Finite Group Databasebeta

← #10696416

Subgroups — #10696416

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×C3×#475C211yesmaximal, characteristic, complemented
2C2×#5088C211yesmaximal, characteristic
3#82476C211yesmaximal, characteristic
4#71778C211yesmaximal, characteristic, complemented
5C2×#4535C211yesmaximal, characteristic, complemented
6C2×#4485C211yesmaximal, characteristic
7#67704C211yesmaximal, characteristic, complemented
8#10071—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×C4×Q8C2211yescharacteristic
11C2×#947—21no—
12C2×#874C2211yescharacteristic
13C3×#475—22no—
14C3×#475—22no—
15C2×C3×#106—21no—
16C2×C3×C4⋊C8C2211yescharacteristic
17C2×C3×#103—21no—
18C2×#835—22no—
19#5088—22no—
20#5088—22no—
21#4991—22no—
22#4988—21no—
23C2×#809—22no—
24#4939—22no—
25C4×C4⋊Dic3C2211yescharacteristic
26C2×C4⋊(C3⋊C8)C2211yescharacteristic
27C2×C4×C3⋊C8C2211yescharacteristic
28#4535—22no—
29#4485—22no—
30C2×#486—31no—
31C2×#475D311yescharacteristic, complemented
32#2548—31no—
33#1981—31no—
34C2×#331—31no—
35C2×Q8⋊C8—31no—
36#1279—31no—
37C22×C3×Q8—21no—
38C22×C3×Q8D411yescharacteristic
39C22×D12—21no—
40C22×D12—21no—
41C22×D12D412yes—
42C22×C3×C8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8D412yes—
47C3×C4×Q8—21no—
48C3×C4×Q8—21no—
49C3×C4×Q8—21no—
50C3×C4×Q8—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4D411yescharacteristic
55C2×C3×C42—21no—
56C2×C3×C42C2311yescharacteristic
57#947—42no—
58#947—42no—
59C2×(C2×D3)⋊C4—24no—
60C2×(C2×D3)⋊C4—24no—
61C2×C4⋊Dic3D412yes—
62C2×C4×Dic3—21no—
63C22×C3⋊C8—21no—
64C22×C3⋊C8—21no—
65#874—21no—
66#874—21no—
67#874—21no—
68#874—21no—
69C3×#106—24no—
70C3×C4⋊C8—21no—
71C3×C4⋊C8D412yes—
72C3×#103—24no—
73#835—44no—
74C3⋊#95—21no—
75#809—28no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)—21no—
78C4×C3⋊C8—21no—
79C4×C3⋊C8D412yes—
80C2×C4⋊Q8C2×D311yescharacteristic
81C2×#127—31no—
82C2×#124—61no—
83C2×C4×Q8C2×D311yescharacteristic
84#486—62no—
85#486—62no—
86#475—22no—
87#475—22no—
88C2×#106—21no—
89C2×C4⋊C8—31no—
90C2×C4⋊C8C2×D311yescharacteristic
91C2×#103—21no—
92C2×#103—62no—
93C2×#102—62no—
94C2×C4×C8—31no—
95#394—62no—
96#390—62no—
97#390—61no—
98C4×C4⋊C4—31no—
99#331—62no—
100Q8⋊C8—62no—
101C23×D3—22no—
102C2×C3×Q8—21no—
103C2×C3×Q8—21no—
104C2×C3×Q8—21no—
105C2×C3×Q8—22no—
106C2×C3×Q8—21no—
107C2×C3×Q8—21no—
108C2×C3×Q8—41no—
109C2×C3×Q8—42no—
110C2×C3×Q8—42no—
111C2×C3×Q8SD1612yes—
112C22×C3×C4—21no—
113C22×C3×C4—21no—
114C22×C3×C4—41no—
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×Dic3—22no—
119C2×D12—22no—
120C2×D12—22no—
121C2×D12—44no—
122C2×D12—41no—
123C2×D12—41no—
124C2×D12—41no—
125C2×D12—41no—
126C2×D12—41no—
127C2×D12—42no—
128C2×D12—41no—
129C2×D12—41no—
130C2×D12—42no—
131C2×D12—41no—
132C2×C3×C8—22no—
133C2×C3×C8—41no—
134C2×C3×C8—41no—
135C3×C4⋊C4—22no—
136C3×C4⋊C4—22no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—41no—
140C3×C4⋊C4—41no—
141C3×C4⋊C4—41no—
142C3×C4⋊C4—41no—
143C3×C42—41no—
144C3×C42—41no—
145C3×C42C2×D411yescharacteristic
146C3×C42C2×D411yescharacteristic
147C3×C42C2×D411yescharacteristic
148C3×C42#4111yescharacteristic
149(C2×D3)⋊C4—48no—
150(C2×D3)⋊C4—84no—
151C4⋊Dic3—22no—
152C4⋊Dic3—22no—
153C4×Dic3—41no—
154C4×Dic3—41no—
155C2×C3⋊C8—22no—
156C2×C3⋊C8—42no—
157C2×C3⋊C8—41no—
158C2×C3⋊C8—42no—
159C22×Q8—21no—
160C22×Q8#6711yescharacteristic
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⋊Q8#6712yes—
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—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—32no—
185C2×C4⋊C4#6711yescharacteristic
186C2×C22⋊C4—64no—
187C2×C22⋊C4—64no—
188C2×C42—21no—
189C2×C42—61no—
190C2×C42C22×D311yescharacteristic
191#106—24no—
192C4⋊C8—21no—
193C4⋊C8—61no—
194C4⋊C8—61no—
195C4⋊C8#6712yes—
196#103—24no—
197#103—124no—
198#102—68no—
199C4×C8—32no—
200C4×C8—61no—
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—
212C3×Q8—44no—
213C3×Q8—81no—
214C3×Q8—81no—
215C2×C3×C4—22no—
216C2×C3×C4—22no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4—42no—
221C2×C3×C4—42no—
222C2×C3×C4—82no—
223C2×C3×C4#12011yescharacteristic
224C2×C3×C4#12011yescharacteristic
225C2×C3×C4#12111yescharacteristic
226C2×C3×C4#12111yescharacteristic
227C2×C3×C4#12111yescharacteristic
228C2×C3×C4#12711yescharacteristic
229C2×C3×C4C2×SD1611yescharacteristic
230C2×C3×C4#13512yes—
231C2×C3×C4#13511yescharacteristic
232C2×C3×C4AGL(1,ℤ/8)12yes—
233C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
234C2×C3×C4#13711yescharacteristic
235C2×Dic3—42no—
236C2×Dic3—42no—
237C2×Dic3—42no—
238D12—44no—
239D12—82no—
240D12—81no—
241D12—81no—
242D12—82no—
243D12—81no—
244D12—81no—
245D12—81no—
246D12—81no—
247C3×C8—42no—
248C3⋊C8—42no—
249C3⋊C8—81no—
250C24—62no—
251C2×Q8—21no—
252C2×Q8—21no—
253C2×Q8—22no—
254C2×Q8—21no—
255C2×Q8—21no—
256C2×Q8—21no—
257C2×Q8—41no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8#21512yes—
261C2×D4—62no—
262C2×D4—62no—
263C2×D4—124no—
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—121no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—41no—
277C22×C4—62no—
278C22×C4D3×D411yescharacteristic
279C22×C4D3×D411yescharacteristic
280C22×C4C2×#6711yescharacteristic
281C2×C8—22no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—62no—
285C2×C8—122no—
286C2×C8—122no—
287C2×C8—121no—
288C4⋊C4—22no—
289C4⋊C4—21no—
290C4⋊C4—22no—
291C4⋊C4—21no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—62no—
297C4⋊C4—62no—
298C22⋊C4—128no—
299C22⋊C4—244no—
300C42—41no—
301C42—41no—
302C42—121no—
303C42—121no—
304C42D3×D411yescharacteristic
305C42#23711yescharacteristic
306C42C2×#6711yescharacteristic
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—82no—
323C3×C4#44811yescharacteristic
324C3×C4#44911yescharacteristic
325C3×C4#46312yes—
326C3×C4#46511yescharacteristic
327C3×C4#46811yescharacteristic
328C3×C4#49511yescharacteristic
329C3×C4#49611yescharacteristic
330Dic3—82no—
331C23—124no—
332C23—124no—
333C23—122no—
334C23—122no—
335C23—122no—
336C23#93711yescharacteristic
337Q8—22no—
338Q8—22no—
339Q8—41no—
340Q8—41no—
341Q8—44no—
342Q8—81no—
343Q8—81no—
344D4—124no—
345D4—242no—
346D4—242no—
347D4—241no—
348D4—241no—
349D4—241no—
350D4—241no—
351D4—241no—
352D4—241no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—82no—
361C2×C4—122no—
362C2×C4—122no—
363C2×C4—122no—
364C2×C4#91412yes—
365C2×C4#91612yes—
366C2×C4#93711yescharacteristic
367C2×C4#93811yescharacteristic
368C2×C4#93911yescharacteristic
369C2×C4#93911yescharacteristic
370C2×C4#94011yescharacteristic
371C2×C4C2×#21511yescharacteristic
372C2×C4#94211yescharacteristic
373C2×C4#94911yescharacteristic
374C2×C4#95011yescharacteristic
375C2×C4#95311yescharacteristic
376C8—42no—
377C8—122no—
378C8—241no—
379C2×C3#164012yes—
380C2×C3#164212yes—
381C2×C3#200011yescharacteristic
382C2×C3#200411yescharacteristic
383C2×C3#201211yescharacteristic
384D3—162no—
385C22—242no—
386C22—242no—
387C22—242no—
388C22—244no—
389C22—244no—
390C22#504611yescharacteristic
391C22#504711yescharacteristic
392C22#517212yes—
393C22#520212yes—
394C22#522311yescharacteristic
395C4—21no—
396C4—21no—
397C4—42no—
398C4—82no—
399C4—81no—
400C4—242no—
401C4#504811yescharacteristic
402C4#504911yescharacteristic
403C4#516512yes—
404C4#516811yescharacteristic
405C4#517611yescharacteristic
406C4#522711yescharacteristic
407C4#523811yescharacteristic
408C3#1007111yescharacteristic, complemented, Sylow
409C2—482no—
410C2#7112412yes—
411C2#7112812yes—
412C2#7169911yescharacteristic
413C2#7205511yescharacteristic
414C2#7210611yescharacteristic
subgroupG/H
centerC23#937
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#475C2
socleC23×C3#120
prime radicalC2×#874C22
hypercenterC2×#475D3
hypocenterC3#10071
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#475

Complements

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

NH ≅ G/Ntype
1. C2×C3×#475C2semidirect
4. #71778C2semidirect
5. C2×#4535C2semidirect
7. #67704C2semidirect
31. C2×#475D3semidirect
408. C3#10071semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4752
C2×#50882
#824762
#717782
C2×#45352
C2×#44852
#677042
#100713