Finite Group Databasebeta

← #10696417

Subgroups — #10696417

Order 768.

654 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5094C211yesmaximal, characteristic, complemented
2C2×#5088C211yesmaximal, characteristic
3#73712C211yesmaximal, characteristic
4C3×#1981C211yesmaximal, characteristic, complemented
5C2×#4485C211yesmaximal, characteristic
6C2×#4484C211yesmaximal, characteristic, complemented
7#67619C211yesmaximal, characteristic, complemented
8#10071—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×#875—21no—
11C2×#874C2211yescharacteristic
12C2×C4×Dic6C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14C2×C3×#103—22no—
15C3×C4×C4⋊C4C2211yescharacteristic
16#5094—22no—
17#5094—22no—
18#5088—22no—
19#5088—22no—
20#4988—22no—
21C2×#810—21no—
22C2×#809—22no—
23C2×#808—21no—
24#4939—21no—
25C2×C4⋊(C3⋊C8)C2211yescharacteristic
26C2×C4×C3⋊C8C2211yescharacteristic
27#4672—22no—
28#4485—22no—
29#4484—22no—
30C2×#486—31no—
31C2×#475—31no—
32#2548—31no—
33#1981D311yescharacteristic, complemented
34C2×#331—31no—
35C2×Q8⋊C8—31no—
36#1279—31no—
37C22×C3×Q8—21no—
38C22×D12—21no—
39C22×D12—21no—
40C22×D12D412yes—
41C22×Dic6D411yescharacteristic
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×C4⋊C4D412yes—
50C2×C3×C42—21no—
51C2×C3×C42C2311yescharacteristic
52C2×(C2×D3)⋊C4—24no—
53C2×(C2×D3)⋊C4—24no—
54C2×C4⋊Dic3D411yescharacteristic
55C2×Dic3⋊C4—21no—
56C2×C4×Dic3—21no—
57C22×C3⋊C8—21no—
58C22×C3⋊C8—21no—
59#875—42no—
60#875—42no—
61#874—21no—
62#874—21no—
63#874—21no—
64#874—21no—
65C4×Dic6—21no—
66C4×Dic6—21no—
67C4×Dic6—21no—
68C4×Dic6—21no—
69C3×C4⋊C8—21no—
70C3×C4⋊C8—21no—
71C3×#103—44no—
72C3×#95—21no—
73#810—24no—
74#809—28no—
75#808—24no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)D412yes—
78C4×C3⋊C8—21no—
79C4×C3⋊C8D412yes—
80C2×C4⋊Q8C2×D311yescharacteristic
81C2×#127—31no—
82C2×#124—61no—
83C2×C4×Q8—31no—
84#486—62no—
85#486—62no—
86#475—62no—
87#475—62no—
88C2×#106—61no—
89C2×C4⋊C8—31no—
90C2×C4⋊C8C2×D311yescharacteristic
91C2×#103—22no—
92C2×#103—61no—
93C2×#102—62no—
94C2×C4×C8—31no—
95#394—62no—
96#390—61no—
97#390—62no—
98C4×C4⋊C4C2×D311yescharacteristic
99#331—62no—
100Q8⋊C8—62no—
101C23×D3—22no—
102C2×C3×Q8—22no—
103C2×C3×Q8—41no—
104C2×C3×Q8—42no—
105C2×C3×Q8—42no—
106C22×C3×C4—22no—
107C22×C3×C4—41no—
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×Dic3—21no—
112C22×Dic3—21no—
113C2×D12—22no—
114C2×D12—22no—
115C2×D12—44no—
116C2×D12—41no—
117C2×D12—41no—
118C2×D12—41no—
119C2×D12—41no—
120C2×D12—41no—
121C2×D12—41no—
122C2×D12—42no—
123C2×D12—41no—
124C2×D12—42no—
125C2×D12—41no—
126C2×Dic6—21no—
127C2×Dic6—21no—
128C2×Dic6—21no—
129C2×Dic6—21no—
130C2×Dic6—21no—
131C2×Dic6SD1612yes—
132C2×C3×C8—41no—
133C2×C3×C8—42no—
134C3×C4⋊C4—22no—
135C3×C4⋊C4—22no—
136C3×C4⋊C4—22no—
137C3×C4⋊C4—22no—
138C3×C4⋊C4—41no—
139C3×C4⋊C4—41no—
140C3×C42—41no—
141C3×C42—41no—
142C3×C42C2×D411yescharacteristic
143C3×C42C2×D411yescharacteristic
144C3×C42C2×D411yescharacteristic
145C3×C42#4111yescharacteristic
146(C2×D3)⋊C4—48no—
147(C2×D3)⋊C4—84no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150Dic3⋊C4—41no—
151Dic3⋊C4—41no—
152C4×Dic3—41no—
153C4×Dic3—41no—
154C2×C3⋊C8—22no—
155C2×C3⋊C8—22no—
156C2×C3⋊C8—42no—
157C2×C3⋊C8—41no—
158C2×C3⋊C8—41no—
159C22×Q8—21no—
160C22×Q8—31no—
161C22×D4—32no—
162C22×D4—61no—
163C22×D4—61no—
164C22×C8—21no—
165C22×C8—61no—
166C22×C8—61no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8D1212yes—
171#127—61no—
172#127—61no—
173#127—61no—
174#127—61no—
175#124—122no—
176#124—122no—
177C4×Q8—61no—
178C4×Q8—61no—
179C4×Q8—61no—
180C4×Q8—61no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—31no—
184C2×C4⋊C4—61no—
185C2×C4⋊C4#6712yes—
186C2×C22⋊C4—64no—
187C2×C22⋊C4—64no—
188C2×C42—21no—
189C2×C42—61no—
190C2×C42C22×D311yescharacteristic
191#106—64no—
192C4⋊C8—21no—
193C4⋊C8—21no—
194C4⋊C8—32no—
195C4⋊C8—61no—
196#103—44no—
197#103—64no—
198#102—68no—
199C4×C8—32no—
200C4×C8—61no—
201#95—21no—
202C23×C3#12011yescharacteristic
203C22×D3—42no—
204C22×D3—44no—
205C22×D3—42no—
206C22×D3—42no—
207C22×D3—44no—
208C3×Q8—44no—
209C3×Q8—81no—
210C3×Q8—81no—
211C2×C3×C4—22no—
212C2×C3×C4—42no—
213C2×C3×C4—42no—
214C2×C3×C4—42no—
215C2×C3×C4—42no—
216C2×C3×C4—82no—
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×C4#13512yes—
226C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
227C2×C3×C4AGL(1,ℤ/8)12yes—
228C2×C3×C4#13711yescharacteristic
229C2×Dic3—22no—
230C2×Dic3—41no—
231C2×Dic3—41no—
232C2×Dic3—41no—
233C2×Dic3—42no—
234D12—44no—
235D12—82no—
236D12—81no—
237D12—81no—
238D12—82no—
239D12—81no—
240D12—81no—
241D12—81no—
242D12—81no—
243Dic6—22no—
244Dic6—22no—
245Dic6—41no—
246Dic6—41no—
247C3×C8—81no—
248C3⋊C8—42no—
249C3⋊C8—42no—
250C24—62no—
251C2×Q8—22no—
252C2×Q8—32no—
253C2×Q8—41no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—61no—
257C2×Q8—61no—
258C2×Q8—61no—
259C2×Q8—61no—
260C2×Q8—61no—
261C2×D4—62no—
262C2×D4—62no—
263C2×D4—124no—
264C2×D4—121no—
265C2×D4—121no—
266C2×D4—121no—
267C2×D4—121no—
268C2×D4—121no—
269C2×D4—121no—
270C2×D4—121no—
271C2×D4—122no—
272C2×D4—121no—
273C2×D4—122no—
274C22×C4—22no—
275C22×C4—41no—
276C22×C4—61no—
277C22×C4—61no—
278C22×C4D3×D411yescharacteristic
279C22×C4D3×D411yescharacteristic
280C22×C4C2×#6711yescharacteristic
281C2×C8—42no—
282C2×C8—41no—
283C2×C8—62no—
284C2×C8—62no—
285C2×C8—122no—
286C2×C8—121no—
287C2×C8—121no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—61no—
295C4⋊C4—61no—
296C4⋊C4—121no—
297C4⋊C4—121no—
298C22⋊C4—128no—
299C22⋊C4—244no—
300C42—41no—
301C42—41no—
302C42—121no—
303C42—121no—
304C42C2×D1211yescharacteristic
305C42#23511yescharacteristic
306C42D3×D411yescharacteristic
307C42D3×D411yescharacteristic
308C22×C3#39211yescharacteristic
309C22×C3#44912yes—
310C22×C3#45011yescharacteristic
311C22×C3#45211yescharacteristic
312C22×C3#45512yes—
313C2×D3—84no—
314C2×D3—84no—
315C2×D3—82no—
316C2×D3—82no—
317C2×D3—82no—
318C3×C4—21no—
319C3×C4—21no—
320C3×C4—82no—
321C3×C4—82no—
322C3×C4#44811yescharacteristic
323C3×C4#44911yescharacteristic
324C3×C4#46312yes—
325C3×C4#46511yescharacteristic
326C3×C4#46811yescharacteristic
327C3×C4#49511yescharacteristic
328C3×C4#49611yescharacteristic
329Dic3—42no—
330Dic3—81no—
331C23—122no—
332C23—122no—
333C23—124no—
334C23—122no—
335C23—124no—
336C23#93711yescharacteristic
337Q8—44no—
338Q8—62no—
339Q8—62no—
340Q8—81no—
341Q8—81no—
342Q8—121no—
343Q8—121no—
344D4—124no—
345D4—242no—
346D4—242no—
347D4—241no—
348D4—241no—
349D4—241no—
350D4—241no—
351D4—241no—
352D4—241no—
353C2×C4—22no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—62no—
359C2×C4—82no—
360C2×C4—121no—
361C2×C4—121no—
362C2×C4—122no—
363C2×C4—121no—
364C2×C4#88211yescharacteristic
365C2×C4#88211yescharacteristic
366C2×C4#88411yescharacteristic
367C2×C4#88411yescharacteristic
368C2×C4D3×SD1611yescharacteristic
369C2×C4#91411yescharacteristic
370C2×C4#91811yescharacteristic
371C2×C4#91911yescharacteristic
372C2×C4#93011yescharacteristic
373C2×C4#93212yes—
374C2×C4#94011yescharacteristic
375C2×C4#95012yes—
376C8—81no—
377C8—122no—
378C8—122no—
379C2×C3#164012yes—
380C2×C3#164212yes—
381C2×C3#200011yescharacteristic
382C2×C3#200411yescharacteristic
383C2×C3#201211yescharacteristic
384D3—162no—
385C22—242no—
386C22—242no—
387C22—244no—
388C22—244no—
389C22—242no—
390C22#496011yescharacteristic
391C22#503712yes—
392C22#516911yescharacteristic
393C22#518711yescharacteristic
394C22#520212yes—
395C4—21no—
396C4—21no—
397C4—82no—
398C4—82no—
399C4—122no—
400C4—241no—
401C4#480611yescharacteristic
402C4#480711yescharacteristic
403C4#481011yescharacteristic
404C4#481211yescharacteristic
405C4#501612yes—
406C4#505911yescharacteristic
407C4#507411yescharacteristic
408C3#1007111yescharacteristic, complemented, Sylow
409C2—482no—
410C2#6965911yescharacteristic
411C2#7027211yescharacteristic
412C2#7034411yescharacteristic
413C2#7104512yes—
414C2#7106012yes—
subgroupG/H
centerC23#937
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1981C2
socleC23×C3#120
prime radicalC2×#874C22
hypercenter#1981D3
hypocenterC3#10071
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1981

Complements

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

NH ≅ G/Ntype
1. C2×#5094C2semidirect
4. C3×#1981C2semidirect
6. C2×#4484C2semidirect
7. #67619C2semidirect
33. #1981D3semidirect
408. C3#10071semidirect

Maximal subgroups

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

subgroupindex
C2×#50942
C2×#50882
#737122
C3×#19812
C2×#44852
C2×#44842
#676192
#100713