Finite Group Databasebeta

← #10697561

Subgroups — #10697561

Order 768.

658 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 105 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×#2542C211yesmaximal, characteristic, complemented
2#83213C211yesmaximal, characteristic, complemented
3#82307C211yesmaximal, characteristic
4#71774C211yesmaximal, characteristic, complemented
5#71597C211yesmaximal, characteristic, complemented
6#69469C211yesmaximal, characteristic, complemented
7#67620C211yesmaximal, characteristic, complemented
8#10240—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×C22⋊Q8—22no—
11C2×C3×C4×D4C2211yescharacteristic
12C2×#932—21no—
13#5477C2211yescharacteristic
14C3×#399—22no—
15C3×#390—21no—
16C3×#384—21no—
17C3×C4×C4⋊C4C2211yescharacteristic
18#5227—22no—
19#5085—21no—
20#5083—22no—
21#5079—21no—
22#5019—21no—
23#5016—22no—
24#5015—21no—
25#4966—21no—
26#4925C2211yescharacteristic
27#4925C2211yescharacteristic
28#4924C2211yescharacteristic
29#4546—22no—
30D4⋊(C3⋊C8)—22no—
31#4451—22no—
32#3206—31no—
33#3036—31no—
34#2542D311yescharacteristic, complemented
35#1548—31no—
36#1502—31no—
37#1478—31no—
38#1303—31no—
39C22×C3×Q8D412yes—
40C22×C3×D4D411yescharacteristic
41C23×C3×C4—21no—
42C2×#235—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—21no—
47C3×C22⋊Q8—48no—
48C3×C4×D4—21no—
49C3×C4×D4—21no—
50C3×C4×D4—21no—
51C3×C4×D4—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—24no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4—21no—
56C2×C3×C4⋊C4D411yescharacteristic
57C2×C3×C22⋊C4—24no—
58C2×C3×C22⋊C4—21no—
59C2×C3×C42—21no—
60C2×C3×C42C2311yescharacteristic
61C2×#215—24no—
62C2×#214—22no—
63#932—42no—
64#932—42no—
65C2×#213—22no—
66#930—21no—
67#930—21no—
68C2×#208—21no—
69#875—21no—
70#874D411yescharacteristic
71C4×D12—21no—
72C4×D12—21no—
73#869D411yescharacteristic
74C3×#95D412yes—
75#835—24no—
76#830—21no—
77#830—21no—
78#810—22no—
79#809—22no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)D412yes—
83C4⋊(C3⋊C8)D412yes—
84C4×C3⋊C8—21no—
85C4×C3⋊C8—21no—
86C2×AGL(1,ℤ/8)—61no—
87#532—31no—
88C2×C4⋊Q8C2×D311yescharacteristic
89C2×C22⋊Q8—22no—
90C2×C4×D4C2×D311yescharacteristic
91#492—61no—
92#487—61no—
93#463—62no—
94#461—62no—
95#461—61no—
96#459—61no—
97#448—62no—
98#431—31no—
99#431—31no—
100#423—31no—
101#417—61no—
102#399—22no—
103#390—21no—
104#384—21no—
105C4×C4⋊C4C2×D311yescharacteristic
106#333—62no—
107#327—62no—
108D4⋊C8—62no—
109C24×C3—21no—
110C2×C3×Q8—22no—
111C2×C3×Q8—22no—
112C2×C3×Q8—44no—
113C2×C3×D4—21no—
114C2×C3×D4—21no—
115C2×C3×D4—21no—
116C2×C3×D4—21no—
117C2×C3×D4—21no—
118C2×C3×D4SD1612yescomplemented
119C22×C3×C4—22no—
120C22×C3×C4—22no—
121C22×C3×C4—21no—
122C22×C3×C4—44no—
123C22×C3×C4—41no—
124C22×C3×C4—41no—
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C2×#67—41no—
129#235—41no—
130#235—41no—
131#235—41no—
132#235—41no—
133C2×D12—21no—
134C2×D12—21no—
135C2×D12—21no—
136C2×C4×D3—41no—
137C2×Dic6—21no—
138C3×C4⋊C4—24no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—48no—
142C3×C4⋊C4—41no—
143C3×C4⋊C4—41no—
144C3×C4⋊C4—41no—
145C3×C4⋊C4—41no—
146C3×C22⋊C4—48no—
147C3×C22⋊C4—41no—
148C3×C22⋊C4—41no—
149C3×C42—41no—
150C3×C42—41no—
151C3×C42C2×D411yescharacteristic
152C3×C42C2×D411yescharacteristic
153C3×C42C2×D411yescharacteristic
154C3×C42#4111yescharacteristic
155#215—84no—
156#214—44no—
157#213—44no—
158(C2×D3)⋊C4—41no—
159(C2×D3)⋊C4—41no—
160C4⋊Dic3—21no—
161C4⋊Dic3—21no—
162#208—41no—
163#208—41no—
164C2×C3⋊C8—22no—
165C2×C3⋊C8—22no—
166C2×C3⋊C8—41no—
167C2×#41—61no—
168C22×Q8#6712yes—
169C22×D4#6711yescharacteristic
170C23×C4—21no—
171AGL(1,ℤ/8)—122no—
172AGL(1,ℤ/8)—122no—
173C2×SD16—62no—
174C2×SD16—64no—
175C2×D8—62no—
176C2×M16—61no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8—31no—
182#127—31no—
183#124—61no—
184C22⋊Q8—48no—
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—21no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4#6711yescharacteristic
198C2×C22⋊C4—24no—
199C2×C22⋊C4—21no—
200C2×C42—21no—
201C2×C42C22×D311yescharacteristic
202C4⋊C8—32no—
203C4⋊C8—32no—
204C4⋊C8—61no—
205C4⋊C8—61no—
206#103—62no—
207#103—64no—
208#102—62no—
209C22⋊C8—61no—
210C22⋊C8—61no—
211C4×C8—61no—
212C4×C8—61no—
213#95#6712yes—
214C23×C3—22no—
215C23×C3—42no—
216C23×C3—42no—
217C23×C3—41no—
218C23×C3—41no—
219C23×C3#12011yescharacteristic
220C22×D3—41no—
221C3×Q8—44no—
222C3×Q8—82no—
223C3×D4—22no—
224C3×D4—22no—
225C3×D4—41no—
226C3×D4—41no—
227C2×C3×C4—22no—
228C2×C3×C4—24no—
229C2×C3×C4—41no—
230C2×C3×C4—42no—
231C2×C3×C4—42no—
232C2×C3×C4—44no—
233C2×C3×C4—42no—
234C2×C3×C4—41no—
235C2×C3×C4—42no—
236C2×C3×C4—82no—
237C2×C3×C4—82no—
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×SD1611yescharacteristic
245C2×C3×C4#13511yescharacteristic
246C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
247C2×C3×C4#13711yescharacteristic
248#67—81no—
249#67—81no—
250C2×Dic3—41no—
251D12—41no—
252D12—41no—
253D12—42no—
254D12—81no—
255C4×D3—81no—
256C4×D3—81no—
257Dic6—41no—
258Dic6—41no—
259C3⋊C8—42no—
260C3⋊C8—42no—
261C3⋊C8—81no—
262C24—21no—
263#41—121no—
264#41—121no—
265#41—121no—
266#41—121no—
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8—44no—
270C2×Q8—61no—
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#21512yescomplemented
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—44no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—121no—
288C22×C4C2×#6711yescharacteristic
289C22×C4C2×#6711yescharacteristic
290C22×C4C2×#6711yescharacteristic
291SD16—124no—
292SD16—244no—
293D8—124no—
294M16—121no—
295M16—121no—
296C2×C8—62no—
297C2×C8—62no—
298C2×C8—121no—
299C4⋊C4—24no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—48no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—61no—
308C4⋊C4—61no—
309C22⋊C4—48no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—121no—
313C22⋊C4—121no—
314C42—41no—
315C42—41no—
316C42D3×D411yescharacteristic
317C42D3×D411yescharacteristic
318C42D3×D411yescharacteristic
319C42#23711yescharacteristic
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—44no—
337C3×C4—81no—
338C3×C4—82no—
339C3×C4#45011yescharacteristic
340C3×C4#45211yescharacteristic
341C3×C4#45312yes—
342C3×C4#46511yescharacteristic
343C3×C4#46811yescharacteristic
344Dic3—81no—
345C23—22no—
346C23—42no—
347C23—42no—
348C23—41no—
349C23—41no—
350C23—121no—
351C23#95311yescharacteristic
352Q8—44no—
353Q8—82no—
354Q8—121no—
355Q8—121no—
356D4—22no—
357D4—22no—
358D4—41no—
359D4—41no—
360D4—121no—
361D4—121no—
362D4—122no—
363D4—241no—
364D4—241no—
365D4—241no—
366C2×C4—22no—
367C2×C4—24no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—44no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—121no—
378C2×C4—241no—
379C2×C4—241no—
380C2×C4#93711yescharacteristic
381C2×C4#93711yescharacteristic
382C2×C4#93811yescharacteristic
383C2×C4#93911yescharacteristic
384C2×C4#93911yescharacteristic
385C2×C4#94011yescharacteristic
386C2×C4C2×#21511yescharacteristic
387C2×C4#94211yescharacteristic
388C2×C4#94911yescharacteristic
389C2×C4#95011yescharacteristic
390C8—122no—
391C8—122no—
392C8—241no—
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#522311yescharacteristic
411C22#522711yescharacteristic
412C22#523811yescharacteristic
413C4—21no—
414C4—21no—
415C4—21no—
416C4—44no—
417C4—81no—
418C4—82no—
419C4—241no—
420C4#504611yescharacteristic
421C4#504711yescharacteristic
422C4#504811yescharacteristic
423C4#504911yescharacteristic
424C4#507712yes—
425C3#1024011yescharacteristic, complemented, Sylow
426C2—21no—
427C2—21no—
428C2—82no—
429C2—481no—
430C2#7169911yescharacteristic
431C2#7176911yescharacteristic
432C2#7186911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2542C2
socleC22×C3#392
prime radical#82307C2
hypercenter#2542D3
hypocenterC3#10240
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2542

Complements

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

NH ≅ G/Ntype
1. C3×#2542C2semidirect
2. #83213C2semidirect
4. #71774C2semidirect
5. #71597C2semidirect
6. #69469C2semidirect
7. #67620C2semidirect
34. #2542D3semidirect
118. C2×C3×D4SD16semidirect
280. C2×D4#215semidirect
425. C3#10240semidirect

Maximal subgroups

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

subgroupindex
C3×#25422
#832132
#823072
#717742
#715972
#694692
#676202
#102403