Finite Group Databasebeta

← #10697731

Subgroups — #10697731

Order 768.

646 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#476C211yesmaximal, characteristic, complemented
2C2×#5088C211yesmaximal, characteristic, complemented
3C2×#5015C211yesmaximal, characteristic
4C2×#4546C211yesmaximal, characteristic, complemented
5#71614C211yesmaximal, characteristic, complemented
6C2×#4485C211yesmaximal, characteristic, complemented
7#67703C211yesmaximal, characteristic, complemented
8#10267—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×C4×D4C2211yescharacteristic
11C22×#213—21no—
12C2×#874C2211yescharacteristic
13C3×#476—22no—
14C3×#476—22no—
15C2×C3×#106—21no—
16C2×C3×C4⋊C8C2211yescharacteristic
17C2×C3×#102—21no—
18C2×#830—22no—
19#5088—22no—
20#5088—22no—
21#5015—22no—
22#5015—22no—
23C2×#809—21no—
24C2×#809—22no—
25C2×C4⋊(C3⋊C8)C2211yescharacteristic
26C2×C4⋊(C3⋊C8)C2211yescharacteristic
27C2×C4×C3⋊C8C2211yescharacteristic
28#4546—22no—
29#4485—22no—
30C2×#486—31no—
31C2×#476D311yescharacteristic, complemented
32C2×#459—31no—
33#1954—31no—
34C2×#333—31no—
35C2×#331—31no—
36#1277—31no—
37C22×C3×Q8—21no—
38C22×C3×D4D411yescharacteristic
39C23×C3×C4—21no—
40C22×D12—21no—
41C22×D12—21no—
42C22×D12—21no—
43C22×C3×C8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—21no—
47C3×C4⋊Q8D412yes—
48C3×C4×D4—21no—
49C3×C4×D4—21no—
50C3×C4×D4—21no—
51C3×C4×D4—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4D411yescharacteristic
55C2×C3×C22⋊C4—21no—
56C2×C3×C42C2311yescharacteristic
57C2×#213—24no—
58C2×#213—42no—
59C2×#213—42no—
60C22×C3⋊C8—21no—
61C22×C3⋊C8—21no—
62C22×C3⋊C8D412yes—
63#874—21no—
64#874—21no—
65#874—21no—
66#874D412yes—
67C3×#106—24no—
68C3×C4⋊C8—21no—
69C3×C4⋊C8D412yes—
70C3×#102—24no—
71#830—44no—
72#809—24no—
73#809—28no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)D412yes—
78C4×C3⋊C8—21no—
79C4×C3⋊C8—21no—
80C22×D8—61no—
81C2×C4⋊Q8C2×D311yescharacteristic
82C2×#127—31no—
83C2×C4×D4C2×D311yescharacteristic
84#486—62no—
85#486—62no—
86#476—22no—
87#476—22no—
88#459—62no—
89#459—62no—
90C2×#106—21no—
91C2×C4⋊C8—31no—
92C2×C4⋊C8—31no—
93C2×C4⋊C8C2×D311yescharacteristic
94C2×#102—21no—
95C2×#102—62no—
96C2×#102—61no—
97C2×C22⋊C8—62no—
98C2×C4×C8—31no—
99#333—62no—
100#331—62no—
101C24×C3—21no—
102C23×D3—41no—
103C2×C3×Q8—22no—
104C2×C3×Q8—41no—
105C2×C3×Q8—82no—
106C2×C3×D4—21no—
107C2×C3×D4—21no—
108C2×C3×D4—21no—
109C2×C3×D4—21no—
110C2×C3×D4—21no—
111C2×C3×D4SD1612yes—
112C22×C3×C4—21no—
113C22×C3×C4—41no—
114C22×C3×C4—41no—
115C22×C3×C4—41no—
116C22×C3×C4—42no—
117C22×C3×C4—42no—
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4#4111yescharacteristic
121C2×D12—22no—
122C2×D12—22no—
123C2×D12—22no—
124C2×D12—41no—
125C2×D12—42no—
126C2×D12—41no—
127C2×D12—41no—
128C2×D12—42no—
129C2×D12—41no—
130C2×D12—41no—
131C2×D12—41no—
132C2×D12—41no—
133C2×D12—82no—
134C2×C3×C8—22no—
135C2×C3×C8—41no—
136C2×C3×C8—41no—
137C3×C4⋊C4—22no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—22no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—41no—
142C3×C4⋊C4—41no—
143C3×C22⋊C4—41no—
144C3×C22⋊C4—41no—
145C3×C42C2×D411yescharacteristic
146C3×C42C2×D411yescharacteristic
147C3×C42C2×D411yescharacteristic
148C3×C42C2×D411yescharacteristic
149#213—48no—
150C2×C3⋊C8—22no—
151C2×C3⋊C8—22no—
152C2×C3⋊C8—22no—
153C2×C3⋊C8—22no—
154C2×C3⋊C8—41no—
155C2×C3⋊C8—41no—
156C2×C3⋊C8—41no—
157C2×C3⋊C8—42no—
158C22×Q8—21no—
159C22×D4—61no—
160C22×D4—61no—
161C22×D4—61no—
162C22×D4#6711yescharacteristic
163C23×C4—21no—
164C2×D8—64no—
165C2×D8—122no—
166C2×D8—122no—
167C22×C8—21no—
168C22×C8—32no—
169C22×C8—61no—
170C22×C8—61no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8#6712yes—
175#127—32no—
176#127—61no—
177#127—61no—
178#127—61no—
179C4×D4—21no—
180C4×D4—21no—
181C4×D4—21no—
182C4×D4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4#6711yescharacteristic
186C2×C22⋊C4—21no—
187C2×C42C22×D311yescharacteristic
188#106—24no—
189C4⋊C8—21no—
190C4⋊C8—32no—
191C4⋊C8—61no—
192C4⋊C8—61no—
193C4⋊C8—61no—
194C4⋊C8#6712yes—
195#102—24no—
196#102—64no—
197#102—68no—
198C22⋊C8—124no—
199C4×C8—61no—
200C4×C8—61no—
201C23×C3—22no—
202C23×C3—41no—
203C23×C3—42no—
204C23×C3—41no—
205C23×C3—42no—
206C23×C3#12111yescharacteristic
207C22×D3—42no—
208C22×D3—82no—
209C22×D3—81no—
210C22×D3—81no—
211C22×D3—82no—
212C3×Q8—82no—
213C3×Q8—82no—
214C3×D4—22no—
215C3×D4—22no—
216C3×D4—41no—
217C3×D4—41no—
218C2×C3×C4—22no—
219C2×C3×C4—41no—
220C2×C3×C4—42no—
221C2×C3×C4—42no—
222C2×C3×C4—82no—
223C2×C3×C4—82no—
224C2×C3×C4—81no—
225C2×C3×C4—81no—
226C2×C3×C4#12011yescharacteristic
227C2×C3×C4#12011yescharacteristic
228C2×C3×C4#12011yescharacteristic
229C2×C3×C4#12111yescharacteristic
230C2×C3×C4#12111yescharacteristic
231C2×C3×C4#12711yescharacteristic
232C2×C3×C4C2×SD1611yescharacteristic
233C2×C3×C4#13512yes—
234C2×C3×C4#13511yescharacteristic
235C2×C3×C4AGL(1,ℤ/8)12yes—
236C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
237C2×C3×C4#13711yescharacteristic
238D12—44no—
239D12—42no—
240D12—42no—
241D12—82no—
242D12—82no—
243D12—81no—
244D12—81no—
245D12—81no—
246D12—81no—
247C3×C8—42no—
248C3⋊C8—42no—
249C3⋊C8—42no—
250C3⋊C8—81no—
251C24—21no—
252C24—121no—
253C2×Q8—22no—
254C2×Q8—41no—
255C2×Q8—82no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—62no—
262C2×D4—62no—
263C2×D4—62no—
264C2×D4—122no—
265C2×D4—122no—
266C2×D4—121no—
267C2×D4—121no—
268C2×D4—121no—
269C2×D4—121no—
270C2×D4—121no—
271C2×D4—121no—
272C2×D4—121no—
273C2×D4—242no—
274C2×D4#21512yes—
275C22×C4—21no—
276C22×C4—41no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4D3×D411yescharacteristic
282C22×C4#23711yescharacteristic
283C22×C4C2×#6711yescharacteristic
284D8—128no—
285C2×C8—22no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—62no—
289C2×C8—62no—
290C2×C8—62no—
291C2×C8—62no—
292C2×C8—121no—
293C2×C8—121no—
294C2×C8—122no—
295C2×C8—121no—
296C4⋊C4—22no—
297C4⋊C4—21no—
298C4⋊C4—22no—
299C4⋊C4—21no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C42D3×D411yescharacteristic
305C42D3×D411yescharacteristic
306C42C2×#6711yescharacteristic
307C42C2×#6711yescharacteristic
308C22×C3—44no—
309C22×C3—42no—
310C22×C3—82no—
311C22×C3—81no—
312C22×C3—81no—
313C22×C3#39211yescharacteristic
314C22×C3#46312yes—
315C22×C3#46511yescharacteristic
316C22×C3#46811yescharacteristic
317C22×C3#47212yes—
318C2×D3—82no—
319C2×D3—82no—
320C2×D3—82no—
321C2×D3—162no—
322C2×D3—162no—
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—81no—
326C3×C4—82no—
327C3×C4#44811yescharacteristic
328C3×C4#44912yes—
329C3×C4#44911yescharacteristic
330C3×C4#45011yescharacteristic
331C3×C4#45211yescharacteristic
332C3×C4#49511yescharacteristic
333C3×C4#49611yescharacteristic
334C23—22no—
335C23—42no—
336C23—42no—
337C23—41no—
338C23—41no—
339C23—122no—
340C23—242no—
341C23—242no—
342C23—241no—
343C23—241no—
344C23#93911yescharacteristic
345Q8—82no—
346Q8—82no—
347D4—22no—
348D4—22no—
349D4—41no—
350D4—41no—
351D4—124no—
352D4—122no—
353D4—122no—
354D4—242no—
355D4—242no—
356D4—241no—
357D4—241no—
358D4—241no—
359D4—241no—
360C2×C4—22no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—82no—
365C2×C4—81no—
366C2×C4—82no—
367C2×C4—81no—
368C2×C4#91412yes—
369C2×C4#91612yes—
370C2×C4#93711yescharacteristic
371C2×C4#93711yescharacteristic
372C2×C4#93811yescharacteristic
373C2×C4#93911yescharacteristic
374C2×C4#94011yescharacteristic
375C2×C4C2×#21511yescharacteristic
376C2×C4#94211yescharacteristic
377C2×C4#94911yescharacteristic
378C2×C4#95011yescharacteristic
379C2×C4#95311yescharacteristic
380C8—42no—
381C8—122no—
382C8—122no—
383C8—241no—
384C2×C3—82no—
385C2×C3#167712yes—
386C2×C3#169712yes—
387C2×C3#200011yescharacteristic
388C2×C3#201211yescharacteristic
389C2×C3#201811yescharacteristic
390D3—162no—
391C22—42no—
392C22—44no—
393C22—81no—
394C22—81no—
395C22—82no—
396C22—242no—
397C22—242no—
398C22—242no—
399C22—482no—
400C22—482no—
401C22#504811yescharacteristic
402C22#504911yescharacteristic
403C22#516512yes—
404C22#519912yes—
405C22#522311yescharacteristic
406C4—21no—
407C4—21no—
408C4—81no—
409C4—82no—
410C4#504611yescharacteristic
411C4#504711yescharacteristic
412C4#516811yescharacteristic
413C4#517212yes—
414C4#517611yescharacteristic
415C4#522711yescharacteristic
416C4#523811yescharacteristic
417C3#1026711yescharacteristic, complemented, Sylow
418C2—82no—
419C2—482no—
420C2#7113412yes—
421C2#7113812yes—
422C2#7169911yescharacteristic
423C2#7205511yescharacteristic
424C2#7209511yescharacteristic
subgroupG/H
centerC23#939
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#476C2
socleC23×C3#121
prime radicalC2×#5015C2
hypercenterC2×#476D3
hypocenterC3#10267
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#476

Complements

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

NH ≅ G/Ntype
1. C2×C3×#476C2semidirect
2. C2×#5088C2semidirect
4. C2×#4546C2semidirect
5. #71614C2semidirect
6. C2×#4485C2semidirect
7. #67703C2semidirect
31. C2×#476D3semidirect
417. C3#10267semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4762
C2×#50882
C2×#50152
C2×#45462
#716142
C2×#44852
#677032
#102673