102—32no—
114C2×C22⋊C8C2×D312yes—
115#391—31no—
116#391—31no—
117#390—31no—
118#390—31no—
119#386—61no—
120#386—61no—
121#384—31no—
122#384—31no—
123#382—61no—
124#380—32no—
125#380—61no—
126C4×C4⋊C4—32no—
127C4×C4⋊C4—61no—
128C4×C22⋊C4—61no—
129C4×C22⋊C4—61no—
130#340—32no—
131#340—32no—
132#336C2×D311yescharacteristic
133#336C2×D311yescharacteristic
134#336C2×D312yes—
135C24×C3C2×D411yescharacteristic
136C23×D3—21no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—21no—
141C22×C3×C4—21no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—41no—
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4C2×D411yescharacteristic
151C22×C3×C4#4114yes—
152C22×C3×C4C2411yescharacteristic
153C2×#67—24no—
154C2×#67—42no—
155C2×#67—42no—
156C22×Dic3—21no—
157C22×Dic3—22no—
158C22×Dic3—22no—
159C22×Dic3—21no—
160C22×Dic3—21no—
161#235—48no—
162#235—48no—
163C2×D12—22no—
164C2×D12—22no—
165C2×D12—21no—
166C2×D12#4112yes—
167C2×C4×D3—24no—
168C2×C4×D3—42no—
169C2×C4×D3—42no—
170C2×Dic6—22no—
171C2×Dic6—22no—
172C2×Dic6—21no—
173C2×Dic6#4112yes—
174C2×C3×C8—22no—
175C2×C3×C8—22no—
176C2×C3×C8—22no—
177C2×C3×C8—22no—
178C2×C3×C8—22no—
179C2×C3×C8—22no—
180C3×C42—22no—
181C3×C42—22no—
182C3×C42—22no—
183C3×C42—22no—
184#217—42no—
185#217—44no—
186(C2×D3)⋊C4—42no—
187(C2×D3)⋊C4—42no—
188(C2×D3)⋊C4—42no—
189(C2×D3)⋊C4—42no—
190C4⋊Dic3—22no—
191C4⋊Dic3—22no—
192C4⋊Dic3—22no—
193C4⋊Dic3—22no—
194C4⋊Dic3—22no—
195C4⋊Dic3—22no—
196C4⋊Dic3—42no—
197C4⋊Dic3—42no—
198C4⋊Dic3—42no—
199C4⋊Dic3—42no—
200C4×Dic3—42no—
201C4×Dic3—44no—
202C2×#41—62no—
203C2×#41—62no—
204C2×#41—64no—
205C2×#41—122no—
206C22×Q8—31no—
207C22×D4—31no—
208C22×D4—61no—
209C23×C4—61no—
210C23×C4D1211yescharacteristic
211C23×C4D1211yescharacteristic
212C23×C4C22×D311yescharacteristic
213C22×C8C22×D312yes—
214C22×C8C22×D312yes—
215#117—122no—
216#117—124no—
217C2×C4⋊C4—31no—
218C2×C4⋊C4—31no—
219C2×C4⋊C4—32no—
220C2×C4⋊C4—32no—
221C2×C4⋊C4—61no—
222C2×C4⋊C4—61no—
223C2×C4⋊C4—61no—
224C2×C4⋊C4—61no—
225C2×C22⋊C4—61no—
226C2×C22⋊C4—62no—
227C2×C22⋊C4—61no—
228C2×C22⋊C4—61no—
229C2×C22⋊C4—61no—
230C2×C22⋊C4—61no—
231C2×C42—24no—
232C2×C42—61no—
233C2×C42—62no—
234C2×C42D1211yescharacteristic
235C2×C42D1211yescharacteristic
236C2×C42C22×D311yescharacteristic
237C2×C42C22×D311yescharacteristic
238#107—122no—
239#106—122no—
240#103—64no—
241#103—122no—
242#102—64no—
243#102—122no—
244C22⋊C8—24no—
245#95—61no—
246#95—61no—
247#95—61no—
248#95—61no—
249#95—62no—
250#95—62no—
251C23×C3—21no—
252C23×C3—21no—
253C23×C3—21no—
254C23×C3—21no—
255C23×C3—21no—
256C23×C3—21no—
257C23×C3#12312yes—
258C23×C3C22×D411yescharacteristic
259C22×D3—22no—
260C22×D3—41no—
261C22×D3—41no—
262C22×D3—41no—
263C22×D3—41no—
264C22×D3—41no—
265C22×D3—41no—
266C2×C3×C4—28no—
267C2×C3×C4—21no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—21no—
271C2×C3×C4—21no—
272C2×C3×C4—21no—
273C2×C3×C4—21no—
274C2×C3×C4—21no—
275C2×C3×C4—24no—
276C2×C3×C4—24no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—44no—
281C2×C3×C4#12312yes—
282C2×C3×C4#13512yes—
283C2×C3×C4AGL(1,ℤ/8)12yes—
284C2×C3×C4#13712yes—
285C2×C3×C4C2×#4112yes—
286C2×C3×C4C2×#4111yescharacteristic
287C2×C3×C4C2×#4111yescharacteristic
288C2×C3×C4H(5,𝔽2)12yes—
289#67—48no—
290C2×Dic3—22no—
291C2×Dic3—41no—
292C2×Dic3—41no—
293C2×Dic3—42no—
294C2×Dic3—41no—
295C2×Dic3—41no—
296C2×Dic3—42no—
297C2×Dic3—42no—
298C2×Dic3—41no—
299C2×Dic3—42no—
300C2×Dic3—42no—
301C2×Dic3—41no—
302C2×Dic3—41no—
303C2×Dic3—42no—
304C2×Dic3—41no—
305D12—24no—
306D12—42no—
307C4×D3—48no—
308Dic6—24no—
309Dic6—42no—
310C3×C8—42no—
311C3×C8—42no—
312C24—61no—
313C24C2×D1211yescharacteristic
314#41—128no—
315#41—128no—
316C2×Q8—32no—
317C2×Q8—61no—
318C2×Q8—62no—
319C2×Q8—62no—
320C2×D4—32no—
321C2×D4—64no—
322C2×D4—62no—
323C2×D4—61no—
324C2×D4—62no—
325C2×D4—122no—
326C2×D4—122no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—41no—
336C22×C4—64no—
337C22×C4—61no—
338C22×C4—61no—
339C22×C4—61no—
340C22×C4—62no—
341C22×C4—62no—
342C22×C4—122no—
343C22×C4—122no—
344C22×C4C2×D1211yescharacteristic
345C22×C4C2×D1211yescharacteristic
346C22×C4C2×D1211yescharacteristic
347C22×C4C2×D1211yescharacteristic
348C22×C4C2×D1211yescharacteristic
349C22×C4#23714yes—
350C22×C4C23×D311yescharacteristic
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—22no—
357C4⋊C4—62no—
358C4⋊C4—62no—
359C4⋊C4—62no—
360C4⋊C4—62no—
361C4⋊C4—62no—
362C4⋊C4—62no—
363C4⋊C4—122no—
364C4⋊C4—122no—
365C4⋊C4—122no—
366C4⋊C4—122no—
367C22⋊C4—122no—
368C22⋊C4—124no—
369C22⋊C4—122no—
370C22⋊C4—122no—
371C22⋊C4—122no—
372C22⋊C4—122no—
373C42—22no—
374C42—22no—
375C42—22no—
376C42—22no—
377C42—122no—
378C42—124no—
379C22×C3—22no—
380C22×C3—22no—
381C22×C3—22no—
382C22×C3—41no—
383C22×C3—41no—
384C22×C3—41no—
385C22×C3—41no—
386C22×C3C2×#12311yescharacteristic
387C22×C3#52511yescharacteristic
388C22×C3#53511yescharacteristic
389C22×C3C2×#13511yescharacteristic
390C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
391C22×C3C2×#13711yescharacteristic
392C22×C3#57511yescharacteristic
393C2×D3—42no—
394C2×D3—42no—
395C2×D3—42no—
396C2×D3—81no—
397C2×D3—81no—
398C2×D3—81no—
399C2×D3—81no—
400C3×C4—22no—
401C3×C4—22no—
402C3×C4—22no—
403C3×C4—22no—
404C3×C4—41no—
405C3×C4#55112yes—
406C3×C4#55112yes—
407Dic3—42no—
408Dic3—82no—
409Dic3—81no—
410Dic3—81no—
411Dic3—82no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—62no—
419C23—121no—
420C23—121no—
421C23—121no—
422C23—121no—
423C23—121no—
424C23—121no—
425C23#88612yes—
426C23C22×D1211yescharacteristic
427Q8—64no—
428Q8—122no—
429D4—64no—
430D4—128no—
431D4—122no—
432C2×C4—28no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—44no—
447C2×C4—62no—
448C2×C4—128no—
449C2×C4—121no—
450C2×C4—121no—
451C2×C4—121no—
452C2×C4—122no—
453C2×C4—122no—
454C2×C4—122no—
455C2×C4—121no—
456C2×C4—121no—
457C2×C4—121no—
458C2×C4—122no—
459C2×C4—122no—
460C2×C4—121no—
461C2×C4—121no—
462C2×C4—122no—
463C2×C4#88612yes—
464C2×C4#90412yes—
465C2×C4#90812yes—
466C2×C4#90912yes—
467C2×C4C2×#23712yes—
468C2×C4D3×#4111yescharacteristic
469C2×C4D3×#4111yescharacteristic
470C2×C4#100912yes—
471C8—42no—
472C8—42no—
473C2×C3—42no—
474C2×C3#248211yescharacteristic
475C2×C3#309111yescharacteristic
476C2×C3#309211yescharacteristic
477C2×C3#310611yescharacteristic
478C2×C3#311611yescharacteristic
479C2×C3#321011yescharacteristic
480C2×C3#321411yescharacteristic
481D3—82no—
482C22—22no—
483C22—22no—
484C22—22no—
485C22—41no—
486C22—41no—
487C22—41no—
488C22—41no—
489C22—122no—
490C22—122no—
491C22—122no—
492C22—241no—
493C22—241no—
494C22—241no—
495C22—241no—
496C22C2×#88611yescharacteristic
497C22#552411yescharacteristic
498C22#552511yescharacteristic
499C22C2×#90411yescharacteristic
500C22C2×#90811yescharacteristic
501C22C2×#90911yescharacteristic
502C22#574811yescharacteristic
503C4—22no—
504C4—22no—
505C4—22no—
506C4—22no—
507C4—41no—
508C4—122no—
509C4—242no—
510C4—241no—
511C4—242no—
512C4—241no—
513C4#565812yes—
514C4#565812yes—
515C3#2758411yescharacteristic, complemented, Sylow
516C2—42no—
517C2—242no—
518C2#7412911yescharacteristic
519C2#7859111yescharacteristic
520C2#7862211yescharacteristic
521C2#7930111yescharacteristic
522C2#7930611yescharacteristic
523C2#7934811yescharacteristic
524C2#7935011yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1817C2
socleC23×C3C22×D4
prime radicalC22×#235C22
hypercenter#1817D3
hypocenterC3#27584
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
2. #82062C2semidirect
3. C3×#1817C2semidirect
5. #69929C2semidirect
6. #68691C2semidirect
7. #68682C2semidirect
8. #68636C2semidirect
9. #68635C2semidirect
10. #68627C2semidirect
11. #68625C2semidirect
53. #1817D3semidirect
515. C3#27584semidirect

Maximal subgroups

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

subgroupindex
#820702
#820622
C3×#18172
#699602
#699292
#686912
#686822
#686362
#686352
#686272
#686252
#275843