Finite Group Databasebeta

← #10696420

Subgroups — #10696420

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
1#82476C211yesmaximal, characteristic
2C2×#5020C211yesmaximal, characteristic, complemented
3C2×#4694C211yesmaximal, characteristic
4C2×C3×Q8⋊C8C211yesmaximal, characteristic, complemented
5#70456C211yesmaximal, characteristic, complemented
6C2×#4485C211yesmaximal, characteristic
7#67655C211yesmaximal, characteristic, complemented
8#10071—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10C2×#947—21no—
11C2×#874C2211yescharacteristic
12C2×#869C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14C2×C3×C4×C8C2211yescharacteristic
15C2×#835—21no—
16C2×#821—22no—
17C2×#816—22no—
18#5020—22no—
19#5020—22no—
20#4991—22no—
21#4988—21no—
22C2×#808—21no—
23#4939—22no—
24C4×C4⋊Dic3C2211yescharacteristic
25C2×C4⋊(C3⋊C8)C2211yescharacteristic
26#4694—22no—
27#4694—22no—
28C3×Q8⋊C8—22no—
29#4485—22no—
30C2×#486—31no—
31C2×#475—31no—
32#2548—31no—
33#1981—31no—
34C2×#331—31no—
35C2×Q8⋊C8D311yescharacteristic, complemented
36#1279—31no—
37C22×C3×Q8D411yescharacteristic
38C22×D12—21no—
39C22×D12—21no—
40C22×D12D412yes—
41C22×Dic6—21no—
42C22×C3×C8—21no—
43C22×C3×C8—21no—
44C3×C4×Q8—21no—
45C3×C4×Q8—21no—
46C3×C4×Q8—21no—
47C3×C4×Q8—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4D411yescharacteristic
50C2×C3×C42—21no—
51C2×C3×C42C2311yescharacteristic
52#947—42no—
53#947—42no—
54C2×(C2×D3)⋊C4—24no—
55C2×(C2×D3)⋊C4—24no—
56C2×C4⋊Dic3—21no—
57C2×C4⋊Dic3—21no—
58C2×C4⋊Dic3D412yes—
59C2×C4×Dic3—21no—
60C22×C3⋊C8—21no—
61#874—21no—
62#874—21no—
63#874—21no—
64#874—21no—
65#869—21no—
66#869—21no—
67#869—21no—
68#869D412yes—
69C3×C4⋊C8—21no—
70C3×C4⋊C8—21no—
71C3×C4×C8—21no—
72C3×C4×C8D412yes—
73#835—24no—
74C3⋊#95—21no—
75#821—28no—
76#816—44no—
77#808—24no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)D412yes—
80C2×C4⋊Q8—31no—
81C2×#127—31no—
82C2×#124—61no—
83C2×C4×Q8C2×D311yescharacteristic
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—61no—
92C2×#103—62no—
93C2×#102—62no—
94C2×C4×C8C2×D311yescharacteristic
95#394—62no—
96#390—62no—
97#390—61no—
98C4×C4⋊C4—31no—
99#331—62no—
100Q8⋊C8—22no—
101C23×D3—22no—
102C2×C3×Q8—21no—
103C2×C3×Q8—21no—
104C2×C3×Q8—21no—
105C2×C3×Q8—21no—
106C2×C3×Q8—21no—
107C2×C3×Q8SD1612yes—
108C22×C3×C4—21no—
109C22×C3×C4—21no—
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×Dic3—22no—
114C22×Dic3—41no—
115C2×D12—22no—
116C2×D12—22no—
117C2×D12—44no—
118C2×D12—41no—
119C2×D12—41no—
120C2×D12—41no—
121C2×D12—41no—
122C2×D12—41no—
123C2×D12—41no—
124C2×D12—42no—
125C2×D12—41no—
126C2×D12—42no—
127C2×D12—41no—
128C2×Dic6—22no—
129C2×Dic6—42no—
130C2×Dic6—42no—
131C2×Dic6—41no—
132C2×C3×C8—22no—
133C2×C3×C8—42no—
134C2×C3×C8—41no—
135C2×C3×C8—42no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
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—22no—
149C4⋊Dic3—22no—
150C4⋊Dic3—22no—
151C4⋊Dic3—22no—
152C4⋊Dic3—41no—
153C4⋊Dic3—41no—
154C4×Dic3—41no—
155C4×Dic3—41no—
156C2×C3⋊C8—22no—
157C2×C3⋊C8—41no—
158C2×C3⋊C8—41no—
159C22×Q8—61no—
160C22×Q8D1211yescharacteristic
161C22×D4—32no—
162C22×D4—61no—
163C22×D4—61no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—61no—
167C4⋊Q8—32no—
168C4⋊Q8—61no—
169C4⋊Q8—61no—
170C4⋊Q8—61no—
171#127—61no—
172#127—61no—
173#127—61no—
174#127—61no—
175#124—122no—
176#124—122no—
177C4×Q8—21no—
178C4×Q8—21no—
179C4×Q8—21no—
180C4×Q8—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—32no—
183C2×C4⋊C4—61no—
184C2×C4⋊C4—61no—
185C2×C4⋊C4D1211yescharacteristic
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—64no—
197#103—124no—
198#102—68no—
199C4×C8—21no—
200C4×C8#6712yes—
201#95—61no—
202C23×C3#12011yescharacteristic
203C22×D3—42no—
204C22×D3—44no—
205C22×D3—42no—
206C22×D3—42no—
207C22×D3—44no—
208C3×Q8—22no—
209C3×Q8—22no—
210C3×Q8—41no—
211C3×Q8—41no—
212C2×C3×C4—22no—
213C2×C3×C4—22no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—42no—
218C2×C3×C4#12011yescharacteristic
219C2×C3×C4#12011yescharacteristic
220C2×C3×C4#12111yescharacteristic
221C2×C3×C4#12111yescharacteristic
222C2×C3×C4#12111yescharacteristic
223C2×C3×C4#12711yescharacteristic
224C2×C3×C4C2×SD1611yescharacteristic
225C2×C3×C4#13511yescharacteristic
226C2×C3×C4#13512yes—
227C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
228C2×C3×C4AGL(1,ℤ/8)12yes—
229C2×C3×C4#13711yescharacteristic
230C2×Dic3—42no—
231C2×Dic3—42no—
232C2×Dic3—42no—
233C2×Dic3—42no—
234C2×Dic3—82no—
235D12—44no—
236D12—82no—
237D12—81no—
238D12—81no—
239D12—82no—
240D12—81no—
241D12—81no—
242D12—81no—
243D12—81no—
244Dic6—44no—
245Dic6—81no—
246Dic6—81no—
247C3×C8—42no—
248C3×C8—81no—
249C3⋊C8—42no—
250C24—62no—
251C2×Q8—21no—
252C2×Q8—21no—
253C2×Q8—21no—
254C2×Q8—21no—
255C2×Q8—21no—
256C2×Q8—62no—
257C2×Q8—121no—
258C2×Q8—122no—
259C2×Q8—122no—
260C2×Q8#20412yes—
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—122no—
271C2×D4—122no—
272C2×D4—121no—
273C2×D4—121no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—62no—
277C22×C4—121no—
278C22×C4C2×D1211yescharacteristic
279C22×C4D3×D411yescharacteristic
280C22×C4D3×D411yescharacteristic
281C2×C8—22no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—41no—
285C2×C8—62no—
286C2×C8—121no—
287C2×C8—121no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—62no—
293C4⋊C4—62no—
294C4⋊C4—62no—
295C4⋊C4—62no—
296C4⋊C4—121no—
297C4⋊C4—121no—
298C22⋊C4—128no—
299C22⋊C4—244no—
300C42—41no—
301C42—41no—
302C42—121no—
303C42—121no—
304C42#23511yescharacteristic
305C42D3×D411yescharacteristic
306C42D3×D411yescharacteristic
307C42C2×#6711yescharacteristic
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—42no—
321C3×C4—81no—
322C3×C4#44811yescharacteristic
323C3×C4#44911yescharacteristic
324C3×C4#46312yes—
325C3×C4#46511yescharacteristic
326C3×C4#46811yescharacteristic
327C3×C4#49511yescharacteristic
328C3×C4#49611yescharacteristic
329Dic3—82no—
330Dic3—82no—
331C23—122no—
332C23—122no—
333C23—122no—
334C23—124no—
335C23—124no—
336C23#88211yescharacteristic
337Q8—22no—
338Q8—22no—
339Q8—41no—
340Q8—41no—
341Q8—124no—
342Q8—241no—
343Q8—241no—
344D4—124no—
345D4—242no—
346D4—241no—
347D4—241no—
348D4—241no—
349D4—241no—
350D4—241no—
351D4—242no—
352D4—241no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—122no—
360C2×C4—122no—
361C2×C4—122no—
362C2×C4—122no—
363C2×C4—242no—
364C2×C4#88211yescharacteristic
365C2×C4#88411yescharacteristic
366C2×C4#88411yescharacteristic
367C2×C4C2×#20411yescharacteristic
368C2×C4#90411yescharacteristic
369C2×C4#90811yescharacteristic
370C2×C4#90911yescharacteristic
371C2×C4#91412yes—
372C2×C4#91912yes—
373C2×C4#93011yescharacteristic
374C2×C4#93711yescharacteristic
375C2×C4#94011yescharacteristic
376C8—42no—
377C8—81no—
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#473111yescharacteristic
391C22#473311yescharacteristic
392C22#481012yes—
393C22#482512yes—
394C22#496011yescharacteristic
395C4—21no—
396C4—21no—
397C4—42no—
398C4—81no—
399C4—242no—
400C4—242no—
401C4#473411yescharacteristic
402C4#473811yescharacteristic
403C4#481912yes—
404C4#511411yescharacteristic
405C4#513411yescharacteristic
406C4#517611yescharacteristic
407C4#519111yescharacteristic
408C3#1007111yescharacteristic, complemented, Sylow
409C2—482no—
410C2#6824812yes—
411C2#6825412yes—
412C2#6996211yescharacteristic
413C2#7027711yescharacteristic
414C2#7038611yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×Q8⋊C8C2
socleC23×C3#120
prime radicalC2×#874C22
hypercenterC2×Q8⋊C8D3
hypocenterC3#10071
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#5020C2semidirect
4. C2×C3×Q8⋊C8C2semidirect
5. #70456C2semidirect
7. #67655C2semidirect
35. C2×Q8⋊C8D3semidirect
408. C3#10071semidirect

Maximal subgroups

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

subgroupindex
#824762
C2×#50202
C2×#46942
C2×C3×Q8⋊C82
#704562
C2×#44852
#676552
#100713