Finite Group Databasebeta

← #10697738

Subgroups — #10697738

Order 768.

658 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#475C211yesmaximal, characteristic, complemented
2C2×#5085C211yesmaximal, characteristic
3C2×#5027C211yesmaximal, characteristic
4C2×#4547C211yesmaximal, characteristic, complemented
5#71634C211yesmaximal, characteristic, complemented
6C2×#4485C211yesmaximal, characteristic
7#67703C211yesmaximal, characteristic, complemented
8#10268—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×C4×Q8C2211yescharacteristic
11C22×#215—21no—
12C22×#215—22no—
13C2×#874C2211yescharacteristic
14C3×#475—22no—
15C3×#475—22no—
16C2×C3×#106—21no—
17C2×C3×C4⋊C8C2211yescharacteristic
18C2×C3×#103—21no—
19#5085—22no—
20#5085—22no—
21#5027—22no—
22#5027—22no—
23C2×#809—21no—
24C2×C4⋊(C3⋊C8)C2211yescharacteristic
25C2×C4⋊(C3⋊C8)C2211yescharacteristic
26C2×C4×C3⋊C8C2211yescharacteristic
27#4547—22no—
28#4524—22no—
29#4485—22no—
30C2×#492—31no—
31C2×#475D311yescharacteristic, complemented
32C2×#460—31no—
33#1984—31no—
34C2×#332—31no—
35C2×#331—31no—
36#1277—31no—
37C22×C3×Q8—21no—
38C22×C3×Q8D411yescharacteristic
39C22×D12—21no—
40C22×D12—21no—
41C22×D12—21no—
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
57C2×#215—24no—
58C2×#215—28no—
59C2×#215—44no—
60C2×#215—42no—
61C2×#215—42no—
62C2×#215—44no—
63C22×C3⋊C8—21no—
64C22×C3⋊C8—21no—
65C22×C3⋊C8D412yes—
66#874—21no—
67#874—21no—
68#874—21no—
69#874D412yes—
70C3×#106—24no—
71C3×C4⋊C8—21no—
72C3×C4⋊C8D412yes—
73C3×#103—24no—
74#809—24no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)D412yes—
79C4×C3⋊C8—21no—
80C4×C3⋊C8—21no—
81C22×SD16—62no—
82C22×SD16—61no—
83C2×C4⋊Q8C2×D311yescharacteristic
84C2×#127—31no—
85C2×C4×Q8C2×D311yescharacteristic
86#492—62no—
87#492—62no—
88#475—22no—
89#475—22no—
90#460—62no—
91#460—62no—
92C2×#106—21no—
93C2×C4⋊C8—31no—
94C2×C4⋊C8—31no—
95C2×C4⋊C8C2×D311yescharacteristic
96C2×#103—21no—
97C2×#102—61no—
98C2×C4×C8—31no—
99#336—62no—
100#332—62no—
101#331—62no—
102C23×D3—41no—
103C2×C3×Q8—21no—
104C2×C3×Q8—21no—
105C2×C3×Q8—21no—
106C2×C3×Q8—22no—
107C2×C3×Q8—21no—
108C2×C3×Q8—21no—
109C2×C3×Q8—41no—
110C2×C3×Q8—42no—
111C2×C3×Q8—42no—
112C2×C3×Q8SD1612yescomplemented
113C22×C3×C4—21no—
114C22×C3×C4—21no—
115C22×C3×C4—41no—
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4#4111yescharacteristic
119C2×D12—22no—
120C2×D12—22no—
121C2×D12—22no—
122C2×D12—42no—
123C2×D12—41no—
124C2×D12—41no—
125C2×D12—42no—
126C2×D12—41no—
127C2×D12—41no—
128C2×D12—41no—
129C2×D12—41no—
130C2×D12—41no—
131C2×D12—82no—
132C2×C3×C8—22no—
133C2×C3×C8—41no—
134C2×C3×C8—41no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—22no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—22no—
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×C42C2×D411yescharacteristic
149#215—416no—
150#215—48no—
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—
158C2×C3⋊C8SD1614yescomplemented
159C22×Q8—21no—
160C22×Q8#6711yescharacteristic
161C22×D4—61no—
162C22×D4—61no—
163C22×D4—61no—
164C2×SD16—64no—
165C2×SD16—68no—
166C2×SD16—124no—
167C2×SD16—124no—
168C2×SD16—122no—
169C2×SD16—122no—
170C22×C8—21no—
171C22×C8—32no—
172C22×C8—61no—
173C22×C8—61no—
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8#6712yes—
178#127—32no—
179#127—61no—
180#127—61no—
181#127—61no—
182C4×Q8—21no—
183C4×Q8—21no—
184C4×Q8—21no—
185C4×Q8—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4#6711yescharacteristic
190C2×C42—21no—
191C2×C42C22×D311yescharacteristic
192#106—24no—
193C4⋊C8—21no—
194C4⋊C8—32no—
195C4⋊C8—61no—
196C4⋊C8—61no—
197C4⋊C8—61no—
198C4⋊C8#6712yes—
199#103—24no—
200#102—64no—
201C4×C8—61no—
202C4×C8—61no—
203C23×C3#12111yescharacteristic
204C22×D3—42no—
205C22×D3—82no—
206C22×D3—81no—
207C22×D3—81no—
208C22×D3—82no—
209C3×Q8—22no—
210C3×Q8—22no—
211C3×Q8—41no—
212C3×Q8—41no—
213C3×Q8—44no—
214C3×Q8—81no—
215C3×Q8—81no—
216C2×C3×C4—22no—
217C2×C3×C4—22no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—42no—
222C2×C3×C4—42no—
223C2×C3×C4—82no—
224C2×C3×C4#12011yescharacteristic
225C2×C3×C4#12011yescharacteristic
226C2×C3×C4#12011yescharacteristic
227C2×C3×C4#12111yescharacteristic
228C2×C3×C4#12111yescharacteristic
229C2×C3×C4#12711yescharacteristic
230C2×C3×C4C2×SD1612yes—
231C2×C3×C4C2×SD1611yescharacteristic
232C2×C3×C4#13511yescharacteristic
233C2×C3×C4AGL(1,ℤ/8)12yes—
234C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
235C2×C3×C4#13711yescharacteristic
236D12—44no—
237D12—42no—
238D12—42no—
239D12—82no—
240D12—82no—
241D12—81no—
242D12—81no—
243D12—81no—
244D12—81no—
245C3×C8—42no—
246C3⋊C8—24no—
247C3⋊C8—42no—
248C3⋊C8—81no—
249C24—121no—
250C2×Q8—21no—
251C2×Q8—21no—
252C2×Q8—22no—
253C2×Q8—21no—
254C2×Q8—21no—
255C2×Q8—21no—
256C2×Q8—41no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8#21512yescomplemented
260C2×D4—62no—
261C2×D4—62no—
262C2×D4—62no—
263C2×D4—122no—
264C2×D4—122no—
265C2×D4—121no—
266C2×D4—121no—
267C2×D4—121no—
268C2×D4—121no—
269C2×D4—121no—
270C2×D4—121no—
271C2×D4—121no—
272C2×D4—242no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—41no—
276C22×C4D3×D411yescharacteristic
277C22×C4#23711yescharacteristic
278C22×C4C2×#6711yescharacteristic
279SD16—1216no—
280SD16—128no—
281C2×C8—22no—
282C2×C8—34no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—62no—
286C2×C8—62no—
287C2×C8—62no—
288C2×C8—121no—
289C2×C8—121no—
290C2×C8—122no—
291C2×C8—121no—
292C4⋊C4—21no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—21no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C42—41no—
301C42—41no—
302C42D3×D411yescharacteristic
303C42D3×D411yescharacteristic
304C42C2×#6711yescharacteristic
305C42C2×#6711yescharacteristic
306C22×C3#39211yescharacteristic
307C22×C3#46012yes—
308C22×C3#46511yescharacteristic
309C22×C3#46811yescharacteristic
310C22×C3#47112yes—
311C2×D3—82no—
312C2×D3—82no—
313C2×D3—82no—
314C2×D3—162no—
315C2×D3—162no—
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—42no—
319C3×C4—82no—
320C3×C4—81no—
321C3×C4#44811yescharacteristic
322C3×C4#44911yescharacteristic
323C3×C4#45012yes—
324C3×C4#45011yescharacteristic
325C3×C4#45211yescharacteristic
326C3×C4#49211yescharacteristic
327C3×C4#49611yescharacteristic
328C23—122no—
329C23—242no—
330C23—242no—
331C23—241no—
332C23—241no—
333C23#93911yescharacteristic
334Q8—22no—
335Q8—22no—
336Q8—41no—
337Q8—41no—
338Q8—44no—
339Q8—81no—
340Q8—81no—
341D4—124no—
342D4—122no—
343D4—122no—
344D4—242no—
345D4—242no—
346D4—241no—
347D4—241no—
348D4—241no—
349D4—241no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—82no—
358C2×C4D3×SD1612yes—
359C2×C4#91412yes—
360C2×C4#93711yescharacteristic
361C2×C4#93711yescharacteristic
362C2×C4#93811yescharacteristic
363C2×C4#93911yescharacteristic
364C2×C4#94011yescharacteristic
365C2×C4C2×#21511yescharacteristic
366C2×C4#94211yescharacteristic
367C2×C4#94911yescharacteristic
368C2×C4#95011yescharacteristic
369C2×C4#95311yescharacteristic
370C8—42no—
371C8—64no—
372C8—122no—
373C8—241no—
374C2×C3#166612yes—
375C2×C3#168612yes—
376C2×C3#200011yescharacteristic
377C2×C3#201411yescharacteristic
378C2×C3#201911yescharacteristic
379D3—162no—
380C22—242no—
381C22—242no—
382C22—242no—
383C22—482no—
384C22—482no—
385C22#504811yescharacteristic
386C22#504911yescharacteristic
387C22#516312yes—
388C22#520312yes—
389C22#522311yescharacteristic
390C4—21no—
391C4—21no—
392C4—42no—
393C4—82no—
394C4—81no—
395C4#504611yescharacteristic
396C4#504711yescharacteristic
397C4#516912yes—
398C4#517511yescharacteristic
399C4#517611yescharacteristic
400C4#522711yescharacteristic
401C4#523811yescharacteristic
402C3#1026811yescharacteristic, complemented, Sylow
403C2—482no—
404C2#7111612yes—
405C2#7112012yes—
406C2#7169911yescharacteristic
407C2#7205611yescharacteristic
408C2#7210711yescharacteristic
subgroupG/H
centerC23#939
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#475C2
socleC23×C3#121
prime radicalC2×#874C22
hypercenterC2×#475D3
hypocenterC3#10268
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. C2×#4547C2semidirect
5. #71634C2semidirect
7. #67703C2semidirect
31. C2×#475D3semidirect
112. C2×C3×Q8SD16semidirect
158. C2×C3⋊C8SD16semidirect
259. C2×Q8#215semidirect
402. C3#10268semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4752
C2×#50852
C2×#50272
C2×#45472
#716342
C2×#44852
#677032
#102683