Finite Group Databasebeta

← #10697569

Subgroups — #10697569

Order 768.

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

normal subgroups only

This group has 590 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#82384C211yesmaximal, characteristic, complemented
2#82307C211yesmaximal, characteristic
3#79269C211yesmaximal, characteristic, complemented
4C3×#1478C211yesmaximal, characteristic, complemented
5#70789C211yesmaximal, characteristic, complemented
6#70780C211yesmaximal, characteristic, complemented
7#67746C211yesmaximal, characteristic, complemented
8#10241—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10C2×#932—21no—
11#5477C2211yescharacteristic
12C2×#869C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×#431C2211yescharacteristic
15#5221—21no—
16#5212—22no—
17#5019—21no—
18#5016—22no—
19#5015—21no—
20#4977—21no—
21#4966—21no—
22#4947—22no—
23C4×C4⋊Dic3C2211yescharacteristic
24#4924C2211yescharacteristic
25#4842—22no—
26#4735—22no—
27#4694—21no—
28#4680—21no—
29C3×D4⋊C8—22no—
30#4482—22no—
31#4470—22no—
32#3227—31no—
33#3036—31no—
34#2563—31no—
35#1549—31no—
36#1502—31no—
37#1478D311yescharacteristic, complemented
38#1303—31no—
39C22×C3×D4D411yescharacteristic
40C23×C3×C4—21no—
41C2×#235—21no—
42C22×Dic6—21no—
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⋊Dic3D412yes—
60C2×C4×Dic3—21no—
61C2×#208—21no—
62#875—21no—
63#874D411yescharacteristic
64C4×D12—21no—
65C4×D12—21no—
66#869—21no—
67#869—21no—
68#869—21no—
69#869—21no—
70#869D411yescharacteristic
71C3×C4⋊C8—21no—
72C3×C4⋊C8D412yes—
73C3×C22⋊C8—21no—
74C3×C22⋊C8—21no—
75C3×C4×C8—21no—
76C3×C4×C8—21no—
77C3⋊#95—24no—
78C3⋊#95D412yes—
79#821—24no—
80#817—24no—
81#810—22no—
82#809—22no—
83C4⋊(C3⋊C8)—21no—
84C4⋊(C3⋊C8)D412yes—
85C2×AGL(1,ℤ/8)—61no—
86#532—31no—
87C2×C4⋊Q8—31no—
88C2×C4×D4C2×D311yescharacteristic
89#498—61no—
90#486—61no—
91#481—62no—
92#476—62no—
93#463—62no—
94#461—61no—
95#459—61no—
96#431C2×D311yescharacteristic
97#431C2×D311yescharacteristic
98#423—31no—
99#417—61no—
100#393—62no—
101#390—61no—
102#382—61no—
103#380—62no—
104C4×C4⋊C4—31no—
105#332—62no—
106#327—62no—
107D4⋊C8—22no—
108C24×C3—21no—
109C2×C3×D4—21no—
110C2×C3×D4—21no—
111C2×C3×D4—21no—
112C2×C3×D4—21no—
113C2×C3×D4—21no—
114C2×C3×D4SD1612yes—
115C22×C3×C4—21no—
116C22×C3×C4—44no—
117C22×C3×C4—41no—
118C22×C3×C4—41no—
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C2×#67—41no—
123C22×Dic3—22no—
124C22×Dic3—22no—
125#235—41no—
126#235—41no—
127#235—41no—
128#235—41no—
129C2×D12—21no—
130C2×D12—21no—
131C2×D12—21no—
132C2×C4×D3—41no—
133C2×Dic6—21no—
134C2×Dic6—41no—
135C2×Dic6—41no—
136C2×Dic6—82no—
137C2×C3×C8—22no—
138C2×C3×C8—41no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—21no—
141C3×C22⋊C4—41no—
142C3×C22⋊C4—41no—
143C3×C42C2×D411yescharacteristic
144C3×C42C2×D411yescharacteristic
145C3×C42C2×D411yescharacteristic
146C3×C42#4111yescharacteristic
147#217—84no—
148#214—44no—
149#213—44no—
150(C2×D3)⋊C4—41no—
151(C2×D3)⋊C4—41no—
152C4⋊Dic3—21no—
153C4⋊Dic3—22no—
154C4⋊Dic3—22no—
155C4⋊Dic3—21no—
156C4⋊Dic3—42no—
157C4⋊Dic3—41no—
158C4⋊Dic3—41no—
159C4×Dic3—41no—
160C4×Dic3—41no—
161#208—41no—
162#208—41no—
163C2×C3⋊C8—22no—
164C2×#41—61no—
165C22×Q8—61no—
166C22×D4D1211yescharacteristic
167C23×C4—21no—
168AGL(1,ℤ/8)—122no—
169AGL(1,ℤ/8)—122no—
170C2×SD16—62no—
171C2×D8—62no—
172C2×M16—61no—
173C4⋊Q8—31no—
174C4⋊Q8—61no—
175C4⋊Q8—61no—
176C4⋊Q8—61no—
177C4⋊Q8—61no—
178#127—31no—
179#124—61no—
180#121—61no—
181#121—61no—
182C4×D4—21no—
183C4×D4—21no—
184C4×D4—21no—
185C4×D4—21no—
186C4×D4—61no—
187C4×D4—61no—
188C2×C4⋊C4—32no—
189C2×C4⋊C4—61no—
190C2×C4⋊C4—61no—
191C2×C4⋊C4D1211yescharacteristic
192C2×C22⋊C4—21no—
193C2×C22⋊C4—64no—
194C2×C42—61no—
195C2×C42C22×D311yescharacteristic
196#106—64no—
197C4⋊C8—21no—
198C4⋊C8—32no—
199C4⋊C8—61no—
200C4⋊C8#6712yes—
201#103—62no—
202#102—62no—
203#102—64no—
204C22⋊C8—21no—
205C22⋊C8—21no—
206C4×C8—21no—
207C4×C8—21no—
208#95—32no—
209#95—64no—
210C23×C3—22no—
211C23×C3—42no—
212C23×C3—42no—
213C23×C3—41no—
214C23×C3—41no—
215C23×C3#12011yescharacteristic
216C22×D3—41no—
217C3×D4—22no—
218C3×D4—22no—
219C3×D4—41no—
220C3×D4—41no—
221C2×C3×C4—22no—
222C2×C3×C4—41no—
223C2×C3×C4—42no—
224C2×C3×C4—41no—
225C2×C3×C4—82no—
226C2×C3×C4—82no—
227C2×C3×C4#12011yescharacteristic
228C2×C3×C4#12011yescharacteristic
229C2×C3×C4#12111yescharacteristic
230C2×C3×C4#12111yescharacteristic
231C2×C3×C4#12111yescharacteristic
232C2×C3×C4#12711yescharacteristic
233C2×C3×C4C2×SD1611yescharacteristic
234C2×C3×C4#13511yescharacteristic
235C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
236C2×C3×C4#13711yescharacteristic
237#67—81no—
238#67—81no—
239C2×Dic3—42no—
240C2×Dic3—42no—
241C2×Dic3—44no—
242C2×Dic3—42no—
243C2×Dic3—41no—
244C2×Dic3—42no—
245D12—41no—
246D12—41no—
247D12—42no—
248D12—81no—
249C4×D3—81no—
250C4×D3—81no—
251Dic6—41no—
252Dic6—41no—
253Dic6—82no—
254Dic6—82no—
255C3×C8—42no—
256C3×C8—81no—
257C3⋊C8—42no—
258C24—21no—
259#41—121no—
260#41—121no—
261#41—121no—
262#41—121no—
263C2×Q8—61no—
264C2×Q8—121no—
265C2×Q8—121no—
266C2×Q8—242no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—61no—
273C2×D4—61no—
274C2×D4—61no—
275C2×D4—121no—
276C2×D4#20412yes—
277C22×C4—21no—
278C22×C4—44no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—62no—
282C22×C4—62no—
283C22×C4—121no—
284C22×C4C2×D1211yescharacteristic
285C22×C4D3×D411yescharacteristic
286C22×C4D3×D411yescharacteristic
287SD16—124no—
288D8—124no—
289M16—121no—
290M16—121no—
291C2×C8—22no—
292C2×C8—41no—
293C2×C8—62no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—62no—
297C4⋊C4—61no—
298C4⋊C4—62no—
299C4⋊C4—61no—
300C4⋊C4—122no—
301C4⋊C4—121no—
302C4⋊C4—121no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—121no—
306C22⋊C4—121no—
307C22⋊C4—244no—
308C42—121no—
309C42—121no—
310C42#23511yescharacteristic
311C42D3×D411yescharacteristic
312C42D3×D411yescharacteristic
313C42C2×#6711yescharacteristic
314C22×C3—21no—
315C22×C3—21no—
316C22×C3—44no—
317C22×C3—42no—
318C22×C3—82no—
319C22×C3—81no—
320C22×C3—81no—
321C22×C3#39211yescharacteristic
322C22×C3#44811yescharacteristic
323C22×C3#44911yescharacteristic
324C2×D3—81no—
325C2×D3—81no—
326C2×D3—81no—
327C3×C4—21no—
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—81no—
331C3×C4#45011yescharacteristic
332C3×C4#45211yescharacteristic
333C3×C4#45412yes—
334C3×C4#46511yescharacteristic
335C3×C4#46811yescharacteristic
336Dic3—82no—
337Dic3—81no—
338Dic3—82no—
339C23—22no—
340C23—42no—
341C23—42no—
342C23—41no—
343C23—41no—
344C23—121no—
345C23#88211yescharacteristic
346Q8—121no—
347Q8—121no—
348Q8—242no—
349Q8—242no—
350D4—22no—
351D4—22no—
352D4—41no—
353D4—41no—
354D4—122no—
355D4—121no—
356D4—121no—
357D4—241no—
358D4—241no—
359D4—241no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—122no—
367C2×C4—124no—
368C2×C4—122no—
369C2×C4—121no—
370C2×C4—122no—
371C2×C4—122no—
372C2×C4—241no—
373C2×C4—241no—
374C2×C4#88211yescharacteristic
375C2×C4#88411yescharacteristic
376C2×C4#88411yescharacteristic
377C2×C4C2×#20411yescharacteristic
378C2×C4#90411yescharacteristic
379C2×C4#90811yescharacteristic
380C2×C4#90911yescharacteristic
381C2×C4#93011yescharacteristic
382C2×C4#93711yescharacteristic
383C2×C4#94011yescharacteristic
384C8—42no—
385C8—81no—
386C8—122no—
387C2×C3—21no—
388C2×C3—21no—
389C2×C3—82no—
390C2×C3#200011yescharacteristic
391C2×C3#200311yescharacteristic
392C2×C3#200811yescharacteristic
393D3—161no—
394C22—21no—
395C22—21no—
396C22—44no—
397C22—42no—
398C22—82no—
399C22—81no—
400C22—81no—
401C22—241no—
402C22—241no—
403C22—241no—
404C22#473411yescharacteristic
405C22#473811yescharacteristic
406C22#496011yescharacteristic
407C4—21no—
408C4—21no—
409C4—21no—
410C4—81no—
411C4—242no—
412C4—242no—
413C4—241no—
414C4#473111yescharacteristic
415C4#473311yescharacteristic
416C4#511411yescharacteristic
417C4#513411yescharacteristic
418C4#520112yes—
419C3#1024111yescharacteristic, complemented, Sylow
420C2—21no—
421C2—21no—
422C2—82no—
423C2—481no—
424C2#6996211yescharacteristic
425C2#7032611yescharacteristic
426C2#7040911yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1478C2
socleC22×C3#392
prime radical#82307C2
hypercenter#1478D3
hypocenterC3#10241
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. #82384C2semidirect
3. #79269C2semidirect
4. C3×#1478C2semidirect
5. #70789C2semidirect
6. #70780C2semidirect
7. #67746C2semidirect
37. #1478D3semidirect
419. C3#10241semidirect

Maximal subgroups

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

subgroupindex
#823842
#823072
#792692
C3×#14782
#707892
#707802
#677462
#102413