Finite Group Databasebeta

← #10697235

Subgroups — #10697235

Order 768.

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

normal subgroups only

This group has 518 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
1C2×#5094C211yesmaximal, characteristic, complemented
2#83216C211yesmaximal, characteristic
3#79067C211yesmaximal, characteristic
4C3×#1548C211yesmaximal, characteristic, complemented
5#70786C211yesmaximal, characteristic, complemented
6#70758C211yesmaximal, characteristic
7#67714C211yesmaximal, characteristic, complemented
8#10189—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×#909—21no—
11#5477C2211yescharacteristic
12C2×C4×Dic6C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×#423C2211yescharacteristic
15#5133—21no—
16#5094—22no—
17#5094—22no—
18#5088—21no—
19#5084—22no—
20#5080—21no—
21C2×#810—21no—
22#4968—22no—
23C2×#808—21no—
24#4925C2211yescharacteristic
25C2×C4⋊(C3⋊C8)C2211yescharacteristic
26#4849—21no—
27#4848—21no—
28#4845—22no—
29C3×#333—22no—
30#4469—22no—
31#4459—22no—
32#3227—31no—
33C2×#475—31no—
34#3037—31no—
35#1548D311yescharacteristic, complemented
36#1480—31no—
37#1461—31no—
38#1301—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⋊Q8D412yes—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C42C2311yescharacteristic
50#930—21no—
51#930—21no—
52C2×C4⋊Dic3D411yescharacteristic
53C2×Dic3⋊C4—21no—
54C2×C4×Dic3—21no—
55C22×C3⋊C8—21no—
56#909—42no—
57#909—42no—
58C2×Dic12—22no—
59C2×#204—22no—
60#875—21no—
61#874D411yescharacteristic
62C4×D12—21no—
63C4×D12—21no—
64#869D411yescharacteristic
65C4×Dic6—21no—
66C4×Dic6—21no—
67C4×Dic6—21no—
68C4×Dic6—21no—
69C3×C4⋊C8—21no—
70C3×C4⋊C8—21no—
71C3×C4⋊C8D412yes—
72C3×C22⋊C8—21no—
73C3×C4×C8—21no—
74#830D412yes—
75#821—22no—
76#816—22no—
77#810—24no—
78#809—24no—
79#808—24no—
80#808—24no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)—21no—
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#475—62no—
94#475—62no—
95#464—62no—
96#462—61no—
97#460—61no—
98#431—31no—
99#431C2×D311yescharacteristic
100C2×#106—61no—
101#423C2×D311yescharacteristic
102C2×C4⋊C8—31no—
103#417—61no—
104C2×#103—61no—
105#333—22no—
106D4⋊C8—62no—
107#324—62no—
108C2×C3×Q8—22no—
109C2×C3×Q8—41no—
110C2×C3×Q8—82no—
111C22×C3×C4—41no—
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4#4111yescharacteristic
115C2×#67—41no—
116C22×Dic3—21no—
117C22×Dic3—21no—
118#235—41no—
119#235—41no—
120#235—41no—
121#235—41no—
122C2×D12—21no—
123C2×D12—21no—
124C2×D12—21no—
125C2×C4×D3—41no—
126C2×Dic6—21no—
127C2×Dic6—21no—
128C2×Dic6—21no—
129C2×Dic6—21no—
130C2×Dic6—21no—
131C2×Dic6—21no—
132C2×Dic6SD1612yes—
133C3×M16—41no—
134C3×M16—41no—
135C2×C3×C8—22no—
136C2×C3×C8—41no—
137C3×C4⋊C4—22no—
138C3×C4⋊C4—22no—
139C3×C4⋊C4—41no—
140C3×C4⋊C4—41no—
141C3×C42C2×D411yescharacteristic
142C3×C42C2×D411yescharacteristic
143C3×C42C2×D411yescharacteristic
144C3×C42C2×D411yescharacteristic
145(C2×D3)⋊C4—41no—
146(C2×D3)⋊C4—41no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151Dic3⋊C4—41no—
152Dic3⋊C4—41no—
153C4×Dic3—41no—
154C4×Dic3—41no—
155C2×C3⋊C8—22no—
156C2×C3⋊C8—22no—
157C2×C3⋊C8—41no—
158C2×C3⋊C8—41no—
159Dic12—44no—
160#204—44no—
161C2×#41—61no—
162C22×Q8—21no—
163C22×Q8—31no—
164#137—122no—
165#137—122no—
166C2×Q16—62no—
167C2×SD16—62no—
168C2×M16—21no—
169C22×C8—61no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—31no—
174C4⋊Q8D1212yes—
175#127—31no—
176#124—61no—
177#121—61no—
178#121—61no—
179C4×Q8—61no—
180C4×Q8—61no—
181C4×Q8—61no—
182C4×Q8—61no—
183C4×D4—61no—
184C4×D4—61no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—31no—
188C2×C4⋊C4—61no—
189C2×C42—61no—
190C2×C42C22×D311yescharacteristic
191#106—64no—
192#106—64no—
193C4⋊C8—21no—
194C4⋊C8—21no—
195C4⋊C8—32no—
196C4⋊C8—61no—
197C4⋊C8—61no—
198C4⋊C8#6712yes—
199#103—64no—
200#103—62no—
201#102—64no—
202#102—62no—
203C22⋊C8—21no—
204C22⋊C8—32no—
205C4×C8—21no—
206C4×C8—61no—
207C23×C3#12111yescharacteristic
208C22×D3—41no—
209C3×Q8—82no—
210C3×Q8—82no—
211C2×C3×C4—22no—
212C2×C3×C4—41no—
213C2×C3×C4—42no—
214C2×C3×C4—82no—
215C2×C3×C4#12011yescharacteristic
216C2×C3×C4#12011yescharacteristic
217C2×C3×C4#12011yescharacteristic
218C2×C3×C4#12111yescharacteristic
219C2×C3×C4#12111yescharacteristic
220C2×C3×C4#12711yescharacteristic
221C2×C3×C4C2×SD1611yescharacteristic
222C2×C3×C4#13511yescharacteristic
223C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
224C2×C3×C4#13711yescharacteristic
225#67—81no—
226#67—81no—
227C2×Dic3—22no—
228C2×Dic3—41no—
229C2×Dic3—41no—
230C2×Dic3—41no—
231C2×Dic3—41no—
232C2×Dic3—81no—
233D12—42no—
234D12—41no—
235D12—41no—
236D12—42no—
237C4×D3—81no—
238C4×D3—81no—
239Dic6—22no—
240Dic6—22no—
241Dic6—41no—
242Dic6—41no—
243Dic6—41no—
244Dic6—41no—
245C3×C8—42no—
246C3×C8—81no—
247C3⋊C8—42no—
248C3⋊C8—42no—
249#41—121no—
250#41—121no—
251#41—121no—
252#41—121no—
253C2×Q8—22no—
254C2×Q8—32no—
255C2×Q8—41no—
256C2×Q8—61no—
257C2×Q8—61no—
258C2×Q8—61no—
259C2×Q8—61no—
260C2×Q8—61no—
261C2×Q8—61no—
262C2×Q8—82no—
263C2×D4—61no—
264C2×D4—61no—
265C2×D4—61no—
266C2×D4—121no—
267C22×C4—41no—
268C22×C4—61no—
269C22×C4—61no—
270C22×C4—121no—
271C22×C4#23511yescharacteristic
272C22×C4D3×D411yescharacteristic
273C22×C4D3×D411yescharacteristic
274Q16—124no—
275SD16—124no—
276M16—41no—
277M16—41no—
278C2×C8—22no—
279C2×C8—41no—
280C2×C8—62no—
281C2×C8—62no—
282C2×C8—121no—
283C2×C8—121no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—61no—
289C4⋊C4—61no—
290C4⋊C4—61no—
291C4⋊C4—61no—
292C4⋊C4—121no—
293C4⋊C4—121no—
294C22⋊C4—121no—
295C22⋊C4—121no—
296C42—121no—
297C42—121no—
298C42C2×D1211yescharacteristic
299C42D3×D411yescharacteristic
300C42D3×D411yescharacteristic
301C42C2×#6711yescharacteristic
302C22×C3—21no—
303C22×C3—21no—
304C22×C3#39211yescharacteristic
305C22×C3#46511yescharacteristic
306C22×C3#46811yescharacteristic
307C2×D3—81no—
308C2×D3—81no—
309C2×D3—81no—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—21no—
313C3×C4—82no—
314C3×C4#44811yescharacteristic
315C3×C4#44911yescharacteristic
316C3×C4#45011yescharacteristic
317C3×C4#45211yescharacteristic
318C3×C4#47112yes—
319Dic3—42no—
320Dic3—81no—
321Dic3—81no—
322C23—121no—
323C23#88411yescharacteristic
324Q8—62no—
325Q8—62no—
326Q8—82no—
327Q8—82no—
328Q8—121no—
329Q8—121no—
330Q8—121no—
331Q8—121no—
332D4—122no—
333D4—121no—
334D4—121no—
335D4—122no—
336D4—241no—
337D4—241no—
338C2×C4—22no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—62no—
342C2×C4—82no—
343C2×C4—121no—
344C2×C4—121no—
345C2×C4—121no—
346C2×C4—121no—
347C2×C4—241no—
348C2×C4—241no—
349C2×C4—241no—
350C2×C4#88211yescharacteristic
351C2×C4#88211yescharacteristic
352C2×C4#88411yescharacteristic
353C2×C4D3×SD1611yescharacteristic
354C2×C4#91411yescharacteristic
355C2×C4#91811yescharacteristic
356C2×C4#91911yescharacteristic
357C2×C4#93011yescharacteristic
358C2×C4#93711yescharacteristic
359C2×C4#94011yescharacteristic
360C8—42no—
361C8—81no—
362C8—122no—
363C8—122no—
364C2×C3—21no—
365C2×C3—21no—
366C2×C3#200011yescharacteristic
367C2×C3#200911yescharacteristic
368C2×C3#201911yescharacteristic
369D3—161no—
370C22—21no—
371C22—21no—
372C22—241no—
373C22—241no—
374C22—241no—
375C22#480711yescharacteristic
376C22#481211yescharacteristic
377C22#496011yescharacteristic
378C4—21no—
379C4—21no—
380C4—21no—
381C4—82no—
382C4—122no—
383C4—241no—
384C4—241no—
385C4#474812yes—
386C4#480611yescharacteristic
387C4#481011yescharacteristic
388C4#516911yescharacteristic
389C4#518711yescharacteristic
390C3#1018911yescharacteristic, complemented, Sylow
391C2—21no—
392C2—21no—
393C2—481no—
394C2#7009111yescharacteristic
395C2#7027211yescharacteristic
396C2#7035111yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1548C2
socleC22×C3#392
prime radical#5477C22
hypercenter#1548D3
hypocenterC3#10189
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
1. C2×#5094C2semidirect
4. C3×#1548C2semidirect
5. #70786C2semidirect
7. #67714C2semidirect
35. #1548D3semidirect
390. C3#10189semidirect

Maximal subgroups

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

subgroupindex
C2×#50942
#832162
#790672
C3×#15482
#707862
#707582
#677142
#101893