Finite Group Databasebeta

← #10698545

Subgroups — #10698545

Order 768.

708 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82378C211yesmaximal, characteristic
2C3×#1802C211yesmaximal, characteristic, complemented
3C2×D4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
4C2×#4485C211yesmaximal, characteristic, complemented
5#69580C211yesmaximal, characteristic, complemented
6#69567C211yesmaximal, characteristic, complemented
7#67704C211yesmaximal, characteristic, complemented
8#10408—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10C2×#940—21no—
11C2×#930—21no—
12C2×#930—21no—
13C2×#874C2211yescharacteristic
14C2×C3×C4⋊C8C2211yescharacteristic
15C2×C3×#102—21no—
16C3×#384C2211yescharacteristic
17#4988—21no—
18C2×#809—22no—
19C2×#807—21no—
20#4960—22no—
21C4×C4⋊Dic3C2211yescharacteristic
22C2×C4⋊(C3⋊C8)C2211yescharacteristic
23C2×C4×C3⋊C8C2211yescharacteristic
24C3×#340—21no—
25D4⋊(C3⋊C8)—22no—
26#4529—21no—
27#4485—22no—
28#2541—31no—
29#1957—31no—
30#1947—31no—
31#1802D311yescharacteristic, complemented
32C2×#331—31no—
33C2×D4⋊C8—31no—
34#1279—31no—
35C22×C3×D4D411yescharacteristic
36C23×C3×C4—21no—
37C22×#67—22no—
38C22×#67—22no—
39C22×D12—21no—
40C22×D12—21no—
41C22×D12D412yes—
42C22×C3×C8—21no—
43C3×C4×D4—21no—
44C3×C4×D4—21no—
45C3×C4×D4—21no—
46C3×C4×D4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4D411yescharacteristic
51C2×C3×C22⋊C4—21no—
52C2×C3×C42C2311yescharacteristic
53#940—42no—
54#940—42no—
55#930—42no—
56#930—42no—
57#930—42no—
58#930—42no—
59C2×(C2×D3)⋊C4—22no—
60C2×(C2×D3)⋊C4—22no—
61C2×C4⋊Dic3D411yescharacteristic
62C2×C4⋊Dic3D411yescharacteristic
63C2×C4×Dic3—21no—
64C22×C3⋊C8—21no—
65C22×C3⋊C8—21no—
66#874—21no—
67#874—21no—
68#874—21no—
69#874—21no—
70C3×C4⋊C8—21no—
71C3×C4⋊C8—21no—
72C3×#102—42no—
73C3⋊#95—21no—
74#809—44no—
75#807—42no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)D412yescomplemented
78C4×C3⋊C8—21no—
79C4×C3⋊C8—21no—
80C2×#127—31no—
81C2×#127—61no—
82C2×#121—61no—
83C2×#121—61no—
84C2×C4×D4C2×D311yescharacteristic
85C2×#107—61no—
86C2×C4⋊C8—31no—
87C2×C4⋊C8C2×D311yescharacteristic
88C2×#102—21no—
89C2×#102—62no—
90C2×C4×C8—31no—
91#392—62no—
92#390—61no—
93#384C2×D311yescharacteristic
94C4×C4⋊C4—31no—
95#340—21no—
96#340—61no—
97#331—62no—
98D4⋊C8—62no—
99C24×C3—21no—
100C23×D3—22no—
101C2×C3×D4—21no—
102C2×C3×D4—21no—
103C2×C3×D4—21no—
104C2×C3×D4—21no—
105C2×C3×D4—21no—
106C2×C3×D4D812yes—
107C22×C3×C4—21no—
108C22×C3×C4—41no—
109C22×C3×C4—41no—
110C22×C3×C4—41no—
111C22×C3×C4—41no—
112C22×C3×C4—41no—
113C22×C3×C4—41no—
114C22×C3×C4—41no—
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C2×#67—44no—
119C2×#67—44no—
120C2×#67—44no—
121C2×#67—44no—
122C2×#67—82no—
123C2×#67—82no—
124C22×Dic3—21no—
125C22×Dic3—21no—
126C2×D12—22no—
127C2×D12—22no—
128C2×D12—42no—
129C2×D12—42no—
130C2×D12—41no—
131C2×D12—41no—
132C2×D12—41no—
133C2×D12—41no—
134C2×D12—41no—
135C2×D12—41no—
136C2×D12—41no—
137C2×D12—41no—
138C2×D12—41no—
139C2×D12—41no—
140C2×D12—41no—
141C2×D12—41no—
142C2×C3×C8—41no—
143C2×C3×C8—41no—
144C2×C3×C8—41no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—21no—
147C3×C4⋊C4—42no—
148C3×C4⋊C4—41no—
149C3×C4⋊C4—41no—
150C3×C4⋊C4—41no—
151C3×C4⋊C4—41no—
152C3×C22⋊C4—41no—
153C3×C22⋊C4—41no—
154C3×C42C2×D411yescharacteristic
155C3×C42#4111yescharacteristic
156C3×C42#4111yescharacteristic
157C3×C42#4111yescharacteristic
158(C2×D3)⋊C4—44no—
159(C2×D3)⋊C4—44no—
160C4⋊Dic3—21no—
161C4⋊Dic3—21no—
162C4⋊Dic3D812yes—
163C4⋊Dic3SD1612yes—
164C4×Dic3—41no—
165C4×Dic3—41no—
166C2×C3⋊C8—22no—
167C2×C3⋊C8—22no—
168C2×C3⋊C8—41no—
169C2×C3⋊C8—41no—
170C2×C3⋊C8—41no—
171C2×C3⋊C8—41no—
172C22×D4—32no—
173C22×D4—62no—
174C22×D4—62no—
175C22×D4—61no—
176C22×D4—61no—
177C22×D4#6711yescharacteristic
178C23×C4—21no—
179C22×C8—21no—
180C22×C8—61no—
181C22×C8—61no—
182#127—61no—
183#127—61no—
184#127—61no—
185#127—61no—
186#127—122no—
187#127—122no—
188#121—122no—
189#121—122no—
190#121—122no—
191#121—122no—
192C4×D4—21no—
193C4×D4—21no—
194C4×D4—21no—
195C4×D4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—31no—
200C2×C4⋊C4—31no—
201C2×C4⋊C4#6711yescharacteristic
202C2×C22⋊C4—21no—
203C2×C22⋊C4—62no—
204C2×C22⋊C4—62no—
205C2×C42—61no—
206C2×C42C22×D311yescharacteristic
207#107—122no—
208C4⋊C8—21no—
209C4⋊C8—21no—
210C4⋊C8—32no—
211C4⋊C8—61no—
212#102—42no—
213#102—124no—
214C4×C8—61no—
215C4×C8—61no—
216#95—61no—
217C23×C3—22no—
218C23×C3—41no—
219C23×C3—41no—
220C23×C3—41no—
221C23×C3—41no—
222C23×C3—41no—
223C23×C3—41no—
224C23×C3#12011yescharacteristic
225C22×D3—42no—
226C22×D3—42no—
227C22×D3—42no—
228C22×D3—42no—
229C22×D3—42no—
230C22×D3—42no—
231C22×D3—42no—
232C3×D4—22no—
233C3×D4—22no—
234C3×D4—41no—
235C3×D4—41no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—81no—
242C2×C3×C4—81no—
243C2×C3×C4—81no—
244C2×C3×C4—81no—
245C2×C3×C4—81no—
246C2×C3×C4—81no—
247C2×C3×C4—81no—
248C2×C3×C4#12111yescharacteristic
249C2×C3×C4#12111yescharacteristic
250C2×C3×C4#12111yescharacteristic
251C2×C3×C4#12311yescharacteristic
252C2×C3×C4#12311yescharacteristic
253C2×C3×C4#12411yescharacteristic
254C2×C3×C4C2×D811yescharacteristic
255C2×C3×C4C2×D811yescharacteristic
256C2×C3×C4C2×SD1611yescharacteristic
257C2×C3×C4#13511yescharacteristic
258C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
259C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
260C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
261C2×C3×C4#13711yescharacteristic
262#67—88no—
263#67—88no—
264C2×Dic3—22no—
265C2×Dic3—22no—
266C2×Dic3—41no—
267C2×Dic3—41no—
268C2×Dic3—41no—
269C2×Dic3—41no—
270D12—44no—
271D12—82no—
272D12—81no—
273D12—81no—
274D12—81no—
275D12—81no—
276D12—81no—
277D12—81no—
278D12—81no—
279D12—81no—
280C3×C8—81no—
281C3⋊C8—42no—
282C3⋊C8—42no—
283C24—21no—
284C24—62no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—62no—
291C2×D4—62no—
292C2×D4—122no—
293C2×D4—122no—
294C2×D4—124no—
295C2×D4—124no—
296C2×D4—124no—
297C2×D4—124no—
298C2×D4—121no—
299C2×D4—121no—
300C2×D4—121no—
301C2×D4—121no—
302C2×D4—121no—
303C2×D4—121no—
304C2×D4—121no—
305C2×D4—121no—
306C2×D4—121no—
307C2×D4—121no—
308C2×D4—121no—
309C2×D4—121no—
310C2×D4—242no—
311C2×D4—242no—
312C2×D4#21312yes—
313C22×C4—21no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—61no—
322C22×C4—61no—
323C22×C4D3×D411yescharacteristic
324C22×C4D3×D411yescharacteristic
325C22×C4C2×#6711yescharacteristic
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—62no—
330C2×C8—62no—
331C2×C8—121no—
332C2×C8—121no—
333C2×C8—121no—
334C2×C8—121no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—32no—
338C4⋊C4—32no—
339C4⋊C4—42no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—61no—
345C4⋊C4—61no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C22⋊C4—124no—
349C22⋊C4—124no—
350C42—121no—
351C42—121no—
352C42#23511yescharacteristic
353C42#23511yescharacteristic
354C42D3×D411yescharacteristic
355C42#23711yescharacteristic
356C22×C3—42no—
357C22×C3—42no—
358C22×C3—42no—
359C22×C3—81no—
360C22×C3—81no—
361C22×C3—81no—
362C22×C3—81no—
363C22×C3#39411yescharacteristic
364C22×C3#44711yescharacteristic
365C22×C3#44711yescharacteristic
366C22×C3#45011yescharacteristic
367C22×C3#45211yescharacteristic
368C22×C3#45311yescharacteristic
369C22×C3#45511yescharacteristic
370C2×D3—82no—
371C2×D3—82no—
372C2×D3—82no—
373C2×D3—82no—
374C2×D3—82no—
375C2×D3—82no—
376C2×D3—82no—
377C3×C4—21no—
378C3×C4—21no—
379C3×C4—81no—
380C3×C4—161no—
381C3×C4#45911yescharacteristic
382C3×C4#46011yescharacteristic
383C3×C4#46611yescharacteristic
384C3×C4#46811yescharacteristic
385C3×C4#48011yescharacteristic
386C3×C4#48211yescharacteristic
387C3×C4#48611yescharacteristic
388C3×C4#49011yescharacteristic
389Dic3—42no—
390Dic3—42no—
391C23—22no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—122no—
399C23—122no—
400C23—122no—
401C23—122no—
402C23—122no—
403C23—122no—
404C23—122no—
405C23#93711yescharacteristic
406D4—22no—
407D4—22no—
408D4—41no—
409D4—41no—
410D4—124no—
411D4—242no—
412D4—248no—
413D4—248no—
414D4—241no—
415D4—241no—
416D4—241no—
417D4—241no—
418D4—241no—
419D4—241no—
420D4—241no—
421D4—241no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—62no—
428C2×C4—62no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—121no—
437C2×C4—121no—
438C2×C4—121no—
439C2×C4—121no—
440C2×C4#88311yescharacteristic
441C2×C4#88411yescharacteristic
442C2×C4#88411yescharacteristic
443C2×C4#88511yescharacteristic
444C2×C4D3×D811yescharacteristic
445C2×C4#91111yescharacteristic
446C2×C4D3×SD1611yescharacteristic
447C2×C4#91411yescharacteristic
448C2×C4#92911yescharacteristic
449C2×C4C2×#21311yescharacteristic
450C2×C4#93911yescharacteristic
451C2×C4#94211yescharacteristic
452C2×C4#94911yescharacteristic
453C2×C4#95011yescharacteristic
454C8—81no—
455C8—122no—
456C8—122no—
457C2×C3—82no—
458C2×C3#161911yescharacteristic
459C2×C3#163111yescharacteristic
460C2×C3#163911yescharacteristic
461C2×C3#164111yescharacteristic
462C2×C3#203311yescharacteristic
463C2×C3#203711yescharacteristic
464C2×C3#204111yescharacteristic
465D3—162no—
466C22—42no—
467C22—42no—
468C22—42no—
469C22—81no—
470C22—81no—
471C22—81no—
472C22—81no—
473C22—242no—
474C22—242no—
475C22—242no—
476C22—242no—
477C22—242no—
478C22—242no—
479C22—242no—
480C22#495911yescharacteristic
481C22#503611yescharacteristic
482C22#504711yescharacteristic
483C22#515611yescharacteristic
484C22#516911yescharacteristic
485C22#519811yescharacteristic
486C22#520211yescharacteristic
487C4—21no—
488C4—21no—
489C4—81no—
490C4—122no—
491C4—122no—
492C4—161no—
493C4#476011yescharacteristic
494C4#476211yescharacteristic
495C4#476411yescharacteristic
496C4#478611yescharacteristic
497C4#496711yescharacteristic
498C4#500311yescharacteristic
499C4#503811yescharacteristic
500C4#504911yescharacteristic
501C3#1040811yescharacteristic, complemented, Sylow
502C2—82no—
503C2—482no—
504C2#6952311yescharacteristic
505C2#7021111yescharacteristic
506C2#7030611yescharacteristic
507C2#7101511yescharacteristic
508C2#7104411yescharacteristic
509C2#7111211yescharacteristic
510C2#7114111yescharacteristic
subgroupG/H
centerC23#937
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1802C2
socleC23×C3#120
prime radical#82378C2
hypercenter#1802D3
hypocenterC3#10408
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
2. C3×#1802C2semidirect
3. C2×D4⋊(C3⋊C8)C2semidirect
4. C2×#4485C2semidirect
5. #69580C2semidirect
6. #69567C2semidirect
7. #67704C2semidirect
31. #1802D3semidirect
77. C4⋊(C3⋊C8)D4semidirect
501. C3#10408semidirect

Maximal subgroups

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

subgroupindex
#823782
C3×#18022
C2×D4⋊(C3⋊C8)2
C2×#44852
#695802
#695672
#677042
#104083