Finite Group Databasebeta

← #10697576

Subgroups — #10697576

Order 768.

658 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 103 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
1#82476C211yesmaximal, characteristic, complemented
2#82441C211yesmaximal, characteristic, complemented
3#79303C211yesmaximal, characteristic
4C3×#1479C211yesmaximal, characteristic, complemented
5#70789C211yesmaximal, characteristic, complemented
6#70774C211yesmaximal, characteristic, complemented
7#67746C211yesmaximal, characteristic, complemented
8#10242—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10C2×#947—21no—
11C2×#942—21no—
12#5477C2211yescharacteristic
13C2×#874C2211yescharacteristic
14C3×#431C2211yescharacteristic
15C3×#431C2211yescharacteristic
16#5031—21no—
17#5028—22no—
18#5027—21no—
19#4991—22no—
20#4988—21no—
21#4966—21no—
22#4939—22no—
23C4×C4⋊Dic3C2211yescharacteristic
24#4924C2211yescharacteristic
25#4843—22no—
26#4732—22no—
27#4695—21no—
28#4694—21no—
29C3×Q8⋊C8—22no—
30#4485—22no—
31#4470—22no—
32#3205—31no—
33#3037—31no—
34#2548—31no—
35#1545—31no—
36#1502—31no—
37#1479D311yescharacteristic, complemented
38#1303—31no—
39C22×C3×Q8D411yescharacteristic
40C2×#235—21no—
41C22×D12—21no—
42C22×D12—21no—
43C22×D12D412yes—
44C3×C4×Q8—21no—
45C3×C4×Q8—21no—
46C3×C4×Q8—21no—
47C3×C4×Q8—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4D411yescharacteristic
50C2×C3×C42—21no—
51C2×C3×C42C2311yescharacteristic
52#947—42no—
53#947—42no—
54C2×#216—22no—
55#942—42no—
56#942—42no—
57C2×#215—22no—
58#930—21no—
59#930—21no—
60C2×(C2×D3)⋊C4—24no—
61C2×(C2×D3)⋊C4—24no—
62C2×C4⋊Dic3—21no—
63C2×C4×Dic3—21no—
64C2×#208—21no—
65C2×D24—24no—
66#875—21no—
67#874—21no—
68#874—21no—
69#874—21no—
70#874—21no—
71#874D411yescharacteristic
72C4×D12—21no—
73C4×D12—21no—
74#869D411yescharacteristic
75C3×C4⋊C8—21no—
76C3×C4⋊C8D412yes—
77C3×C22⋊C8—21no—
78C3×C22⋊C8—21no—
79C3×C4×C8—21no—
80C3×C4×C8—21no—
81C3⋊#95D412yes—
82#821—24no—
83#810—22no—
84#809—22no—
85C4⋊(C3⋊C8)—21no—
86C4⋊(C3⋊C8)D412yes—
87C2×#137—61no—
88#532—31no—
89C2×#127—31no—
90C2×#124—61no—
91C2×C4×Q8C2×D311yescharacteristic
92#493—61no—
93#486—61no—
94#464—62no—
95#462—61no—
96#460—61no—
97#459—62no—
98#447—62no—
99#431C2×D311yescharacteristic
100#431C2×D311yescharacteristic
101#423—31no—
102#417—61no—
103#394—62no—
104#390—62no—
105#390—61no—
106C4×C4⋊C4—31no—
107#331—62no—
108#327—62no—
109Q8⋊C8—22no—
110C23×D3—22no—
111C2×C3×Q8—21no—
112C2×C3×Q8—21no—
113C2×C3×Q8—21no—
114C2×C3×Q8—21no—
115C2×C3×Q8—21no—
116C2×C3×Q8D812yescomplemented
117C22×C3×C4—21no—
118C22×C3×C4—21no—
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C2×#67—41no—
123C22×Dic3—22no—
124#235—41no—
125#235—41no—
126#235—41no—
127#235—41no—
128C2×D12—21no—
129C2×D12—22no—
130C2×D12—21no—
131C2×D12—21no—
132C2×D12—22no—
133C2×D12—44no—
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×D12—82no—
143C2×C4×D3—41no—
144C2×Dic6—21no—
145C2×C3×C8—22no—
146C2×C3×C8—41no—
147C3×C4⋊C4—21no—
148C3×C4⋊C4—21no—
149C3×C4⋊C4—41no—
150C3×C4⋊C4—41no—
151C3×C42—41no—
152C3×C42—41no—
153C3×C42C2×D411yescharacteristic
154C3×C42C2×D411yescharacteristic
155C3×C42C2×D411yescharacteristic
156C3×C42#4111yescharacteristic
157#216—44no—
158#215—44no—
159(C2×D3)⋊C4—48no—
160(C2×D3)⋊C4—41no—
161(C2×D3)⋊C4—41no—
162(C2×D3)⋊C4—84no—
163C4⋊Dic3—21no—
164C4⋊Dic3—21no—
165C4⋊Dic3—42no—
166C4×Dic3—41no—
167C4×Dic3—41no—
168#208—41no—
169#208—41no—
170C2×C3⋊C8—22no—
171D24—84no—
172C2×#41—61no—
173C22×Q8D1211yescharacteristic
174C22×D4—32no—
175C22×D4—61no—
176C22×D4—61no—
177#137—122no—
178#137—122no—
179C2×Q16—62no—
180C2×SD16—62no—
181C2×D8—64no—
182C2×M16—61no—
183C4⋊Q8—31no—
184#127—31no—
185#127—61no—
186#127—61no—
187#127—61no—
188#127—61no—
189#124—61no—
190#124—122no—
191#124—122no—
192#121—61no—
193#121—61no—
194C4×Q8—21no—
195C4×Q8—21no—
196C4×Q8—21no—
197C4×Q8—21no—
198C4×D4—61no—
199C4×D4—61no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—61no—
202C2×C4⋊C4D1211yescharacteristic
203C2×C22⋊C4—64no—
204C2×C22⋊C4—64no—
205C2×C42—21no—
206C2×C42—61no—
207C2×C42C22×D311yescharacteristic
208C4⋊C8—21no—
209C4⋊C8—32no—
210C4⋊C8—61no—
211C4⋊C8#6712yes—
212#103—62no—
213#102—62no—
214#102—64no—
215C22⋊C8—21no—
216C22⋊C8—21no—
217C4×C8—21no—
218C4×C8—21no—
219#95—32no—
220C23×C3#12011yescharacteristic
221C22×D3—44no—
222C22×D3—44no—
223C22×D3—41no—
224C22×D3—42no—
225C22×D3—42no—
226C22×D3—42no—
227C3×Q8—22no—
228C3×Q8—22no—
229C3×Q8—41no—
230C3×Q8—41no—
231C2×C3×C4—22no—
232C2×C3×C4—22no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—42no—
237C2×C3×C4—41no—
238C2×C3×C4#12011yescharacteristic
239C2×C3×C4#12011yescharacteristic
240C2×C3×C4#12111yescharacteristic
241C2×C3×C4#12111yescharacteristic
242C2×C3×C4#12111yescharacteristic
243C2×C3×C4#12711yescharacteristic
244C2×C3×C4C2×D811yescharacteristic
245C2×C3×C4#13511yescharacteristic
246C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
247C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
248#67—81no—
249#67—81no—
250C2×Dic3—42no—
251C2×Dic3—42no—
252C2×Dic3—41no—
253C2×Dic3—42no—
254D12—41no—
255D12—41no—
256D12—42no—
257D12—44no—
258D12—81no—
259D12—82no—
260D12—82no—
261D12—82no—
262D12—81no—
263D12—81no—
264D12—81no—
265D12—81no—
266C4×D3—81no—
267C4×D3—81no—
268Dic6—41no—
269Dic6—41no—
270C3×C8—42no—
271C3×C8—81no—
272C3⋊C8—42no—
273C24—62no—
274#41—121no—
275#41—121no—
276#41—121no—
277#41—121no—
278C2×Q8—21no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—61no—
284C2×Q8D2412yescomplemented
285C2×D4—61no—
286C2×D4—61no—
287C2×D4—62no—
288C2×D4—62no—
289C2×D4—61no—
290C2×D4—124no—
291C2×D4—121no—
292C2×D4—121no—
293C2×D4—121no—
294C2×D4—121no—
295C2×D4—121no—
296C2×D4—121no—
297C2×D4—121no—
298C2×D4—121no—
299C2×D4—121no—
300C2×D4—242no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—62no—
304C22×C4—121no—
305C22×C4C2×D1211yescharacteristic
306C22×C4D3×D411yescharacteristic
307C22×C4D3×D411yescharacteristic
308Q16—124no—
309SD16—124no—
310D8—244no—
311M16—121no—
312M16—121no—
313C2×C8—22no—
314C2×C8—41no—
315C2×C8—62no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—61no—
321C4⋊C4—61no—
322C4⋊C4—122no—
323C22⋊C4—128no—
324C22⋊C4—121no—
325C22⋊C4—121no—
326C22⋊C4—244no—
327C42—41no—
328C42—41no—
329C42—121no—
330C42—121no—
331C42#23511yescharacteristic
332C42D3×D411yescharacteristic
333C42D3×D411yescharacteristic
334C42C2×#6711yescharacteristic
335C22×C3—21no—
336C22×C3—21no—
337C22×C3#39211yescharacteristic
338C22×C3#44711yescharacteristic
339C22×C3#44911yescharacteristic
340C2×D3—84no—
341C2×D3—84no—
342C2×D3—81no—
343C2×D3—82no—
344C2×D3—82no—
345C2×D3—81no—
346C2×D3—82no—
347C2×D3—81no—
348C3×C4—21no—
349C3×C4—21no—
350C3×C4—21no—
351C3×C4—42no—
352C3×C4—81no—
353C3×C4#44711yescharacteristic
354C3×C4#44911yescharacteristic
355C3×C4#45512yes—
356C3×C4#46611yescharacteristic
357C3×C4#46911yescharacteristic
358Dic3—82no—
359Dic3—81no—
360C23—124no—
361C23—124no—
362C23—121no—
363C23—122no—
364C23—122no—
365C23—122no—
366C23#88211yescharacteristic
367Q8—22no—
368Q8—22no—
369Q8—41no—
370Q8—41no—
371Q8—121no—
372Q8—121no—
373D4—122no—
374D4—124no—
375D4—121no—
376D4—121no—
377D4—242no—
378D4—241no—
379D4—242no—
380D4—242no—
381D4—241no—
382D4—241no—
383D4—241no—
384D4—241no—
385D4—241no—
386D4—241no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—122no—
395C2×C4—122no—
396C2×C4—121no—
397C2×C4—122no—
398C2×C4—241no—
399C2×C4—241no—
400C2×C4#88211yescharacteristic
401C2×C4#88411yescharacteristic
402C2×C4#88411yescharacteristic
403C2×C4C2×D2411yescharacteristic
404C2×C4#90411yescharacteristic
405C2×C4#90811yescharacteristic
406C2×C4#90811yescharacteristic
407C2×C4#93011yescharacteristic
408C2×C4#93711yescharacteristic
409C2×C4#94011yescharacteristic
410C8—42no—
411C8—81no—
412C8—122no—
413C2×C3—21no—
414C2×C3—21no—
415C2×C3#199911yescharacteristic
416C2×C3#200411yescharacteristic
417C2×C3#200811yescharacteristic
418D3—161no—
419D3—162no—
420C22—21no—
421C22—21no—
422C22—244no—
423C22—244no—
424C22—242no—
425C22—241no—
426C22—241no—
427C22—242no—
428C22—242no—
429C22—241no—
430C22#473211yescharacteristic
431C22#473411yescharacteristic
432C22#496011yescharacteristic
433C4—21no—
434C4—21no—
435C4—21no—
436C4—42no—
437C4—81no—
438C4—242no—
439C4—241no—
440C4#473211yescharacteristic
441C4#473411yescharacteristic
442C4#511511yescharacteristic
443C4#513511yescharacteristic
444C4#520212yes—
445C3#1024211yescharacteristic, complemented, Sylow
446C2—21no—
447C2—21no—
448C2—482no—
449C2—481no—
450C2#6996111yescharacteristic
451C2#7034411yescharacteristic
452C2#7040911yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1479C2
socleC22×C3#392
prime radical#79303C2
hypercenter#1479D3
hypocenterC3#10242
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1479

Complements

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

NH ≅ G/Ntype
1. #82476C2semidirect
2. #82441C2semidirect
4. C3×#1479C2semidirect
5. #70789C2semidirect
6. #70774C2semidirect
7. #67746C2semidirect
37. #1479D3semidirect
116. C2×C3×Q8D8semidirect
284. C2×Q8D24semidirect
445. C3#10242semidirect

Maximal subgroups

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

subgroupindex
#824762
#824412
#793032
C3×#14792
#707892
#707742
#677462
#102423