Finite Group Databasebeta

← #10695942

Subgroups — #10695942

Order 768.

632 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 99 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×#3101C211yesmaximal, characteristic, complemented
2#83350C211yesmaximal, characteristic, complemented
3#82415C211yesmaximal, characteristic
4#71838C211yesmaximal, characteristic, complemented
5#71598C211yesmaximal, characteristic, complemented
6#70757C211yesmaximal, characteristic, complemented
7#67738C211yesmaximal, characteristic, complemented
8#10000—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4×D4C2211yescharacteristic
11C2×#930—21no—
12C2×C4×D12C2211yescharacteristic
13C3×#483—21no—
14C3×#476—21no—
15C3×#461—21no—
16C3×#450—21no—
17C3×C4×SD16—22no—
18C3×#431C2211yescharacteristic
19#5223—21no—
20#5212—21no—
21#5087—21no—
22#5084—21no—
23#5066—22no—
24#5001—21no—
25#4995—21no—
26#4992—21no—
27#4956—22no—
28#4936C2211yescharacteristic
29#4925C2211yescharacteristic
30#4924C2211yescharacteristic
31#4538—22no—
32D4⋊(C3⋊C8)—22no—
33#4459—22no—
34#3101D311yescharacteristic, complemented
35#3077—31no—
36#2535—31no—
37#1518—31no—
38#1486—31no—
39#1478—31no—
40#1286—31no—
41C22×C3×D4D411yescharacteristic
42C23×C3×C4—21no—
43C22×#67—22no—
44C22×D12D411yescharacteristic
45C22×C4×D3—21no—
46C2×C3×SD16—22no—
47C3×C4⋊Q8D411yescharacteristic, complemented
48C3×C4⋊Q8D411yescharacteristic, complemented
49C3×#125—21no—
50C3×C22⋊Q8—21no—
51C3×C22⋊Q8—21no—
52C3×C4×Q8—21no—
53C3×C4×Q8—21no—
54C3×C4×D4—21no—
55C3×C4×D4—21no—
56C3×C4×D4—21no—
57C3×C4×D4—21no—
58C3×#117—21no—
59C2×C3×C4⋊C4D411yescharacteristic
60C2×C3×C22⋊C4—21no—
61C2×C3×C42C2311yescharacteristic
62C2×#217—22no—
63#930—42no—
64#930—42no—
65C2×(C2×D3)⋊C4—22no—
66C2×(C2×D3)⋊C4—21no—
67C2×C4⋊Dic3—21no—
68C2×C4⋊Dic3D411yescharacteristic
69C2×#208—21no—
70C4×D12—21no—
71C4×D12—21no—
72C4×D12—21no—
73C4×D12—21no—
74C3×#106—22no—
75C3×C4⋊C8—21no—
76C3×#103—22no—
77C3×#102—22no—
78C3×C22⋊C8—21no—
79C3×C4×C8D412yescomplemented
80C3⋊#95—22no—
81C3⋊#95—21no—
82C3⋊#95D412yes—
83#830—21no—
84#809—22no—
85#809—22no—
86#808—22no—
87#807—22no—
88C4⋊(C3⋊C8)—21no—
89C4⋊(C3⋊C8)—21no—
90C4⋊(C3⋊C8)D412yescomplemented
91C4×C3⋊C8—21no—
92#533C2×D311yescharacteristic
93C2×#121—61no—
94C2×C4×D4—31no—
95C2×C4×D4C2×D311yescharacteristic
96#483—21no—
97#478—62no—
98#476—21no—
99#476—61no—
100#474—61no—
101#461—21no—
102#450—21no—
103C4×SD16—22no—
104#431—31no—
105#431C2×D311yescharacteristic
106#428—61no—
107#423—31no—
108#418—61no—
109#397—61no—
110#392—61no—
111#386—62no—
112#382—31no—
113#380—61no—
114#328—62no—
115D4⋊C8—62no—
116D4⋊C8—62no—
117C24×C3—21no—
118C23×D3—21no—
119C2×C3×Q8—21no—
120C2×C3×Q8—21no—
121C2×C3×D4—21no—
122C2×C3×D4—21no—
123C2×C3×D4—42no—
124C22×C3×C4—21no—
125C22×C3×C4—41no—
126C22×C3×C4—41no—
127C22×C3×C4—41no—
128C22×C3×C4—41no—
129C22×C3×C4—41no—
130C22×C3×C4—41no—
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C2×#67—44no—
135C2×#67—44no—
136C2×#67—82no—
137C22×Dic3—22no—
138C22×Dic3—21no—
139C2×D12—21no—
140C2×D12—21no—
141C2×D12—21no—
142C2×D12—21no—
143C2×D12—21no—
144C2×D12SD1612yescomplemented
145C2×C4×D3—41no—
146C2×C4×D3—41no—
147C2×C4×D3—82no—
148C3×SD16—44no—
149C2×C3×C8—22no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4—21no—
152C3×C4⋊C4—21no—
153C3×C4⋊C4—21no—
154C3×C4⋊C4—21no—
155C3×C4⋊C4—21no—
156C3×C4⋊C4—41no—
157C3×C4⋊C4—41no—
158C3×C22⋊C4—41no—
159C3×C22⋊C4—41no—
160C3×C22⋊C4—41no—
161C3×C42—41no—
162C3×C42C2×D411yescharacteristic
163C3×C42C2×D411yescharacteristic
164C3×C42C2×D411yescharacteristic
165C3×C42#4111yescharacteristic
166#217—82no—
167(C2×D3)⋊C4—44no—
168(C2×D3)⋊C4—41no—
169(C2×D3)⋊C4—41no—
170C4⋊Dic3—21no—
171C4⋊Dic3—21no—
172C4⋊Dic3—41no—
173C4⋊Dic3—41no—
174#208—41no—
175#208—41no—
176C2×C3⋊C8—22no—
177C2×C3⋊C8—41no—
178C22×D4—31no—
179C22×D4—62no—
180C22×D4#6711yescharacteristic
181C23×C4—21no—
182C23×C4—61no—
183C2×SD16—22no—
184C2×M16—61no—
185C4⋊Q8#6711yescharacteristic, complemented
186C4⋊Q8#6711yescharacteristic, complemented
187#125—21no—
188C22⋊Q8—21no—
189C22⋊Q8—21no—
190#121—122no—
191#121—122no—
192C4×Q8—21no—
193C4×Q8—21no—
194C4×D4—21no—
195C4×D4—21no—
196C4×D4—21no—
197C4×D4—21no—
198C4×D4—61no—
199C4×D4—61no—
200C4×D4—61no—
201C4×D4—61no—
202#117—21no—
203C2×C4⋊C4—31no—
204C2×C4⋊C4—61no—
205C2×C4⋊C4#6711yescharacteristic
206C2×C22⋊C4—21no—
207C2×C22⋊C4—62no—
208C2×C22⋊C4—61no—
209C2×C22⋊C4—62no—
210C2×C42C22×D311yescharacteristic
211#107—62no—
212#106—22no—
213#106—62no—
214C4⋊C8—21no—
215C4⋊C8—32no—
216C4⋊C8—61no—
217C4⋊C8—61no—
218#103—22no—
219#102—22no—
220#102—62no—
221#102—62no—
222C22⋊C8—21no—
223C22⋊C8—61no—
224C4×C8—61no—
225C4×C8#6712yescomplemented
226#95—32no—
227#95—62no—
228#95—61no—
229C23×C3—22no—
230C23×C3—41no—
231C23×C3—41no—
232C23×C3—41no—
233C23×C3—41no—
234C23×C3—41no—
235C23×C3—41no—
236C23×C3#12011yescharacteristic
237C22×D3—22no—
238C22×D3—42no—
239C22×D3—42no—
240C22×D3—41no—
241C22×D3—41no—
242C3×Q8—42no—
243C3×Q8—81no—
244C3×D4—22no—
245C3×D4—22no—
246C3×D4—41no—
247C3×D4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—41no—
255C2×C3×C4—81no—
256C2×C3×C4—81no—
257C2×C3×C4—81no—
258C2×C3×C4—81no—
259C2×C3×C4#12011yescharacteristic
260C2×C3×C4#12011yescharacteristic
261C2×C3×C4#12111yescharacteristic
262C2×C3×C4#12111yescharacteristic
263C2×C3×C4#12111yescharacteristic
264C2×C3×C4#12711yescharacteristic
265C2×C3×C4C2×SD1611yescharacteristic
266C2×C3×C4#13511yescharacteristic
267C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
268C2×C3×C4#13711yescharacteristic
269#67—88no—
270C2×Dic3—42no—
271C2×Dic3—42no—
272C2×Dic3—42no—
273C2×Dic3—41no—
274C2×Dic3—42no—
275D12—22no—
276D12—22no—
277D12—41no—
278D12—41no—
279C4×D3—82no—
280C4×D3—82no—
281C3×C8—42no—
282C3⋊C8—42no—
283C3⋊C8—81no—
284C24—21no—
285C24—61no—
286C2×Q8—21no—
287C2×Q8—21no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—32no—
291C2×D4—42no—
292C2×D4—61no—
293C2×D4—61no—
294C2×D4—61no—
295C2×D4—61no—
296C2×D4—61no—
297C2×D4—124no—
298C2×D4—124no—
299C2×D4—242no—
300C22×C4—21no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—62no—
308C22×C4—61no—
309C22×C4—121no—
310C22×C4—121no—
311C22×C4—242no—
312C22×C4D3×D411yescharacteristic
313C22×C4D3×D411yescharacteristic
314C22×C4C2×#6711yescharacteristic
315SD16—44no—
316M16—121no—
317M16—121no—
318C2×C8—22no—
319C2×C8—62no—
320C2×C8—121no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—61no—
330C4⋊C4—61no—
331C4⋊C4—121no—
332C4⋊C4—121no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—124no—
337C22⋊C4—121no—
338C22⋊C4—121no—
339C22⋊C4—242no—
340C42—41no—
341C42D3×D411yescharacteristic
342C42#23711yescharacteristic
343C42C2×#6711yescharacteristic
344C42C2×#6711yescharacteristic
345C22×C3—21no—
346C22×C3—21no—
347C22×C3—42no—
348C22×C3—42no—
349C22×C3—42no—
350C22×C3—81no—
351C22×C3—81no—
352C22×C3—81no—
353C22×C3—81no—
354C22×C3#39211yescharacteristic
355C22×C3#45011yescharacteristic
356C22×C3#45211yescharacteristic
357C2×D3—44no—
358C2×D3—42no—
359C2×D3—82no—
360C2×D3—81no—
361C2×D3—81no—
362C3×C4—21no—
363C3×C4—21no—
364C3×C4—21no—
365C3×C4—21no—
366C3×C4—81no—
367C3×C4—81no—
368C3×C4—81no—
369C3×C4#44811yescharacteristic
370C3×C4#44911yescharacteristic
371C3×C4#46511yescharacteristic
372C3×C4#46811yescharacteristic
373Dic3—82no—
374Dic3—81no—
375C23—22no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—62no—
383C23—122no—
384C23—122no—
385C23—121no—
386C23—121no—
387C23#93711yescharacteristic
388Q8—42no—
389Q8—81no—
390D4—22no—
391D4—22no—
392D4—41no—
393D4—41no—
394D4—62no—
395D4—62no—
396D4—121no—
397D4—121no—
398D4—248no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—122no—
411C2×C4—122no—
412C2×C4—122no—
413C2×C4—122no—
414C2×C4—121no—
415C2×C4—242no—
416C2×C4—242no—
417C2×C4D3×SD1611yescharacteristic
418C2×C4#91411yescharacteristic
419C2×C4#91511yescharacteristic
420C2×C4#91611yescharacteristic
421C2×C4#93711yescharacteristic
422C2×C4#93811yescharacteristic
423C2×C4#93911yescharacteristic
424C2×C4#93911yescharacteristic
425C2×C4#94011yescharacteristic
426C2×C4#95311yescharacteristic
427C8—42no—
428C8—122no—
429C8—241no—
430C2×C3—21no—
431C2×C3—21no—
432C2×C3—82no—
433C2×C3#200011yescharacteristic
434C2×C3#200811yescharacteristic
435C2×C3#201011yescharacteristic
436D3—82no—
437C22—21no—
438C22—21no—
439C22—42no—
440C22—42no—
441C22—42no—
442C22—81no—
443C22—81no—
444C22—81no—
445C22—81no—
446C22—124no—
447C22—122no—
448C22—242no—
449C22—241no—
450C22—241no—
451C22#516911yescharacteristic
452C22#517311yescharacteristic
453C22#522311yescharacteristic
454C4—21no—
455C4—21no—
456C4—21no—
457C4—21no—
458C4—81no—
459C4—81no—
460C4—81no—
461C4—242no—
462C4—241no—
463C4#517011yescharacteristic
464C4#517111yescharacteristic
465C4#517211yescharacteristic
466C4#517411yescharacteristic
467C3#1000011yescharacteristic, complemented, Sylow
468C2—21no—
469C2—21no—
470C2—82no—
471C2—242no—
472C2#7171511yescharacteristic
473C2#7205411yescharacteristic
474C2#7211911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3101C2
socleC22×C3#392
prime radical#82415C2
hypercenter#3101D3
hypocenterC3#10000
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3101

Complements

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

NH ≅ G/Ntype
1. C3×#3101C2semidirect
2. #83350C2semidirect
4. #71838C2semidirect
5. #71598C2semidirect
6. #70757C2semidirect
7. #67738C2semidirect
34. #3101D3semidirect
47. C3×C4⋊Q8D4semidirect
48. C3×C4⋊Q8D4semidirect
79. C3×C4×C8D4semidirect
90. C4⋊(C3⋊C8)D4semidirect
144. C2×D12SD16semidirect
185. C4⋊Q8#67semidirect
186. C4⋊Q8#67semidirect
225. C4×C8#67semidirect
467. C3#10000semidirect

Maximal subgroups

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

subgroupindex
C3×#31012
#833502
#824152
#718382
#715982
#707572
#677382
#100003