Finite Group Databasebeta

← #10695818

Subgroups — #10695818

Order 768.

822 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 822 conjugacy classes of subgroups — too many to draw. Only its 81 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3006C211yesmaximal, characteristic, complemented
2#84082C211yesmaximal, characteristic, complemented
3#82722C211yesmaximal, characteristic, complemented
4#72139C211yesmaximal, characteristic, complemented
5#71674C211yesmaximal, characteristic, complemented
6#69978C211yesmaximal, characteristic, complemented
7#67686C211yesmaximal, characteristic, complemented
8#9980—31nomaximal, Sylow
9C22×C3×#41C2211yescharacteristic
10C2×C3×AGL(1,ℤ/8)—21no—
11C2×C3×#121C2211yescharacteristic
12C2×#949—21no—
13C2×#930C2211yescharacteristic
14C3×#449—22no—
15C3×#448—21no—
16C3×#447—21no—
17C3×#417—21no—
18C3×#407C2211yescharacteristic
19#5238—22no—
20#5232—21no—
21#5227—21no—
22#5223—21no—
23#5223—22no—
24C2×#834—22no—
25#5217—21no—
26#5211C2211yescharacteristic
27#5211C2211yescharacteristic
28#5207C2211yescharacteristic
29#4992—21no—
30#4960—21no—
31#4953—21no—
32#4537—22no—
33#4470—22no—
34#3006—31no—
35#3006D311yescharacteristic, complemented
36#2815—31no—
37#1782—31no—
38#1510—31no—
39#1510—31no—
40#1306—31no—
41C2×C3×#41—21no—
42C2×C3×#41—21no—
43C2×C3×#41—21no—
44C2×C3×#41—21no—
45C2×C3×#41—44no—
46C22×C3×Q8D411yescharacteristic, complemented
47C22×C3×D4—21no—
48C22×C3×D4—21no—
49C22×C3×D4D411yescharacteristic
50C22×C3×D4D411yescharacteristic, complemented
51C23×C3×C4—21no—
52C23×C3×C4C2311yescharacteristic
53C22×#67—21no—
54C22×#67—22no—
55C22×D12D411yescharacteristic
56C3×AGL(1,ℤ/8)—44no—
57C2×C3×SD16—22no—
58C2×C3×D8—22no—
59C2×C3×M16—21no—
60C3×#121—21no—
61C3×#121—21no—
62C3×#121—21no—
63C3×#121—21no—
64C2×C3×C4⋊C4D411yescharacteristic
65C2×C3×C22⋊C4—21no—
66C2×#217—21no—
67C2×#217—22no—
68C2×#217D412yes—
69#949—44no—
70C2×#215—22no—
71C2×#213—22no—
72#930—21no—
73#930—21no—
74#930—21no—
75#930—21no—
76C2×(C2×D3)⋊C4—21no—
77C2×(C2×D3)⋊C4—22no—
78C2×C4⋊Dic3D411yescharacteristic
79C2×#208—21no—
80C3×#103—22no—
81C3×#102—22no—
82C3×C22⋊C8—21no—
83C3×C22⋊C8D412yescomplemented
84#835—22no—
85#834—44no—
86#832—22no—
87C3⋊#95—22no—
88C3⋊#95—21no—
89C3⋊#95—21no—
90C3⋊#95—21no—
91C3⋊#95—21no—
92#830—21no—
93#830D412yescomplemented
94C22×#41C2×D311yescharacteristic
95C2×AGL(1,ℤ/8)—21no—
96C2×AGL(1,ℤ/8)—61no—
97C2×#121—31no—
98C2×#121C2×D311yescharacteristic
99#449—22no—
100#449—62no—
101#448—21no—
102#448—61no—
103#447—21no—
104#447—61no—
105#417—21no—
106#417—61no—
107C2×#99—62no—
108#407—31no—
109#407C2×D311yescharacteristic
110#397—61no—
111#397—61no—
112#392—61no—
113#392—62no—
114#392—61no—
115#381—31no—
116#381—31no—
117#327—62no—
118#327—62no—
119C24×C3—21no—
120C24×C3—21no—
121C24×C3C2×D411yescharacteristic
122C23×D3—21no—
123C3×#41—24no—
124C3×#41—24no—
125C3×#41—42no—
126C3×#41—42no—
127C3×#41—42no—
128C3×#41—42no—
129C3×#41—42no—
130C3×#41—42no—
131C2×C3×Q8—21no—
132C2×C3×Q8—21no—
133C2×C3×Q8—42no—
134C2×C3×D4—24no—
135C2×C3×D4—21no—
136C2×C3×D4—21no—
137C2×C3×D4—21no—
138C2×C3×D4—21no—
139C2×C3×D4—42no—
140C2×C3×D4—42no—
141C2×C3×D4—42no—
142C2×C3×D4—41no—
143C2×C3×D4—41no—
144C2×C3×D4—41no—
145C2×C3×D4—41no—
146C2×C3×D4—82no—
147C22×C3×C4—21no—
148C22×C3×C4—21no—
149C22×C3×C4—42no—
150C22×C3×C4—42no—
151C22×C3×C4—42no—
152C22×C3×C4—41no—
153C22×C3×C4—41no—
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4#4111yescharacteristic
160C2×#67—41no—
161C2×#67—41no—
162C2×#67—82no—
163C2×#67—82no—
164C2×#67—82no—
165C2×#67—82no—
166C22×Dic3—22no—
167C22×Dic3—21no—
168C22×Dic3—41no—
169C2×D12—21no—
170C2×D12—21no—
171C2×D12—42no—
172C3×SD16—44no—
173C3×D8—44no—
174C3×M16—42no—
175C2×C3×C8—22no—
176C3×C4⋊C4—21no—
177C3×C4⋊C4—21no—
178C3×C22⋊C4—41no—
179C3×C22⋊C4—41no—
180#217—42no—
181#217—42no—
182#217—82no—
183#215—44no—
184#213—44no—
185(C2×D3)⋊C4—41no—
186(C2×D3)⋊C4—41no—
187(C2×D3)⋊C4—82no—
188C4⋊Dic3—21no—
189C4⋊Dic3—21no—
190#208—42no—
191C2×C3⋊C8—22no—
192C2×#41—21no—
193C2×#41—21no—
194C2×#41—21no—
195C2×#41—21no—
196C2×#41—44no—
197C22×Q8#6711yescharacteristic, complemented
198C22×D4—21no—
199C22×D4—21no—
200C22×D4—31no—
201C22×D4—61no—
202C22×D4—62no—
203C22×D4#6711yescharacteristic
204C22×D4#6711yescharacteristic, complemented
205C23×C4—21no—
206C23×C4C22×D311yescharacteristic
207AGL(1,ℤ/8)—44no—
208AGL(1,ℤ/8)—124no—
209C2×SD16—22no—
210C2×SD16—62no—
211C2×D8—22no—
212C2×D8—62no—
213C2×M16—21no—
214C2×M16—61no—
215#121—21no—
216#121—21no—
217#121—21no—
218#121—21no—
219#121—61no—
220#121—61no—
221#121—61no—
222#121—61no—
223C2×C4⋊C4—31no—
224C2×C4⋊C4#6711yescharacteristic
225C2×C22⋊C4—21no—
226C2×C22⋊C4—32no—
227C2×C22⋊C4—61no—
228C2×C22⋊C4—61no—
229C2×C22⋊C4—62no—
230C2×C22⋊C4—62no—
231#103—22no—
232#103—62no—
233#102—22no—
234#102—62no—
235#99—124no—
236C22⋊C8—21no—
237C22⋊C8—32no—
238C22⋊C8—61no—
239C22⋊C8#6712yescomplemented
240#95—62no—
241#95—61no—
242#95—61no—
243#95—61no—
244#95—61no—
245C23×C3—22no—
246C23×C3—22no—
247C23×C3—41no—
248C23×C3—41no—
249C23×C3—41no—
250C23×C3—41no—
251C23×C3—41no—
252C23×C3—42no—
253C23×C3—41no—
254C23×C3—41no—
255C23×C3—42no—
256C23×C3—81no—
257C23×C3—81no—
258C23×C3#12011yescharacteristic
259C23×C3#12111yescharacteristic
260C23×C3#12711yescharacteristic
261C22×D3—41no—
262C22×D3—41no—
263C22×D3—42no—
264C22×D3—41no—
265C22×D3—42no—
266C3×Q8—22no—
267C3×Q8—22no—
268C3×Q8—41no—
269C3×Q8—41no—
270C3×D4—22no—
271C3×D4—22no—
272C3×D4—41no—
273C3×D4—41no—
274C3×D4—42no—
275C3×D4—44no—
276C3×D4—44no—
277C3×D4—81no—
278C3×D4—82no—
279C3×D4—82no—
280C2×C3×C4—22no—
281C2×C3×C4—22no—
282C2×C3×C4—22no—
283C2×C3×C4—41no—
284C2×C3×C4—41no—
285C2×C3×C4—41no—
286C2×C3×C4—41no—
287C2×C3×C4—44no—
288C2×C3×C4—44no—
289C2×C3×C4—41no—
290C2×C3×C4—41no—
291C2×C3×C4—42no—
292C2×C3×C4—41no—
293C2×C3×C4#12011yescharacteristic
294C2×C3×C4#12011yescharacteristic
295C2×C3×C4#12111yescharacteristic
296C2×C3×C4#12111yescharacteristic
297#67—82no—
298#67—82no—
299#67—88no—
300C2×Dic3—42no—
301C2×Dic3—42no—
302C2×Dic3—42no—
303C2×Dic3—41no—
304C2×Dic3—42no—
305C2×Dic3—81no—
306C2×Dic3—81no—
307C2×Dic3—81no—
308D12—42no—
309D12—81no—
310C3×C8—42no—
311C3⋊C8—42no—
312C24—21no—
313C24—21no—
314C24—61no—
315C24D3×D411yescharacteristic
316#41—24no—
317#41—24no—
318#41—42no—
319#41—42no—
320#41—42no—
321#41—42no—
322#41—42no—
323#41—42no—
324C2×Q8—21no—
325C2×Q8—21no—
326C2×Q8—42no—
327C2×D4—24no—
328C2×D4—21no—
329C2×D4—21no—
330C2×D4—21no—
331C2×D4—21no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—41no—
339C2×D4—61no—
340C2×D4—61no—
341C2×D4—82no—
342C2×D4—122no—
343C2×D4—121no—
344C2×D4—121no—
345C2×D4—242no—
346C2×D4—242no—
347C2×D4—242no—
348C2×D4—242no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—62no—
357C22×C4—61no—
358C22×C4—121no—
359C22×C4D3×D411yescharacteristic
360C22×C4D3×D411yescharacteristic
361C22×C4#23711yescharacteristic
362C22×C4C2×#6711yescharacteristic
363C22×C4C2×#6711yescharacteristic
364C22×C4C2×#6711yescharacteristic
365SD16—44no—
366SD16—124no—
367D8—44no—
368D8—124no—
369M16—42no—
370M16—122no—
371C2×C8—22no—
372C2×C8—62no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—61no—
376C4⋊C4—61no—
377C22⋊C4—41no—
378C22⋊C4—41no—
379C22⋊C4—121no—
380C22⋊C4—121no—
381C22⋊C4—122no—
382C22⋊C4—122no—
383C22⋊C4—242no—
384C22⋊C4—242no—
385C22×C3—21no—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—21no—
389C22×C3—21no—
390C22×C3—21no—
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—42no—
394C22×C3—82no—
395C22×C3—82no—
396C22×C3—82no—
397C22×C3—82no—
398C22×C3—82no—
399C22×C3—81no—
400C22×C3—81no—
401C22×C3—81no—
402C22×C3—81no—
403C22×C3—81no—
404C22×C3#39211yescharacteristic
405C22×C3B(4,𝔽2)12yes—
406C2×D3—82no—
407C2×D3—82no—
408C2×D3—81no—
409C2×D3—81no—
410C2×D3—81no—
411C3×C4—21no—
412C3×C4—21no—
413C3×C4—21no—
414C3×C4—21no—
415C3×C4—42no—
416C3×C4—81no—
417Dic3—82no—
418Dic3—81no—
419Dic3—161no—
420C23—22no—
421C23—22no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—42no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—42no—
431C23—81no—
432C23—81no—
433C23—121no—
434C23—121no—
435C23—122no—
436C23—121no—
437C23—122no—
438C23#93711yescharacteristic
439C23#93911yescharacteristic
440C23#94011yescharacteristic
441Q8—22no—
442Q8—22no—
443Q8—41no—
444Q8—41no—
445D4—22no—
446D4—22no—
447D4—41no—
448D4—41no—
449D4—42no—
450D4—44no—
451D4—44no—
452D4—81no—
453D4—82no—
454D4—82no—
455D4—122no—
456D4—241no—
457D4—242no—
458D4—242no—
459D4—248no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—122no—
474C2×C4—122no—
475C2×C4—122no—
476C2×C4—121no—
477C2×C4—122no—
478C2×C4—241no—
479C2×C4—241no—
480C2×C4—241no—
481C2×C4#93711yescharacteristic
482C2×C4#93811yescharacteristic
483C2×C4#93911yescharacteristic
484C2×C4#95311yescharacteristic
485C8—42no—
486C8—122no—
487C2×C3—22no—
488C2×C3—41no—
489C2×C3—41no—
490C2×C3—82no—
491C2×C3—161no—
492C2×C3#200811yescharacteristic
493C2×C3#200811yescharacteristic
494C2×C3#202111yescharacteristic
495D3—161no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—42no—
505C22—82no—
506C22—82no—
507C22—82no—
508C22—82no—
509C22—82no—
510C22—81no—
511C22—81no—
512C22—81no—
513C22—81no—
514C22—81no—
515C22—242no—
516C22—242no—
517C22—241no—
518C22—241no—
519C22—241no—
520C22#503212yes—
521C22#522311yescharacteristic
522C4—21no—
523C4—21no—
524C4—21no—
525C4—21no—
526C4—42no—
527C4—81no—
528C4—242no—
529C4—241no—
530C4—481no—
531C3#998011yescharacteristic, complemented, Sylow
532C2—22no—
533C2—41no—
534C2—41no—
535C2—82no—
536C2—161no—
537C2—481no—
538C2#7167111yescharacteristic
539C2#7209911yescharacteristic
540C2#7211911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#3006C2
socleC22×C3#392
hypercenter#3006D3
hypocenterC3#9980
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3006

Complements

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

NH ≅ G/Ntype
1. C3×#3006C2semidirect
2. #84082C2semidirect
3. #82722C2semidirect
4. #72139C2semidirect
5. #71674C2semidirect
6. #69978C2semidirect
7. #67686C2semidirect
35. #3006D3semidirect
46. C22×C3×Q8D4semidirect
50. C22×C3×D4D4semidirect
83. C3×C22⋊C8D4semidirect
93. #830D4semidirect
197. C22×Q8#67semidirect
204. C22×D4#67semidirect
239. C22⋊C8#67semidirect
531. C3#9980semidirect

Maximal subgroups

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

subgroupindex
C3×#30062
#840822
#827222
#721392
#716742
#699782
#676862
#99803