Finite Group Databasebeta

← #10696026

Subgroups — #10696026

Order 768.

734 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 734 conjugacy classes of subgroups — too many to draw. Only its 101 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×#3100C211yesmaximal, characteristic, complemented
2#83298C211yesmaximal, characteristic, complemented
3#82347C211yesmaximal, characteristic, complemented
4#71816C211yesmaximal, characteristic, complemented
5#71598C211yesmaximal, characteristic, complemented
6#70762C211yesmaximal, characteristic, complemented
7#67737C211yesmaximal, characteristic, complemented
8#10012—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C3×C4×D4C2211yescharacteristic
11C2×C4×#67—22no—
12C2×C4×D12C2211yescharacteristic
13C3×#480—21no—
14C3×#474—21no—
15C3×#459—21no—
16C3×#447—21no—
17C3×C4×D8—22no—
18C3×#431C2211yescharacteristic
19C2×#837—22no—
20#5223—21no—
21#5087—21no—
22#5072—21no—
23#5036—21no—
24#5033—21no—
25C4×#213—22no—
26#4993—21no—
27#4960—21no—
28#4949—21no—
29#4932C2211yescharacteristic
30#4925C2211yescharacteristic
31#4924C2211yescharacteristic
32D4⋊(C3⋊C8)—22no—
33#4459—22no—
34#3100—31no—
35#3100D311yescharacteristic, complemented
36#2397—31no—
37#1963—31no—
38#1486—31no—
39#1486—31no—
40#1274—31no—
41C2×C3×#41—21no—
42C22×C3×D4D411yescharacteristic
43C23×C3×C4—21no—
44C22×#67—22no—
45C22×D12D411yescharacteristic
46C22×C4×D3—21no—
47C2×C3×D8—22no—
48C3×C4⋊Q8D411yescharacteristic, complemented
49C3×#127D411yescharacteristic, complemented
50C3×#124—21no—
51C3×#121—21no—
52C3×#121—21no—
53C3×C4×D4—21no—
54C3×C4×D4—21no—
55C3×C4×D4—21no—
56C3×C4×D4—21no—
57C3×C4×D4—21no—
58C3×C4×D4—21no—
59C2×C3×C4⋊C4D411yescharacteristic
60C2×C3×C22⋊C4—21no—
61C2×C3×C42C2311yescharacteristic
62C2×#217—22no—
63C2×#213—22no—
64C4×#67—44no—
65C4×#67—44no—
66C2×(C2×D3)⋊C4—22no—
67C2×(C2×D3)⋊C4—21no—
68C2×C4⋊Dic3D411yescharacteristic
69C2×Dic3⋊C4—22no—
70C2×C4×Dic3D412yes—
71C2×#208—21no—
72C4×D12—21no—
73C4×D12—21no—
74C4×D12—21no—
75C4×D12—21no—
76C3×#107—22no—
77C3×C4⋊C8—21no—
78C3×#102—22no—
79C3×#102—22no—
80C3×C22⋊C8—21no—
81C3×C4×C8D412yescomplemented
82#837—44no—
83#832—22no—
84C3⋊#95—21no—
85C3⋊#95—21no—
86#830—21no—
87#809—22no—
88#807—22no—
89C4⋊(C3⋊C8)—21no—
90C4⋊(C3⋊C8)—21no—
91C4⋊(C3⋊C8)—21no—
92C4×C3⋊C8D412yescomplemented
93#532C2×D311yescharacteristic
94C2×C4×D4—31no—
95C2×C4×D4—62no—
96C2×C4×D4C2×D311yescharacteristic
97#480—21no—
98#480—61no—
99#474—21no—
100#474—61no—
101#459—21no—
102#459—61no—
103#447—21no—
104#447—61no—
105C4×D8—22no—
106C4×D8—62no—
107#431—31no—
108#431C2×D311yescharacteristic
109#423—31no—
110C2×#104—62no—
111#399—61no—
112#399—61no—
113#392—61no—
114#392—61no—
115#376—31no—
116D4⋊C8—62no—
117D4⋊C8—62no—
118C24×C3—21no—
119C23×D3—21no—
120C3×#41—42no—
121C3×#41—42no—
122C2×C3×Q8—21no—
123C2×C3×D4—21no—
124C2×C3×D4—21no—
125C2×C3×D4—21no—
126C2×C3×D4—21no—
127C2×C3×D4—21no—
128C2×C3×D4—42no—
129C2×C3×D4—41no—
130C22×C3×C4—21no—
131C22×C3×C4—41no—
132C22×C3×C4—41no—
133C22×C3×C4—41no—
134C22×C3×C4—42no—
135C22×C3×C4—42no—
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C2×#67—44no—
140C2×#67—44no—
141C2×#67—44no—
142C22×Dic3—22no—
143C22×Dic3—21no—
144C2×D12—21no—
145C2×D12—21no—
146C2×D12—42no—
147C2×C4×D3—41no—
148C2×C4×D3—41no—
149C2×C4×D3—42no—
150C2×C4×D3—42no—
151C3×D8—44no—
152C2×C3×C8—22no—
153C3×C4⋊C4—21no—
154C3×C4⋊C4—21no—
155C3×C4⋊C4—21no—
156C3×C4⋊C4—21no—
157C3×C22⋊C4—41no—
158C3×C22⋊C4—41no—
159C3×C22⋊C4—41no—
160C3×C22⋊C4—41no—
161C3×C42C2×D411yescharacteristic
162C3×C42C2×D411yescharacteristic
163C3×C42C2×D411yescharacteristic
164C3×C42#4111yescharacteristic
165#217—44no—
166#213—44no—
167(C2×D3)⋊C4—44no—
168(C2×D3)⋊C4—41no—
169(C2×D3)⋊C4—41no—
170C4⋊Dic3—21no—
171C4⋊Dic3—21no—
172Dic3⋊C4—44no—
173C4×Dic3—22no—
174C4×Dic3D814yescomplemented
175#208—42no—
176C2×C3⋊C8—22no—
177C2×C3⋊C8—41no—
178C2×#41—21no—
179C22×D4—31no—
180C22×D4—62no—
181C22×D4#6711yescharacteristic
182C23×C4—21no—
183C23×C4—61no—
184C2×D8—22no—
185C2×D8—62no—
186C2×M16—61no—
187C4⋊Q8#6711yescharacteristic, complemented
188#127#6711yescharacteristic, complemented
189#124—21no—
190#121—21no—
191#121—21no—
192C4×D4—21no—
193C4×D4—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—
202C4×D4—124no—
203C4×D4—124no—
204C2×C4⋊C4—31no—
205C2×C4⋊C4—62no—
206C2×C4⋊C4#6711yescharacteristic
207C2×C22⋊C4—21no—
208C2×C22⋊C4—62no—
209C2×C22⋊C4—62no—
210C2×C22⋊C4—61no—
211C2×C42—32no—
212C2×C42C22×D311yescharacteristic
213#107—22no—
214#107—62no—
215C4⋊C8—21no—
216C4⋊C8—61no—
217C4⋊C8—61no—
218C4⋊C8—61no—
219#104—124no—
220#102—22no—
221#102—22no—
222#102—62no—
223#102—62no—
224C22⋊C8—21no—
225C22⋊C8—61no—
226C4×C8—32no—
227C4×C8#6712yescomplemented
228#95—61no—
229#95—61no—
230C23×C3—22no—
231C23×C3—42no—
232C23×C3—41no—
233C23×C3—41no—
234C23×C3—42no—
235C23×C3—41no—
236C23×C3#12011yescharacteristic
237C22×D3—22no—
238C22×D3—41no—
239C22×D3—42no—
240C22×D3—41no—
241C22×D3—42no—
242C3×Q8—42no—
243C3×D4—22no—
244C3×D4—22no—
245C3×D4—41no—
246C3×D4—41no—
247C3×D4—42no—
248C3×D4—42no—
249C3×D4—42no—
250C3×D4—82no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—41no—
255C2×C3×C4—42no—
256C2×C3×C4—82no—
257C2×C3×C4—81no—
258C2×C3×C4—81no—
259C2×C3×C4—81no—
260C2×C3×C4—81no—
261C2×C3×C4#12011yescharacteristic
262C2×C3×C4#12011yescharacteristic
263C2×C3×C4#12111yescharacteristic
264C2×C3×C4#12111yescharacteristic
265C2×C3×C4#12111yescharacteristic
266C2×C3×C4#12711yescharacteristic
267C2×C3×C4C2×D812yes—
268C2×C3×C4AGL(1,ℤ/8)12yes—
269#67—416no—
270C2×Dic3—24no—
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—81no—
281C4×D3—81no—
282C3×C8—42no—
283C3⋊C8—42no—
284C3⋊C8—81no—
285C24—21no—
286C24—61no—
287#41—42no—
288#41—42no—
289C2×Q8—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—42no—
296C2×D4—41no—
297C2×D4—61no—
298C2×D4—61no—
299C2×D4—122no—
300C2×D4—124no—
301C2×D4—124no—
302C2×D4—124no—
303C22×C4—21no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—62no—
310C22×C4—61no—
311C22×C4—122no—
312C22×C4—122no—
313C22×C4—121no—
314C22×C4—121no—
315C22×C4D3×D411yescharacteristic
316C22×C4D3×D411yescharacteristic
317C22×C4C2×#6711yescharacteristic
318D8—44no—
319D8—124no—
320M16—122no—
321C2×C8—22no—
322C2×C8—62no—
323C2×C8—121no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—61no—
329C4⋊C4—61no—
330C4⋊C4—124no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—124no—
336C22⋊C4—124no—
337C22⋊C4—121no—
338C22⋊C4—121no—
339C42—34no—
340C42—62no—
341C42D3×D411yescharacteristic
342C42#23711yescharacteristic
343C42C2×#6711yescharacteristic
344C42C2×#6711yescharacteristic
345C22×C3—21no—
346C22×C3—21no—
347C22×C3—44no—
348C22×C3—42no—
349C22×C3—82no—
350C22×C3—81no—
351C22×C3—81no—
352C22×C3—81no—
353C22×C3—81no—
354C22×C3—81no—
355C22×C3#39211yescharacteristic
356C22×C3#44712yes—
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#45912yes—
369C3×C4#49311yescharacteristic
370C3×C4#49611yescharacteristic
371Dic3—44no—
372Dic3—81no—
373C23—22no—
374C23—41no—
375C23—42no—
376C23—42no—
377C23—41no—
378C23—41no—
379C23—62no—
380C23—122no—
381C23—121no—
382C23—122no—
383C23—121no—
384C23#93711yescharacteristic
385Q8—42no—
386D4—22no—
387D4—22no—
388D4—41no—
389D4—41no—
390D4—42no—
391D4—42no—
392D4—42no—
393D4—62no—
394D4—62no—
395D4—82no—
396D4—121no—
397D4—121no—
398D4—1216no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—64no—
405C2×C4—82no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—122no—
411C2×C4—122no—
412C2×C4—122no—
413C2×C4—121no—
414C2×C4—242no—
415C2×C4—241no—
416C2×C4—241no—
417C2×C4D3×D812yes—
418C2×C4#91112yes—
419C2×C4#93711yescharacteristic
420C2×C4#93811yescharacteristic
421C2×C4#93911yescharacteristic
422C2×C4#93911yescharacteristic
423C2×C4#94011yescharacteristic
424C2×C4#95311yescharacteristic
425C8—42no—
426C8—122no—
427C8—241no—
428C2×C3—22no—
429C2×C3—82no—
430C2×C3—161no—
431C2×C3#201011yescharacteristic
432C2×C3#201011yescharacteristic
433C2×C3#201111yescharacteristic
434D3—82no—
435C22—21no—
436C22—21no—
437C22—44no—
438C22—42no—
439C22—82no—
440C22—81no—
441C22—81no—
442C22—81no—
443C22—81no—
444C22—81no—
445C22—124no—
446C22—122no—
447C22—242no—
448C22—241no—
449C22—241no—
450C22#515612yes—
451C22#522311yescharacteristic
452C4—21no—
453C4—21no—
454C4—21no—
455C4—21no—
456C4—81no—
457C4—81no—
458C4—124no—
459C4—241no—
460C4#515012yes—
461C4#515111yescharacteristic
462C4#515411yescharacteristic
463C3#1001211yescharacteristic, complemented, Sylow
464C2—22no—
465C2—82no—
466C2—161no—
467C2—242no—
468C2#7171511yescharacteristic
469C2#7204411yescharacteristic
470C2#7211811yescharacteristic
subgroupG/H
centerC2×C4#940
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3100C2
socleC22×C3#392
hypercenter#3100D3
hypocenterC3#10012
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3100

Complements

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

NH ≅ G/Ntype
1. C3×#3100C2semidirect
2. #83298C2semidirect
3. #82347C2semidirect
4. #71816C2semidirect
5. #71598C2semidirect
6. #70762C2semidirect
7. #67737C2semidirect
35. #3100D3semidirect
48. C3×C4⋊Q8D4semidirect
49. C3×#127D4semidirect
81. C3×C4×C8D4semidirect
92. C4×C3⋊C8D4semidirect
174. C4×Dic3D8semidirect
187. C4⋊Q8#67semidirect
188. #127#67semidirect
227. C4×C8#67semidirect
463. C3#10012semidirect

Maximal subgroups

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

subgroupindex
C3×#31002
#832982
#823472
#718162
#715982
#707622
#677372
#100123