Finite Group Databasebeta

← #10697589

Subgroups — #10697589

Order 768.

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

normal subgroups only

This group has 554 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×#2568C211yesmaximal, characteristic, complemented
2#83213C211yesmaximal, characteristic
3#82441C211yesmaximal, characteristic
4#71774C211yesmaximal, characteristic, complemented
5#71621C211yesmaximal, characteristic, complemented
6#69469C211yesmaximal, characteristic
7#67620C211yesmaximal, characteristic, complemented
8#10244—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×C4×Q8C2211yescharacteristic
11C2×#942—21no—
12#5477C2211yescharacteristic
13C3×#400—22no—
14C3×#391—22no—
15C3×#391—21no—
16C3×#382—21no—
17C3×C4×C4⋊C4C2211yescharacteristic
18#5227—22no—
19#5085—21no—
20#5083—22no—
21#5079—21no—
22#5031—21no—
23#5028—22no—
24#5027—21no—
25#4966—21no—
26#4925C2211yescharacteristic
27#4925C2211yescharacteristic
28#4924C2211yescharacteristic
29#4546—22no—
30#4535—22no—
31#4451—22no—
32#3206—31no—
33#3037—31no—
34#2568D311yescharacteristic, complemented
35#1548—31no—
36#1502—31no—
37#1479—31no—
38#1303—31no—
39C22×C3×Q8D411yescharacteristic
40C22×C3×Q8D412yes—
41C2×#235—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4×Q8—21no—
47C3×C4×Q8—21no—
48C3×C4×Q8—21no—
49C3×C4×Q8—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4D411yescharacteristic
55C2×C3×C42—21no—
56C2×C3×C42—21no—
57C2×C3×C42C2311yescharacteristic
58C2×#216—22no—
59#942—42no—
60#942—42no—
61C2×#215—22no—
62C2×#215—24no—
63#930—21no—
64#930—21no—
65C2×#208—21no—
66#875—21no—
67#874D411yescharacteristic
68C4×D12—21no—
69C4×D12—21no—
70#869D411yescharacteristic
71C3×#95—24no—
72C3×#95—24no—
73C3×#95D412yes—
74#835—24no—
75#830—21no—
76#830—21no—
77#810—22no—
78#809—22no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)D412yes—
82C4⋊(C3⋊C8)D412yes—
83C4×C3⋊C8—21no—
84C4×C3⋊C8—21no—
85C2×#137—61no—
86#532—31no—
87C2×C4⋊Q8C2×D311yescharacteristic
88C2×C4×Q8C2×D311yescharacteristic
89#492—61no—
90#487—61no—
91#464—62no—
92#462—61no—
93#461—62no—
94#460—61no—
95#448—62no—
96#431—31no—
97#431—31no—
98#423—31no—
99#417—61no—
100#400—22no—
101#391—22no—
102#391—21no—
103#382—21no—
104C4×C4⋊C4C2×D311yescharacteristic
105#333—62no—
106#327—62no—
107Q8⋊C8—62no—
108C2×C3×Q8—21no—
109C2×C3×Q8—21no—
110C2×C3×Q8—22no—
111C2×C3×Q8—21no—
112C2×C3×Q8—22no—
113C2×C3×Q8—21no—
114C2×C3×Q8—21no—
115C2×C3×Q8—44no—
116C2×C3×Q8SD1612yescomplemented
117C22×C3×C4—21no—
118C22×C3×C4—22no—
119C22×C3×C4—21no—
120C22×C3×C4—22no—
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C2×#67—41no—
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—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—42no—
137C3×C4⋊C4—41no—
138C3×C4⋊C4—41no—
139C3×C4⋊C4—41no—
140C3×C4⋊C4—41no—
141C3×C4⋊C4—41no—
142C3×C4⋊C4—41no—
143C3×C42—41no—
144C3×C42—41no—
145C3×C42—41no—
146C3×C42—41no—
147C3×C42C2×D411yescharacteristic
148C3×C42C2×D411yescharacteristic
149C3×C42C2×D411yescharacteristic
150C3×C42#4111yescharacteristic
151#216—44no—
152#215—44no—
153#215—84no—
154(C2×D3)⋊C4—41no—
155(C2×D3)⋊C4—41no—
156C4⋊Dic3—21no—
157C4⋊Dic3—21no—
158#208—41no—
159#208—41no—
160C2×C3⋊C8—22no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—41no—
163C2×#41—61no—
164C22×Q8#6711yescharacteristic
165C22×Q8#6712yes—
166#137—122no—
167#137—122no—
168C2×Q16—62no—
169C2×SD16—62no—
170C2×SD16—64no—
171C2×M16—61no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C4⋊Q8—31no—
177#127—31no—
178#124—61no—
179#121—61no—
180#121—61no—
181C4×Q8—21no—
182C4×Q8—21no—
183C4×Q8—21no—
184C4×Q8—21no—
185C4×D4—61no—
186C4×D4—61no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4#6711yescharacteristic
192C2×C42—21no—
193C2×C42—21no—
194C2×C42C22×D311yescharacteristic
195C4⋊C8—32no—
196C4⋊C8—32no—
197C4⋊C8—61no—
198C4⋊C8—61no—
199#103—62no—
200#103—64no—
201#102—62no—
202C22⋊C8—61no—
203C22⋊C8—61no—
204C4×C8—61no—
205C4×C8—61no—
206#95—24no—
207#95—24no—
208#95#6712yes—
209C23×C3#12011yescharacteristic
210C22×D3—41no—
211C3×Q8—22no—
212C3×Q8—22no—
213C3×Q8—41no—
214C3×Q8—41no—
215C3×Q8—44no—
216C3×Q8—82no—
217C2×C3×C4—22no—
218C2×C3×C4—22no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—42no—
223C2×C3×C4—42no—
224C2×C3×C4—42no—
225C2×C3×C4—44no—
226C2×C3×C4—41no—
227C2×C3×C4—42no—
228C2×C3×C4—42no—
229C2×C3×C4#12011yescharacteristic
230C2×C3×C4#12011yescharacteristic
231C2×C3×C4#12111yescharacteristic
232C2×C3×C4#12111yescharacteristic
233C2×C3×C4#12111yescharacteristic
234C2×C3×C4#12711yescharacteristic
235C2×C3×C4C2×SD1611yescharacteristic
236C2×C3×C4#13511yescharacteristic
237C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
238C2×C3×C4#13711yescharacteristic
239#67—81no—
240#67—81no—
241C2×Dic3—41no—
242D12—41no—
243D12—41no—
244D12—42no—
245D12—81no—
246C4×D3—81no—
247C4×D3—81no—
248Dic6—41no—
249Dic6—41no—
250C3⋊C8—42no—
251C3⋊C8—42no—
252C3⋊C8—81no—
253#41—121no—
254#41—121no—
255#41—121no—
256#41—121no—
257C2×Q8—21no—
258C2×Q8—21no—
259C2×Q8—22no—
260C2×Q8—21no—
261C2×Q8—22no—
262C2×Q8—21no—
263C2×Q8—21no—
264C2×Q8—44no—
265C2×Q8—61no—
266C2×Q8#21512yescomplemented
267C2×D4—61no—
268C2×D4—61no—
269C2×D4—61no—
270C2×D4—121no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—121no—
276C22×C4C2×#6711yescharacteristic
277C22×C4C2×#6711yescharacteristic
278C22×C4C2×#6711yescharacteristic
279Q16—124no—
280SD16—124no—
281SD16—244no—
282M16—121no—
283M16—121no—
284C2×C8—62no—
285C2×C8—62no—
286C2×C8—121no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—42no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—61no—
297C4⋊C4—61no—
298C22⋊C4—121no—
299C22⋊C4—121no—
300C42—41no—
301C42—41no—
302C42—41no—
303C42—41no—
304C42D3×D411yescharacteristic
305C42D3×D411yescharacteristic
306C42D3×D411yescharacteristic
307C42#23711yescharacteristic
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—42no—
320C3×C4—82no—
321C3×C4—82no—
322C3×C4—81no—
323C3×C4#45011yescharacteristic
324C3×C4#45211yescharacteristic
325C3×C4#45512yes—
326C3×C4#46511yescharacteristic
327C3×C4#46811yescharacteristic
328Dic3—81no—
329C23—121no—
330C23#95311yescharacteristic
331Q8—22no—
332Q8—22no—
333Q8—41no—
334Q8—41no—
335Q8—44no—
336Q8—82no—
337Q8—121no—
338Q8—121no—
339D4—121no—
340D4—121no—
341D4—122no—
342D4—241no—
343D4—241no—
344D4—241no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—44no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—121no—
358C2×C4—241no—
359C2×C4—241no—
360C2×C4#93711yescharacteristic
361C2×C4#93711yescharacteristic
362C2×C4#93811yescharacteristic
363C2×C4#93911yescharacteristic
364C2×C4#93911yescharacteristic
365C2×C4#94011yescharacteristic
366C2×C4C2×#21511yescharacteristic
367C2×C4#94211yescharacteristic
368C2×C4#94911yescharacteristic
369C2×C4#95011yescharacteristic
370C8—122no—
371C8—122no—
372C8—241no—
373C2×C3—21no—
374C2×C3—21no—
375C2×C3#200011yescharacteristic
376C2×C3#200411yescharacteristic
377C2×C3#200911yescharacteristic
378D3—161no—
379C22—21no—
380C22—21no—
381C22—241no—
382C22—241no—
383C22—241no—
384C22#522311yescharacteristic
385C22#522711yescharacteristic
386C22#523811yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—42no—
391C4—82no—
392C4—82no—
393C4—81no—
394C4—241no—
395C4#504611yescharacteristic
396C4#504711yescharacteristic
397C4#504811yescharacteristic
398C4#504911yescharacteristic
399C4#506112yes—
400C3#1024411yescharacteristic, complemented, Sylow
401C2—21no—
402C2—21no—
403C2—481no—
404C2#7169911yescharacteristic
405C2#7173111yescharacteristic
406C2#7186911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2568C2
socleC22×C3#392
prime radical#5477C22
hypercenter#2568D3
hypocenterC3#10244
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2568

Complements

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

NH ≅ G/Ntype
1. C3×#2568C2semidirect
4. #71774C2semidirect
5. #71621C2semidirect
7. #67620C2semidirect
34. #2568D3semidirect
116. C2×C3×Q8SD16semidirect
266. C2×Q8#215semidirect
400. C3#10244semidirect

Maximal subgroups

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

subgroupindex
C3×#25682
#832132
#824412
#717742
#716212
#694692
#676202
#102443