Finite Group Databasebeta

← #10698548

Subgroups — #10698548

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#73799C211yesmaximal, characteristic
2#70479C211yesmaximal, characteristic, complemented
3C2×#4485C211yesmaximal, characteristic, complemented
4#69772C211yesmaximal, characteristic, complemented
5C2×#4459C211yesmaximal, characteristic, complemented
6#69466C211yesmaximal, characteristic, complemented
7C3×#1279C211yesmaximal, characteristic, complemented
8#10408—31nomaximal, Sylow
9C2×#895—21no—
10C2×#895—21no—
11C2×#874—21no—
12C2×#874C2211yescharacteristic
13C2×C4×D12C2211yescharacteristic
14C2×C3×C4⋊C8C2211yescharacteristic
15C2×C3×C4×C8C2211yescharacteristic
16C3×C4×C4⋊C4C2211yescharacteristic
17C2×#821—21no—
18C2×#821—22no—
19C2×#807—21no—
20#4939—21no—
21#4935C2211yescharacteristic
22C2×C4⋊(C3⋊C8)C2211yescharacteristic
23#4671—22no—
24#4551—21no—
25#4529—21no—
26#4485—22no—
27#4459—22no—
28#2541—31no—
29#1957—31no—
30#1947—31no—
31#1802—31no—
32C2×#331—31no—
33C2×D4⋊C8—31no—
34#1279D311yescharacteristic, complemented
35C22×D12—21no—
36C22×D12—21no—
37C22×D12—22no—
38C22×D12—22no—
39C22×D12D411yescharacteristic
40C22×D12D412yes—
41C22×C4×D3—21no—
42C22×C3×C8—21no—
43C22×C3×C8—21no—
44C2×C3×C4⋊C4D411yescharacteristic
45C2×C3×C4⋊C4D411yescharacteristic
46C2×C3×C42—21no—
47C2×C3×C42C2311yescharacteristic
48C2×(C2×D3)⋊C4—22no—
49C2×(C2×D3)⋊C4—22no—
50C2×(C2×D3)⋊C4—21no—
51C2×C4⋊Dic3—21no—
52C2×C4⋊Dic3—21no—
53C2×C4⋊Dic3—21no—
54C2×C4⋊Dic3D411yescharacteristic
55C22×C3⋊C8—21no—
56#895—42no—
57#895—42no—
58#895—42no—
59#895—42no—
60#874—21no—
61#874—21no—
62#874—21no—
63#874—21no—
64#874—42no—
65#874—42no—
66C4×D12—21no—
67C4×D12—21no—
68C4×D12—21no—
69C4×D12—21no—
70C3×C4⋊C8—21no—
71C3×C4⋊C8—21no—
72C3×C4×C8—21no—
73C3×C4×C8—21no—
74C3×#95—21no—
75#821—44no—
76#821—42no—
77#807—42no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)D412yescomplemented
80C2×#127—31no—
81C2×#127—61no—
82C2×#121—61no—
83C2×#121—61no—
84C2×C4×D4—31no—
85C2×#107—61no—
86C2×C4⋊C8—31no—
87C2×C4⋊C8C2×D311yescharacteristic
88C2×#102—61no—
89C2×#102—62no—
90C2×C4×C8C2×D311yescharacteristic
91#392—62no—
92#390—61no—
93#384—31no—
94C4×C4⋊C4C2×D311yescharacteristic
95#340—61no—
96#340—61no—
97#331—62no—
98D4⋊C8—62no—
99C23×D3—21no—
100C23×D3—22no—
101C22×C3×C4—21no—
102C22×C3×C4—21no—
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×Dic3—21no—
107C22×Dic3—41no—
108C2×D12—21no—
109C2×D12—21no—
110C2×D12—21no—
111C2×D12—21no—
112C2×D12—22no—
113C2×D12—21no—
114C2×D12—22no—
115C2×D12—42no—
116C2×D12—42no—
117C2×D12—44no—
118C2×D12—44no—
119C2×D12—44no—
120C2×D12—44no—
121C2×D12—41no—
122C2×D12—41no—
123C2×D12—41no—
124C2×D12—41no—
125C2×D12—41no—
126C2×D12—41no—
127C2×D12—41no—
128C2×D12—41no—
129C2×D12—41no—
130C2×D12—41no—
131C2×D12—41no—
132C2×D12—41no—
133C2×D12—82no—
134C2×D12—82no—
135C2×D12D812yes—
136C2×C4×D3—41no—
137C2×C4×D3—41no—
138C2×C4×D3—41no—
139C2×C4×D3—41no—
140C2×C4×D3—41no—
141C2×C4×D3—41no—
142C2×C3×C8—22no—
143C2×C3×C8—41no—
144C2×C3×C8—41no—
145C2×C3×C8—41no—
146C2×C3×C8—41no—
147C2×C3×C8—41no—
148C3×C4⋊C4—21no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4D812yes—
151C3×C4⋊C4SD1612yes—
152C3×C42—41no—
153C3×C42—41no—
154C3×C42C2×D411yescharacteristic
155C3×C42#4111yescharacteristic
156C3×C42#4111yescharacteristic
157C3×C42#4111yescharacteristic
158(C2×D3)⋊C4—44no—
159(C2×D3)⋊C4—44no—
160(C2×D3)⋊C4—41no—
161(C2×D3)⋊C4—41no—
162C4⋊Dic3—21no—
163C4⋊Dic3—21no—
164C4⋊Dic3—42no—
165C4⋊Dic3—41no—
166C4⋊Dic3—41no—
167C4⋊Dic3—41no—
168C4⋊Dic3—41no—
169C2×C3⋊C8—22no—
170C2×C3⋊C8—41no—
171C2×C3⋊C8—41no—
172C22×D4—31no—
173C22×D4—32no—
174C22×D4—62no—
175C22×D4—62no—
176C22×D4—61no—
177C22×D4—61no—
178C23×C4—61no—
179C22×C8—21no—
180C22×C8—21no—
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—61no—
193C4×D4—61no—
194C4×D4—61no—
195C4×D4—61no—
196C2×C4⋊C4—31no—
197C2×C4⋊C4—61no—
198C2×C4⋊C4—61no—
199C2×C4⋊C4—61no—
200C2×C4⋊C4D1211yescharacteristic
201C2×C4⋊C4D1211yescharacteristic
202C2×C22⋊C4—62no—
203C2×C22⋊C4—62no—
204C2×C22⋊C4—61no—
205C2×C42—21no—
206C2×C42C22×D311yescharacteristic
207#107—122no—
208C4⋊C8—21no—
209C4⋊C8—21no—
210C4⋊C8—32no—
211C4⋊C8—61no—
212#102—124no—
213#102—122no—
214C4×C8—21no—
215C4×C8—21no—
216#95—21no—
217C23×C3#12011yescharacteristic
218C22×D3—22no—
219C22×D3—41no—
220C22×D3—41no—
221C22×D3—41no—
222C22×D3—41no—
223C22×D3—41no—
224C22×D3—41no—
225C22×D3—42no—
226C22×D3—42no—
227C22×D3—42no—
228C22×D3—42no—
229C22×D3—42no—
230C22×D3—42no—
231C22×D3—42no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—22no—
235C2×C3×C4—22no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4#12111yescharacteristic
241C2×C3×C4#12111yescharacteristic
242C2×C3×C4#12111yescharacteristic
243C2×C3×C4#12311yescharacteristic
244C2×C3×C4#12311yescharacteristic
245C2×C3×C4#12411yescharacteristic
246C2×C3×C4C2×D811yescharacteristic
247C2×C3×C4C2×D811yescharacteristic
248C2×C3×C4C2×SD1611yescharacteristic
249C2×C3×C4#13511yescharacteristic
250C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
251C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
252C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
253C2×C3×C4#13711yescharacteristic
254C2×Dic3—41no—
255C2×Dic3—41no—
256C2×Dic3—41no—
257C2×Dic3—81no—
258C2×Dic3—81no—
259C2×Dic3—81no—
260D12—22no—
261D12—22no—
262D12—44no—
263D12—41no—
264D12—41no—
265D12—82no—
266D12—88no—
267D12—88no—
268D12—81no—
269D12—81no—
270D12—81no—
271D12—81no—
272D12—81no—
273D12—81no—
274D12—81no—
275D12—81no—
276C4×D3—81no—
277C4×D3—81no—
278C4×D3—81no—
279C4×D3—81no—
280C3×C8—42no—
281C3×C8—81no—
282C3⋊C8—42no—
283C24—61no—
284C24—62no—
285C2×D4—32no—
286C2×D4—61no—
287C2×D4—62no—
288C2×D4—61no—
289C2×D4—62no—
290C2×D4—61no—
291C2×D4—61no—
292C2×D4—61no—
293C2×D4—122no—
294C2×D4—122no—
295C2×D4—124no—
296C2×D4—124no—
297C2×D4—124no—
298C2×D4—124no—
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—121no—
311C2×D4—242no—
312C2×D4—242no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—61no—
316C22×C4—121no—
317C22×C4—121no—
318C22×C4—121no—
319C22×C4—121no—
320C22×C4—121no—
321C22×C4—121no—
322C22×C4—121no—
323C22×C4C2×D1211yescharacteristic
324C22×C4D3×D411yescharacteristic
325C22×C4D3×D411yescharacteristic
326C2×C8—22no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—62no—
333C2×C8—121no—
334C2×C8—121no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—61no—
338C4⋊C4—61no—
339C4⋊C4—122no—
340C4⋊C4—121no—
341C4⋊C4—121no—
342C4⋊C4—121no—
343C4⋊C4—121no—
344C4⋊C4#20412yes—
345C4⋊C4D2412yes—
346C22⋊C4—124no—
347C22⋊C4—124no—
348C22⋊C4—121no—
349C22⋊C4—121no—
350C42—41no—
351C42—41no—
352C42#23511yescharacteristic
353C42#23511yescharacteristic
354C42D3×D411yescharacteristic
355C42#23911yescharacteristic
356C22×C3#39411yescharacteristic
357C22×C3#44711yescharacteristic
358C22×C3#44711yescharacteristic
359C22×C3#45011yescharacteristic
360C22×C3#45211yescharacteristic
361C22×C3#45311yescharacteristic
362C22×C3#45511yescharacteristic
363C2×D3—42no—
364C2×D3—42no—
365C2×D3—42no—
366C2×D3—81no—
367C2×D3—81no—
368C2×D3—81no—
369C2×D3—81no—
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—42no—
380C3×C4—42no—
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—81no—
390Dic3—161no—
391C23—62no—
392C23—121no—
393C23—121no—
394C23—121no—
395C23—121no—
396C23—121no—
397C23—121no—
398C23—122no—
399C23—122no—
400C23—122no—
401C23—122no—
402C23—122no—
403C23—122no—
404C23—122no—
405C23#88211yescharacteristic
406D4—62no—
407D4—62no—
408D4—124no—
409D4—121no—
410D4—121no—
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—22no—
425C2×C4—22no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—121no—
431C2×C4—121no—
432C2×C4—121no—
433C2×C4—241no—
434C2×C4—241no—
435C2×C4—241no—
436C2×C4—241no—
437C2×C4—241no—
438C2×C4—241no—
439C2×C4—241no—
440C2×C4#87511yescharacteristic
441C2×C4#88411yescharacteristic
442C2×C4#88411yescharacteristic
443C2×C4#89411yescharacteristic
444C2×C4#89411yescharacteristic
445C2×C4#89511yescharacteristic
446C2×C4C2×#20411yescharacteristic
447C2×C4C2×D2411yescharacteristic
448C2×C4#90811yescharacteristic
449C2×C4#90811yescharacteristic
450C2×C4D3×D811yescharacteristic
451C2×C4#91411yescharacteristic
452C2×C4#91611yescharacteristic
453C2×C4#91811yescharacteristic
454C8—42no—
455C8—81no—
456C8—122no—
457C2×C3#161911yescharacteristic
458C2×C3#163111yescharacteristic
459C2×C3#163911yescharacteristic
460C2×C3#164111yescharacteristic
461C2×C3#203311yescharacteristic
462C2×C3#203711yescharacteristic
463C2×C3#204111yescharacteristic
464D3—82no—
465D3—162no—
466C22—122no—
467C22—122no—
468C22—122no—
469C22—241no—
470C22—241no—
471C22—241no—
472C22—241no—
473C22—242no—
474C22—242no—
475C22—242no—
476C22—242no—
477C22—242no—
478C22—242no—
479C22—242no—
480C22#467211yescharacteristic
481C22#473111yescharacteristic
482C22#473211yescharacteristic
483C22#477311yescharacteristic
484C22#480811yescharacteristic
485C22#482211yescharacteristic
486C22#482511yescharacteristic
487C4—21no—
488C4—21no—
489C4—42no—
490C4—42no—
491C4—241no—
492C4—481no—
493C4#469411yescharacteristic
494C4#471311yescharacteristic
495C4#477511yescharacteristic
496C4#481211yescharacteristic
497C4#484311yescharacteristic
498C4#484811yescharacteristic
499C4#486611yescharacteristic
500C4#488211yescharacteristic
501C3#1040811yescharacteristic, complemented, Sylow
502C2—242no—
503C2—482no—
504C2#6818211yescharacteristic
505C2#6819411yescharacteristic
506C2#6826111yescharacteristic
507C2#6826311yescharacteristic
508C2#6862911yescharacteristic
509C2#6873311yescharacteristic
510C2#6882211yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1279C2
socleC23×C3#120
prime radical#73799C2
hypercenter#1279D3
hypocenterC3#10408
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1279

Complements

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

NH ≅ G/Ntype
2. #70479C2semidirect
3. C2×#4485C2semidirect
4. #69772C2semidirect
5. C2×#4459C2semidirect
6. #69466C2semidirect
7. C3×#1279C2semidirect
34. #1279D3semidirect
79. 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
#737992
#704792
C2×#44852
#697722
C2×#44592
#694662
C3×#12792
#104083