Finite Group Databasebeta

← #10697598

Subgroups — #10697598

Order 768.

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

normal subgroups only

This group has 558 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
1#83243C211yesmaximal, characteristic, complemented
2#83216C211yesmaximal, characteristic
3#79067C211yesmaximal, characteristic
4C3×#1549C211yesmaximal, characteristic, complemented
5#70789C211yesmaximal, characteristic
6#70767C211yesmaximal, characteristic, complemented
7#67715C211yesmaximal, characteristic, complemented
8#10245—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×#909—21no—
11#5477C2211yescharacteristic
12C2×C4×Dic6C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×#423C2211yescharacteristic
15#5230—22no—
16#5133—21no—
17#5088—21no—
18#5084—22no—
19#5080—21no—
20#4975—21no—
21#4973—21no—
22#4972—22no—
23#4968—22no—
24C4×C4⋊Dic3C2211yescharacteristic
25#4925C2211yescharacteristic
26#4849—21no—
27#4848—21no—
28#4845—22no—
29C3×#332—22no—
30#4470—22no—
31#4456—22no—
32#3227—31no—
33#3037—31no—
34#2567—31no—
35#1549D311yescharacteristic, complemented
36#1502—31no—
37#1479—31no—
38#1303—31no—
39C22×C3×Q8—21no—
40C2×#235—21no—
41C22×Dic6D411yescharacteristic
42C2×C3×M16—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4D412yes—
49C2×C3×C42C2311yescharacteristic
50#930—21no—
51#930—21no—
52C2×C4⋊Dic3—21no—
53C2×C4⋊Dic3D411yescharacteristic
54C2×Dic3⋊C4—21no—
55C2×Dic3⋊C4—24no—
56C2×C4×Dic3—21no—
57C2×C4×Dic3—21no—
58#909—42no—
59#909—42no—
60C2×Dic12—22no—
61C2×#204—22no—
62#875—21no—
63#874D411yescharacteristic
64C4×D12—21no—
65C4×D12—21no—
66#869D411yescharacteristic
67C4×Dic6—21no—
68C4×Dic6—21no—
69C4×Dic6—21no—
70C4×Dic6—21no—
71C3×C4⋊C8—21no—
72C3×C4⋊C8—21no—
73C3×C4⋊C8D412yes—
74C3×C22⋊C8—21no—
75C3×C4×C8—21no—
76C3⋊#95—24no—
77C3⋊#95D412yes—
78#830—21no—
79#821—22no—
80#816—22no—
81#809—24no—
82#808—24no—
83C4⋊(C3⋊C8)D412yes—
84C4×C3⋊C8—21no—
85C2×#137—61no—
86#532—31no—
87C2×C4⋊Q8C2×D311yescharacteristic
88C2×C4×Q8—31no—
89#498—61no—
90#486—61no—
91#481—62no—
92#476—62no—
93#464—62no—
94#462—61no—
95#460—61no—
96#431—31no—
97#431C2×D311yescharacteristic
98#423C2×D311yescharacteristic
99#417—61no—
100#395—62no—
101#391—61no—
102#389—62no—
103#384—61no—
104C4×C4⋊C4—31no—
105#332—22no—
106#327—62no—
107Q8⋊C8—62no—
108C2×C3×Q8—41no—
109C2×C3×Q8—41no—
110C2×C3×Q8—82no—
111C22×C3×C4—22no—
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C2×#67—41no—
116C22×Dic3—21no—
117C22×Dic3—22no—
118C22×Dic3—21no—
119#235—41no—
120#235—41no—
121#235—41no—
122#235—41no—
123C2×D12—21no—
124C2×D12—21no—
125C2×D12—21no—
126C2×C4×D3—41no—
127C2×Dic6—21no—
128C2×Dic6—21no—
129C2×Dic6—21no—
130C2×Dic6—21no—
131C2×Dic6—21no—
132C2×Dic6—21no—
133C2×Dic6SD1612yes—
134C3×M16—41no—
135C3×M16—41no—
136C2×C3×C8—22no—
137C2×C3×C8—41no—
138C3×C4⋊C4—22no—
139C3×C4⋊C4—22no—
140C3×C4⋊C4—41no—
141C3×C4⋊C4—41no—
142C3×C42C2×D411yescharacteristic
143C3×C42C2×D411yescharacteristic
144C3×C42C2×D411yescharacteristic
145C3×C42#4111yescharacteristic
146(C2×D3)⋊C4—41no—
147(C2×D3)⋊C4—41no—
148C4⋊Dic3—24no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151C4⋊Dic3—21no—
152C4⋊Dic3—21no—
153Dic3⋊C4—41no—
154Dic3⋊C4—41no—
155Dic3⋊C4—84no—
156C4×Dic3—41no—
157C4×Dic3—41no—
158C4×Dic3—41no—
159C4×Dic3—41no—
160C2×C3⋊C8—22no—
161Dic12—44no—
162#204—44no—
163C2×#41—61no—
164C22×Q8—21no—
165C22×Q8—31no—
166#137—122no—
167#137—122no—
168C2×Q16—62no—
169C2×SD16—62no—
170C2×M16—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175C4⋊Q8—31no—
176#127—31no—
177#124—61no—
178#121—61no—
179#121—61no—
180C4×Q8—61no—
181C4×Q8—61no—
182C4×Q8—61no—
183C4×Q8—61no—
184C4×D4—61no—
185C4×D4—61no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—31no—
188C2×C4⋊C4—61no—
189C2×C4⋊C4—61no—
190C2×C4⋊C4—64no—
191C2×C4⋊C4D1212yes—
192C2×C42—61no—
193C2×C42—61no—
194C2×C42C22×D311yescharacteristic
195#106—64no—
196C4⋊C8—21no—
197C4⋊C8—21no—
198C4⋊C8—32no—
199C4⋊C8#6712yes—
200#103—62no—
201#102—64no—
202#102—62no—
203C22⋊C8—21no—
204C22⋊C8—61no—
205C4×C8—21no—
206C4×C8—61no—
207#95—32no—
208#95—64no—
209C23×C3#12011yescharacteristic
210C22×D3—41no—
211C3×Q8—82no—
212C3×Q8—82no—
213C2×C3×C4—22no—
214C2×C3×C4—41no—
215C2×C3×C4—44no—
216C2×C3×C4—42no—
217C2×C3×C4#12011yescharacteristic
218C2×C3×C4#12011yescharacteristic
219C2×C3×C4#12111yescharacteristic
220C2×C3×C4#12111yescharacteristic
221C2×C3×C4#12111yescharacteristic
222C2×C3×C4#12711yescharacteristic
223C2×C3×C4C2×SD1611yescharacteristic
224C2×C3×C4#13511yescharacteristic
225C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
226C2×C3×C4#13711yescharacteristic
227#67—81no—
228#67—81no—
229C2×Dic3—24no—
230C2×Dic3—22no—
231C2×Dic3—42no—
232C2×Dic3—41no—
233C2×Dic3—42no—
234C2×Dic3—41no—
235C2×Dic3—42no—
236C2×Dic3—41no—
237C2×Dic3—41no—
238D12—41no—
239D12—41no—
240D12—42no—
241D12—81no—
242C4×D3—81no—
243C4×D3—81no—
244Dic6—22no—
245Dic6—22no—
246Dic6—41no—
247Dic6—41no—
248Dic6—41no—
249Dic6—41no—
250C3×C8—42no—
251C3×C8—81no—
252C3⋊C8—42no—
253#41—121no—
254#41—121no—
255#41—121no—
256#41—121no—
257C2×Q8—32no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—61no—
261C2×Q8—61no—
262C2×Q8—61no—
263C2×Q8—61no—
264C2×Q8—61no—
265C2×Q8—61no—
266C2×Q8—82no—
267C2×D4—61no—
268C2×D4—61no—
269C2×D4—61no—
270C2×D4—121no—
271C22×C4—22no—
272C22×C4—62no—
273C22×C4—61no—
274C22×C4—61no—
275C22×C4—121no—
276C22×C4C2×D1211yescharacteristic
277C22×C4D3×D411yescharacteristic
278C22×C4D3×D411yescharacteristic
279Q16—124no—
280SD16—124no—
281M16—41no—
282M16—41no—
283C2×C8—22no—
284C2×C8—41no—
285C2×C8—62no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—64no—
291C4⋊C4—61no—
292C4⋊C4—61no—
293C4⋊C4—61no—
294C4⋊C4—61no—
295C4⋊C4—121no—
296C4⋊C4—121no—
297C4⋊C4—244no—
298C22⋊C4—121no—
299C22⋊C4—121no—
300C42—121no—
301C42—121no—
302C42—121no—
303C42—121no—
304C42#23511yescharacteristic
305C42D3×D411yescharacteristic
306C42D3×D411yescharacteristic
307C42C2×#6711yescharacteristic
308C22×C3—21no—
309C22×C3—21no—
310C22×C3#39211yescharacteristic
311C22×C3#44811yescharacteristic
312C22×C3#44911yescharacteristic
313C2×D3—81no—
314C2×D3—81no—
315C2×D3—81no—
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—21no—
319C3×C4—82no—
320C3×C4#45011yescharacteristic
321C3×C4#45211yescharacteristic
322C3×C4#45612yes—
323C3×C4#46511yescharacteristic
324C3×C4#46811yescharacteristic
325Dic3—44no—
326Dic3—42no—
327Dic3—81no—
328Dic3—81no—
329C23—121no—
330C23#88211yescharacteristic
331Q8—62no—
332Q8—62no—
333Q8—82no—
334Q8—82no—
335Q8—121no—
336Q8—121no—
337Q8—121no—
338Q8—121no—
339D4—121no—
340D4—121no—
341D4—122no—
342D4—241no—
343D4—241no—
344D4—241no—
345C2×C4—22no—
346C2×C4—41no—
347C2×C4—44no—
348C2×C4—42no—
349C2×C4—64no—
350C2×C4—62no—
351C2×C4—122no—
352C2×C4—121no—
353C2×C4—122no—
354C2×C4—121no—
355C2×C4—122no—
356C2×C4—121no—
357C2×C4—121no—
358C2×C4—241no—
359C2×C4—241no—
360C2×C4#88211yescharacteristic
361C2×C4#88411yescharacteristic
362C2×C4#88411yescharacteristic
363C2×C4D3×SD1611yescharacteristic
364C2×C4#91411yescharacteristic
365C2×C4#91811yescharacteristic
366C2×C4#91911yescharacteristic
367C2×C4#93011yescharacteristic
368C2×C4#93711yescharacteristic
369C2×C4#94011yescharacteristic
370C8—42no—
371C8—81no—
372C8—122no—
373C2×C3—21no—
374C2×C3—21no—
375C2×C3#200011yescharacteristic
376C2×C3#200511yescharacteristic
377C2×C3#200811yescharacteristic
378D3—161no—
379C22—21no—
380C22—21no—
381C22—241no—
382C22—241no—
383C22—241no—
384C22#480611yescharacteristic
385C22#481011yescharacteristic
386C22#496011yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—82no—
391C4—124no—
392C4—122no—
393C4—241no—
394C4—241no—
395C4#475212yes—
396C4#480711yescharacteristic
397C4#481211yescharacteristic
398C4#516911yescharacteristic
399C4#518711yescharacteristic
400C3#1024511yescharacteristic, complemented, Sylow
401C2—21no—
402C2—21no—
403C2—481no—
404C2#7011911yescharacteristic
405C2#7027211yescharacteristic
406C2#7033111yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1549C2
socleC22×C3#392
prime radical#5477C22
hypercenter#1549D3
hypocenterC3#10245
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1549

Complements

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

NH ≅ G/Ntype
1. #83243C2semidirect
4. C3×#1549C2semidirect
6. #70767C2semidirect
7. #67715C2semidirect
35. #1549D3semidirect
400. C3#10245semidirect

Maximal subgroups

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

subgroupindex
#832432
#832162
#790672
C3×#15492
#707892
#707672
#677152
#102453