Finite Group Databasebeta

← #10697761

Subgroups — #10697761

Order 768.

594 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5084C211yesmaximal, characteristic
2C2×#5080C211yesmaximal, characteristic, complemented
3C2×#4841C211yesmaximal, characteristic
4C2×C3×#333C211yesmaximal, characteristic, complemented
5#70470C211yesmaximal, characteristic
6C2×#4486C211yesmaximal, characteristic, complemented
7#67703C211yesmaximal, characteristic, complemented
8#10271—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C22×#204—21no—
11C2×C4×D12C2211yescharacteristic
12C2×#869C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14C2×C3×C4⋊C8C2211yescharacteristic
15C2×(C2×D3)⋊C8—22no—
16C2×#816—21no—
17#5084—22no—
18#5084—22no—
19#5080—22no—
20#5080—22no—
21C2×#809—21no—
22C2×#808—21no—
23C2×#808—22no—
24C2×C4⋊(C3⋊C8)C2211yescharacteristic
25C2×C4×C3⋊C8C2211yescharacteristic
26#4841—22no—
27#4841—22no—
28C3×#333—22no—
29#4486—22no—
30C2×#498—31no—
31C2×#476—31no—
32C2×#461—31no—
33#1954—31no—
34C2×#333D311yescharacteristic, complemented
35C2×#332—31no—
36#1277—31no—
37C22×C3×Q8—21no—
38C22×D12D411yescharacteristic
39C22×C4×D3—21no—
40C22×Dic6—21no—
41C22×C3×C8—21no—
42C22×C3×C8—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
50C2×(C2×D3)⋊C4—21no—
51C2×C4⋊Dic3—21no—
52C2×C4⋊Dic3—21no—
53C2×C4⋊Dic3D411yescharacteristic
54C22×C3⋊C8—21no—
55C22×C3⋊C8D412yes—
56C2×#204—24no—
57C2×#204—42no—
58C2×#204—42no—
59C4×D12—21no—
60C4×D12—21no—
61C4×D12—21no—
62C4×D12—21no—
63#869—21no—
64#869—21no—
65#869—21no—
66#869D412yes—
67C3×C4⋊C8—21no—
68C3×C4⋊C8—21no—
69C3×C4⋊C8—21no—
70C3×C4⋊C8D412yes—
71(C2×D3)⋊C8—44no—
72#816—24no—
73#809—24no—
74#808—28no—
75#808—24no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)D412yes—
78C4×C3⋊C8—21no—
79C4×C3⋊C8—21no—
80C22×SD16—61no—
81C2×C4⋊Q8—31no—
82C2×C4⋊Q8C2×D311yescharacteristic
83C2×C4×D4—31no—
84#498—62no—
85#498—62no—
86#476—62no—
87#476—62no—
88#461—62no—
89#461—62no—
90C2×#106—61no—
91C2×#106—62no—
92C2×C4⋊C8—31no—
93C2×C4⋊C8C2×D311yescharacteristic
94C2×C4⋊C8C2×D311yescharacteristic
95C2×#103—61no—
96C2×#102—61no—
97C2×C22⋊C8—62no—
98C2×C4×C8—31no—
99#333—22no—
100#332—62no—
101C23×D3—21no—
102C2×C3×Q8—22no—
103C2×C3×Q8—41no—
104C2×C3×Q8—82no—
105C22×C3×C4—41no—
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4#4111yescharacteristic
109C22×Dic3—21no—
110C22×Dic3—41no—
111C2×D12—21no—
112C2×D12—21no—
113C2×D12—21no—
114C2×D12—21no—
115C2×D12—21no—
116C2×D12SD1612yes—
117C2×C4×D3—41no—
118C2×C4×D3—41no—
119C2×C4×D3—42no—
120C2×C4×D3—42no—
121C2×Dic6—22no—
122C2×Dic6—41no—
123C2×Dic6—41no—
124C2×Dic6—41no—
125C2×Dic6—41no—
126C2×Dic6—41no—
127C2×C3×C8—22no—
128C2×C3×C8—41no—
129C2×C3×C8—41no—
130C2×C3×C8—41no—
131C2×C3×C8—42no—
132C3×C4⋊C4—22no—
133C3×C4⋊C4—22no—
134C3×C4⋊C4—41no—
135C3×C4⋊C4—41no—
136C3×C42C2×D411yescharacteristic
137C3×C42C2×D411yescharacteristic
138C3×C42C2×D411yescharacteristic
139C3×C42C2×D411yescharacteristic
140(C2×D3)⋊C4—41no—
141(C2×D3)⋊C4—41no—
142C4⋊Dic3—22no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—22no—
146C4⋊Dic3—41no—
147C4⋊Dic3—41no—
148C2×C3⋊C8—22no—
149C2×C3⋊C8—22no—
150C2×C3⋊C8—22no—
151C2×C3⋊C8—41no—
152C2×C3⋊C8—41no—
153C2×C3⋊C8SD1614yescomplemented
154#204—48no—
155C22×Q8—21no—
156C22×Q8—61no—
157C22×D4—31no—
158C23×C4—61no—
159C2×SD16—64no—
160C2×SD16—122no—
161C2×SD16—122no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—32no—
165C22×C8—61no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—32no—
170C4⋊Q8—61no—
171C4⋊Q8—61no—
172C4⋊Q8—61no—
173C4⋊Q8D1212yes—
174C4×D4—61no—
175C4×D4—61no—
176C4×D4—61no—
177C4×D4—61no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—31no—
181C2×C4⋊C4—61no—
182C2×C4⋊C4—61no—
183C2×C22⋊C4—61no—
184C2×C42C22×D311yescharacteristic
185#106—64no—
186#106—68no—
187C4⋊C8—21no—
188C4⋊C8—21no—
189C4⋊C8—21no—
190C4⋊C8—32no—
191C4⋊C8—61no—
192C4⋊C8#6712yes—
193#103—64no—
194#102—64no—
195C22⋊C8—124no—
196C4×C8—61no—
197C4×C8—61no—
198C23×C3#12111yescharacteristic
199C22×D3—22no—
200C22×D3—41no—
201C22×D3—42no—
202C22×D3—41no—
203C22×D3—42no—
204C3×Q8—82no—
205C3×Q8—82no—
206C2×C3×C4—22no—
207C2×C3×C4—42no—
208C2×C3×C4—82no—
209C2×C3×C4#12011yescharacteristic
210C2×C3×C4#12011yescharacteristic
211C2×C3×C4#12011yescharacteristic
212C2×C3×C4#12111yescharacteristic
213C2×C3×C4#12111yescharacteristic
214C2×C3×C4#12711yescharacteristic
215C2×C3×C4C2×SD1611yescharacteristic
216C2×C3×C4C2×SD1612yes—
217C2×C3×C4#13511yescharacteristic
218C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
219C2×C3×C4#13711yescharacteristic
220C2×C3×C4#13712yes—
221C2×Dic3—42no—
222C2×Dic3—41no—
223C2×Dic3—42no—
224C2×Dic3—81no—
225C2×Dic3—81no—
226D12—22no—
227D12—22no—
228D12—41no—
229D12—41no—
230C4×D3—82no—
231C4×D3—81no—
232C4×D3—81no—
233Dic6—42no—
234Dic6—42no—
235Dic6—81no—
236Dic6—81no—
237C3×C8—42no—
238C3×C8—81no—
239C3⋊C8—24no—
240C3⋊C8—42no—
241C24—61no—
242C2×Q8—22no—
243C2×Q8—41no—
244C2×Q8—62no—
245C2×Q8—82no—
246C2×Q8—121no—
247C2×Q8—121no—
248C2×Q8—121no—
249C2×Q8—121no—
250C2×Q8—121no—
251C2×D4—32no—
252C2×D4—61no—
253C2×D4—61no—
254C2×D4—61no—
255C2×D4—61no—
256C2×D4—61no—
257C22×C4—41no—
258C22×C4—61no—
259C22×C4—121no—
260C22×C4—121no—
261C22×C4—122no—
262C22×C4—122no—
263C22×C4—121no—
264C22×C4#23511yescharacteristic
265C22×C4D3×D411yescharacteristic
266C22×C4D3×D411yescharacteristic
267SD16—128no—
268C2×C8—22no—
269C2×C8—34no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—42no—
273C2×C8—41no—
274C2×C8—62no—
275C2×C8—62no—
276C2×C8—62no—
277C2×C8—121no—
278C2×C8—121no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—61no—
284C4⋊C4—62no—
285C4⋊C4—62no—
286C4⋊C4—61no—
287C4⋊C4—121no—
288C4⋊C4—121no—
289C22⋊C4—121no—
290C22⋊C4—121no—
291C42C2×D1211yescharacteristic
292C42D3×D411yescharacteristic
293C42D3×D411yescharacteristic
294C42C2×#6711yescharacteristic
295C22×C3#39211yescharacteristic
296C22×C3#46112yes—
297C22×C3#46511yescharacteristic
298C22×C3#46811yescharacteristic
299C22×C3#47112yes—
300C2×D3—44no—
301C2×D3—42no—
302C2×D3—82no—
303C2×D3—81no—
304C2×D3—81no—
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—82no—
308C3×C4#44811yescharacteristic
309C3×C4#44812yes—
310C3×C4#44911yescharacteristic
311C3×C4#45011yescharacteristic
312C3×C4#45211yescharacteristic
313C3×C4#49211yescharacteristic
314C3×C4#49711yescharacteristic
315Dic3—82no—
316Dic3—81no—
317C23—62no—
318C23—121no—
319C23—122no—
320C23—121no—
321C23—122no—
322C23#88411yescharacteristic
323Q8—82no—
324Q8—82no—
325Q8—122no—
326Q8—122no—
327Q8—241no—
328Q8—241no—
329D4—62no—
330D4—62no—
331D4—121no—
332D4—121no—
333C2×C4—22no—
334C2×C4—42no—
335C2×C4—82no—
336C2×C4—121no—
337C2×C4—122no—
338C2×C4—122no—
339C2×C4—241no—
340C2×C4—241no—
341C2×C4—241no—
342C2×C4—241no—
343C2×C4—242no—
344C2×C4#88211yescharacteristic
345C2×C4#88211yescharacteristic
346C2×C4#88411yescharacteristic
347C2×C4D3×SD1611yescharacteristic
348C2×C4D3×SD1612yes—
349C2×C4#91411yescharacteristic
350C2×C4#91811yescharacteristic
351C2×C4#91812yes—
352C2×C4#91911yescharacteristic
353C2×C4#93011yescharacteristic
354C2×C4#93711yescharacteristic
355C2×C4#94011yescharacteristic
356C8—42no—
357C8—64no—
358C8—81no—
359C8—122no—
360C2×C3#167112yes—
361C2×C3#169112yes—
362C2×C3#200011yescharacteristic
363C2×C3#201311yescharacteristic
364C2×C3#201911yescharacteristic
365D3—82no—
366C22—122no—
367C22—124no—
368C22—241no—
369C22—242no—
370C22—241no—
371C22#474812yes—
372C22#480711yescharacteristic
373C22#481211yescharacteristic
374C22#481812yes—
375C22#496011yescharacteristic
376C4—21no—
377C4—21no—
378C4—82no—
379C4—242no—
380C4—241no—
381C4#480611yescharacteristic
382C4#480612yes—
383C4#481011yescharacteristic
384C4#516911yescharacteristic
385C4#517511yescharacteristic
386C4#518711yescharacteristic
387C4#519011yescharacteristic
388C3#1027111yescharacteristic, complemented, Sylow
389C2—242no—
390C2#6849612yes—
391C2#6850012yes—
392C2#7009111yescharacteristic
393C2#7027211yescharacteristic
394C2#7028211yescharacteristic
subgroupG/H
centerC23#884
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#333C2
socleC23×C3#121
prime radicalC22×D12D4
hypercenterC2×#333D3
hypocenterC3#10271
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#333

Complements

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

NH ≅ G/Ntype
2. C2×#5080C2semidirect
4. C2×C3×#333C2semidirect
6. C2×#4486C2semidirect
7. #67703C2semidirect
34. C2×#333D3semidirect
153. C2×C3⋊C8SD16semidirect
388. C3#10271semidirect

Maximal subgroups

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

subgroupindex
C2×#50842
C2×#50802
C2×#48412
C2×C3×#3332
#704702
C2×#44862
#677032
#102713