Finite Group Databasebeta

← #10697562

Subgroups — #10697562

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#82379C211yesmaximal, characteristic, complemented
2#82307C211yesmaximal, characteristic
3#79312C211yesmaximal, characteristic, complemented
4C3×#1478C211yesmaximal, characteristic, complemented
5#70789C211yesmaximal, characteristic, complemented
6#70779C211yesmaximal, characteristic, complemented
7#67746C211yesmaximal, characteristic, complemented
8#10240—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10C2×#932—21no—
11C2×#924—22no—
12#5477C2211yescharacteristic
13C2×#869C2211yescharacteristic
14C3×#431C2211yescharacteristic
15C3×#431C2211yescharacteristic
16#5221—21no—
17#5019—21no—
18#5016—22no—
19#5015—21no—
20#4975—21no—
21#4966—21no—
22#4949—22no—
23C4×C4⋊Dic3C2211yescharacteristic
24#4924C2211yescharacteristic
25#4841—22no—
26#4738—22no—
27#4693—21no—
28#4681—21no—
29C3×D4⋊C8—22no—
30#4481—22no—
31#4470—22no—
32#3206—31no—
33#3036—31no—
34#2542—31no—
35#1548—31no—
36#1502—31no—
37#1478D311yescharacteristic, complemented
38#1303—31no—
39C22×C3×D4D411yescharacteristic
40C23×C3×C4—21no—
41C2×#235—21no—
42C22×Dic6D412yes—
43C3×C4×D4—21no—
44C3×C4×D4—21no—
45C3×C4×D4—21no—
46C3×C4×D4—21no—
47C2×C3×C4⋊C4D411yescharacteristic
48C2×C3×C22⋊C4—21no—
49C2×C3×C42C2311yescharacteristic
50C2×#217—24no—
51C2×#214—22no—
52#932—42no—
53#932—42no—
54C2×#213—22no—
55#930—21no—
56#930—21no—
57C2×C4⋊Dic3—21no—
58C2×C4⋊Dic3—21no—
59C2×C4⋊Dic3—21no—
60#924—48no—
61C2×Dic3⋊C4—24no—
62C2×C4×Dic3—21no—
63C2×#208—21no—
64C2×#204—24no—
65#875—21no—
66#874D411yescharacteristic
67C4×D12—21no—
68C4×D12—21no—
69#869—21no—
70#869—21no—
71#869—21no—
72#869—21no—
73#869D411yescharacteristic
74C3×C4⋊C8—21no—
75C3×C4⋊C8D412yes—
76C3×C22⋊C8—21no—
77C3×C22⋊C8—21no—
78C3×C4×C8—21no—
79C3×C4×C8—21no—
80C3⋊#95D412yes—
81#816—24no—
82#810—22no—
83#809—22no—
84C4⋊(C3⋊C8)—21no—
85C4⋊(C3⋊C8)D412yes—
86C2×AGL(1,ℤ/8)—61no—
87#532—31no—
88C2×C4⋊Q8—31no—
89C2×C22⋊Q8—62no—
90C2×C4×D4C2×D311yescharacteristic
91#492—61no—
92#487—61no—
93#463—62no—
94#461—61no—
95#461—62no—
96#459—61no—
97#448—62no—
98#431C2×D311yescharacteristic
99#431C2×D311yescharacteristic
100#423—31no—
101#417—61no—
102#399—62no—
103#390—61no—
104#384—61no—
105C4×C4⋊C4—31no—
106#333—62no—
107#327—62no—
108D4⋊C8—22no—
109C24×C3—21no—
110C2×C3×D4—21no—
111C2×C3×D4—21no—
112C2×C3×D4—21no—
113C2×C3×D4—21no—
114C2×C3×D4—21no—
115C2×C3×D4SD1612yescomplemented
116C22×C3×C4—21no—
117C22×C3×C4—44no—
118C22×C3×C4—41no—
119C22×C3×C4—41no—
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C2×#67—41no—
124C22×Dic3—22no—
125C22×Dic3—22no—
126#235—41no—
127#235—41no—
128#235—41no—
129#235—41no—
130C2×D12—21no—
131C2×D12—21no—
132C2×D12—21no—
133C2×C4×D3—41no—
134C2×Dic6—21no—
135C2×Dic6—22no—
136C2×Dic6—22no—
137C2×Dic6—44no—
138C2×C3×C8—22no—
139C2×C3×C8—41no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—21no—
142C3×C22⋊C4—41no—
143C3×C22⋊C4—41no—
144C3×C42C2×D411yescharacteristic
145C3×C42C2×D411yescharacteristic
146C3×C42C2×D411yescharacteristic
147C3×C42#4111yescharacteristic
148#217—48no—
149#214—44no—
150#213—44no—
151(C2×D3)⋊C4—41no—
152(C2×D3)⋊C4—41no—
153C4⋊Dic3—24no—
154C4⋊Dic3—21no—
155C4⋊Dic3—21no—
156C4⋊Dic3—41no—
157C4⋊Dic3—41no—
158C4⋊Dic3—41no—
159C4⋊Dic3—41no—
160Dic3⋊C4—48no—
161C4×Dic3—41no—
162C4×Dic3—41no—
163#208—41no—
164#208—41no—
165C2×C3⋊C8—22no—
166#204—84no—
167C2×#41—61no—
168C22×Q8—32no—
169C22×D4D1211yescharacteristic
170C23×C4—21no—
171AGL(1,ℤ/8)—122no—
172AGL(1,ℤ/8)—122no—
173C2×SD16—64no—
174C2×SD16—62no—
175C2×D8—62no—
176C2×M16—61no—
177C4⋊Q8—31no—
178C4⋊Q8—61no—
179C4⋊Q8—61no—
180C4⋊Q8—61no—
181C4⋊Q8—61no—
182#127—31no—
183#124—61no—
184C22⋊Q8—128no—
185#121—61no—
186#121—61no—
187C4×D4—21no—
188C4×D4—21no—
189C4×D4—21no—
190C4×D4—21no—
191C4×D4—61no—
192C4×D4—61no—
193C2×C4⋊C4—61no—
194C2×C4⋊C4—64no—
195C2×C4⋊C4—61no—
196C2×C4⋊C4—61no—
197C2×C4⋊C4D1211yescharacteristic
198C2×C22⋊C4—21no—
199C2×C22⋊C4—64no—
200C2×C42—61no—
201C2×C42C22×D311yescharacteristic
202C4⋊C8—21no—
203C4⋊C8—32no—
204C4⋊C8—61no—
205C4⋊C8#6712yes—
206#103—62no—
207#103—64no—
208#102—62no—
209C22⋊C8—21no—
210C22⋊C8—21no—
211C4×C8—21no—
212C4×C8—21no—
213#95—32no—
214C23×C3—22no—
215C23×C3—42no—
216C23×C3—42no—
217C23×C3—41no—
218C23×C3—41no—
219C23×C3#12011yescharacteristic
220C22×D3—41no—
221C3×D4—22no—
222C3×D4—22no—
223C3×D4—41no—
224C3×D4—41no—
225C2×C3×C4—22no—
226C2×C3×C4—41no—
227C2×C3×C4—42no—
228C2×C3×C4—41no—
229C2×C3×C4—82no—
230C2×C3×C4—82no—
231C2×C3×C4#12011yescharacteristic
232C2×C3×C4#12011yescharacteristic
233C2×C3×C4#12111yescharacteristic
234C2×C3×C4#12111yescharacteristic
235C2×C3×C4#12111yescharacteristic
236C2×C3×C4#12711yescharacteristic
237C2×C3×C4C2×SD1611yescharacteristic
238C2×C3×C4#13511yescharacteristic
239C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
240C2×C3×C4#13711yescharacteristic
241#67—81no—
242#67—81no—
243C2×Dic3—24no—
244C2×Dic3—42no—
245C2×Dic3—42no—
246C2×Dic3—44no—
247C2×Dic3—41no—
248C2×Dic3—42no—
249D12—41no—
250D12—41no—
251D12—42no—
252D12—81no—
253C4×D3—81no—
254C4×D3—81no—
255Dic6—41no—
256Dic6—41no—
257Dic6—44no—
258Dic6—82no—
259C3×C8—42no—
260C3×C8—81no—
261C3⋊C8—42no—
262C24—21no—
263#41—121no—
264#41—121no—
265#41—121no—
266#41—121no—
267C2×Q8—61no—
268C2×Q8—62no—
269C2×Q8—62no—
270C2×Q8—124no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—61no—
277C2×D4—61no—
278C2×D4—61no—
279C2×D4—121no—
280C2×D4#20412yescomplemented
281C22×C4—21no—
282C22×C4—44no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—62no—
286C22×C4—62no—
287C22×C4—121no—
288C22×C4C2×D1211yescharacteristic
289C22×C4D3×D411yescharacteristic
290C22×C4D3×D411yescharacteristic
291SD16—124no—
292SD16—244no—
293D8—124no—
294M16—121no—
295M16—121no—
296C2×C8—22no—
297C2×C8—41no—
298C2×C8—62no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—64no—
302C4⋊C4—61no—
303C4⋊C4—61no—
304C4⋊C4—128no—
305C4⋊C4—121no—
306C4⋊C4—121no—
307C4⋊C4—121no—
308C4⋊C4—121no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—128no—
312C22⋊C4—121no—
313C22⋊C4—121no—
314C42—121no—
315C42—121no—
316C42#23511yescharacteristic
317C42D3×D411yescharacteristic
318C42D3×D411yescharacteristic
319C42C2×#6711yescharacteristic
320C22×C3—21no—
321C22×C3—21no—
322C22×C3—44no—
323C22×C3—42no—
324C22×C3—82no—
325C22×C3—81no—
326C22×C3—81no—
327C22×C3#39211yescharacteristic
328C22×C3#44811yescharacteristic
329C22×C3#44911yescharacteristic
330C2×D3—81no—
331C2×D3—81no—
332C2×D3—81no—
333C3×C4—21no—
334C3×C4—21no—
335C3×C4—21no—
336C3×C4—81no—
337C3×C4#45011yescharacteristic
338C3×C4#45211yescharacteristic
339C3×C4#45312yes—
340C3×C4#46511yescharacteristic
341C3×C4#46811yescharacteristic
342Dic3—44no—
343Dic3—81no—
344Dic3—82no—
345C23—22no—
346C23—42no—
347C23—42no—
348C23—41no—
349C23—41no—
350C23—121no—
351C23#88211yescharacteristic
352Q8—124no—
353Q8—121no—
354Q8—121no—
355Q8—242no—
356D4—22no—
357D4—22no—
358D4—41no—
359D4—41no—
360D4—122no—
361D4—121no—
362D4—121no—
363D4—241no—
364D4—241no—
365D4—241no—
366C2×C4—22no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—64no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—122no—
374C2×C4—122no—
375C2×C4—124no—
376C2×C4—121no—
377C2×C4—122no—
378C2×C4—241no—
379C2×C4—241no—
380C2×C4#88211yescharacteristic
381C2×C4#88411yescharacteristic
382C2×C4#88411yescharacteristic
383C2×C4C2×#20411yescharacteristic
384C2×C4#90411yescharacteristic
385C2×C4#90811yescharacteristic
386C2×C4#90911yescharacteristic
387C2×C4#93011yescharacteristic
388C2×C4#93711yescharacteristic
389C2×C4#94011yescharacteristic
390C8—42no—
391C8—81no—
392C8—122no—
393C2×C3—21no—
394C2×C3—21no—
395C2×C3—82no—
396C2×C3#200011yescharacteristic
397C2×C3#200211yescharacteristic
398C2×C3#200911yescharacteristic
399D3—161no—
400C22—21no—
401C22—21no—
402C22—44no—
403C22—42no—
404C22—82no—
405C22—81no—
406C22—81no—
407C22—241no—
408C22—241no—
409C22—241no—
410C22#473411yescharacteristic
411C22#473811yescharacteristic
412C22#496011yescharacteristic
413C4—21no—
414C4—21no—
415C4—21no—
416C4—81no—
417C4—124no—
418C4—241no—
419C4—242no—
420C4#473111yescharacteristic
421C4#473311yescharacteristic
422C4#511411yescharacteristic
423C4#513411yescharacteristic
424C4#519812yes—
425C3#1024011yescharacteristic, complemented, Sylow
426C2—21no—
427C2—21no—
428C2—82no—
429C2—481no—
430C2#6996211yescharacteristic
431C2#7032411yescharacteristic
432C2#7041211yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1478C2
socleC22×C3#392
prime radical#82307C2
hypercenter#1478D3
hypocenterC3#10240
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1478

Complements

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

NH ≅ G/Ntype
1. #82379C2semidirect
3. #79312C2semidirect
4. C3×#1478C2semidirect
5. #70789C2semidirect
6. #70779C2semidirect
7. #67746C2semidirect
37. #1478D3semidirect
115. C2×C3×D4SD16semidirect
280. C2×D4#204semidirect
425. C3#10240semidirect

Maximal subgroups

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

subgroupindex
#823792
#823072
#793122
C3×#14782
#707892
#707792
#677462
#102403