Finite Group Databasebeta

← #10697707

Subgroups — #10697707

Order 768.

626 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#462C211yesmaximal, characteristic, complemented
2C2×#5088C211yesmaximal, characteristic
3C2×#5027C211yesmaximal, characteristic
4C2×#4547C211yesmaximal, characteristic, complemented
5#71634C211yesmaximal, characteristic, complemented
6C2×#4485C211yesmaximal, characteristic
7#67703C211yesmaximal, characteristic, complemented
8#10263—31nomaximal, Sylow
9C22×C3×Q16—21no—
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×C3×C4×Q8C2211yescharacteristic
12C22×#215—21no—
13C2×#874C2211yescharacteristic
14C3×#462—22no—
15C3×#462—22no—
16C2×C3×C4⋊C8C2211yescharacteristic
17C2×C3×#103—21no—
18#5088—22no—
19#5088—22no—
20#5027—22no—
21#5027—22no—
22C2×#809—21no—
23C2×#809—22no—
24C2×C4⋊(C3⋊C8)C2211yescharacteristic
25C2×C4⋊(C3⋊C8)C2211yescharacteristic
26C2×C4×C3⋊C8C2211yescharacteristic
27#4547—22no—
28#4524—22no—
29#4485—22no—
30C2×#486—31no—
31C2×#462D311yescharacteristic, 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—
42C2×C3×Q16—24no—
43C2×C3×Q16—42no—
44C2×C3×Q16—42no—
45C22×C3×C8—21no—
46C3×C4⋊Q8—21no—
47C3×C4⋊Q8—21no—
48C3×C4⋊Q8—21no—
49C3×C4⋊Q8D412yes—
50C3×C4×Q8—21no—
51C3×C4×Q8—21no—
52C3×C4×Q8—21no—
53C3×C4×Q8—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4—21no—
56C2×C3×C4⋊C4—21no—
57C2×C3×C4⋊C4D411yescharacteristic
58C2×C3×C42—21no—
59C2×C3×C42C2311yescharacteristic
60C2×#215—24no—
61C2×#215—42no—
62C2×#215—42no—
63C22×C3⋊C8—21no—
64C22×C3⋊C8—21no—
65C22×C3⋊C8D412yes—
66#874—21no—
67#874—21no—
68#874—21no—
69#874D412yes—
70C3×C4⋊C8—21no—
71C3×C4⋊C8D412yes—
72C3×#103—24no—
73#809—24no—
74#809—28no—
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×Q16—21no—
82C22×SD16—61no—
83C2×C4⋊Q8C2×D311yescharacteristic
84C2×#127—31no—
85C2×C4×Q8C2×D311yescharacteristic
86#486—62no—
87#486—62no—
88#462—22no—
89#462—22no—
90#460—62no—
91#460—62no—
92C2×C4⋊C8—31no—
93C2×C4⋊C8—31no—
94C2×C4⋊C8C2×D311yescharacteristic
95C2×#103—21no—
96C2×#102—62no—
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—41no—
111C2×C3×Q8—41no—
112C2×C3×Q8—41no—
113C2×C3×Q8—41no—
114C2×C3×Q8D812yes—
115C22×C3×C4—21no—
116C22×C3×C4—21no—
117C22×C3×C4—41no—
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—41no—
129C2×D12—42no—
130C2×D12—41no—
131C2×D12—41no—
132C2×D12—41no—
133C2×D12—82no—
134C3×Q16—48no—
135C2×C3×C8—22no—
136C2×C3×C8—41no—
137C2×C3×C8—41no—
138C3×C4⋊C4—22no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—22no—
142C3×C4⋊C4—41no—
143C3×C4⋊C4—41no—
144C3×C4⋊C4—41no—
145C3×C4⋊C4—41no—
146C3×C42—41no—
147C3×C42—41no—
148C3×C42C2×D411yescharacteristic
149C3×C42C2×D411yescharacteristic
150C3×C42C2×D411yescharacteristic
151C3×C42C2×D411yescharacteristic
152#215—48no—
153C2×C3⋊C8—22no—
154C2×C3⋊C8—22no—
155C2×C3⋊C8—22no—
156C2×C3⋊C8—22no—
157C2×C3⋊C8—41no—
158C2×C3⋊C8—41no—
159C2×C3⋊C8—41no—
160C2×C3⋊C8—42no—
161C22×Q8—21no—
162C22×Q8#6711yescharacteristic
163C22×D4—61no—
164C22×D4—61no—
165C22×D4—61no—
166C2×Q16—24no—
167C2×Q16—42no—
168C2×Q16—42no—
169C2×SD16—64no—
170C2×SD16—122no—
171C2×SD16—122no—
172C22×C8—21no—
173C22×C8—32no—
174C22×C8—61no—
175C22×C8—61no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8#6712yes—
180#127—32no—
181#127—61no—
182#127—61no—
183#127—61no—
184C4×Q8—21no—
185C4×Q8—21no—
186C4×Q8—21no—
187C4×Q8—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4#6711yescharacteristic
192C2×C42—21no—
193C2×C42C22×D311yescharacteristic
194C4⋊C8—21no—
195C4⋊C8—32no—
196C4⋊C8—61no—
197C4⋊C8—61no—
198C4⋊C8—61no—
199C4⋊C8#6712yes—
200#103—24no—
201#102—64no—
202#102—68no—
203C4×C8—61no—
204C4×C8—61no—
205C23×C3#12111yescharacteristic
206C22×D3—42no—
207C22×D3—82no—
208C22×D3—81no—
209C22×D3—81no—
210C22×D3—82no—
211C3×Q8—22no—
212C3×Q8—22no—
213C3×Q8—41no—
214C3×Q8—41no—
215C3×Q8—42no—
216C3×Q8—42no—
217C3×Q8—81no—
218C3×Q8—81no—
219C2×C3×C4—22no—
220C2×C3×C4—22no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—42no—
225C2×C3×C4—42no—
226C2×C3×C4—81no—
227C2×C3×C4—81no—
228C2×C3×C4#12011yescharacteristic
229C2×C3×C4#12011yescharacteristic
230C2×C3×C4#12011yescharacteristic
231C2×C3×C4#12111yescharacteristic
232C2×C3×C4#12111yescharacteristic
233C2×C3×C4#12711yescharacteristic
234C2×C3×C4C2×D811yescharacteristic
235C2×C3×C4#13512yes—
236C2×C3×C4#13511yescharacteristic
237C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
238C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
239C2×C3×C4#13712yes—
240D12—42no—
241D12—42no—
242D12—44no—
243D12—82no—
244D12—82no—
245D12—81no—
246D12—81no—
247D12—81no—
248D12—81no—
249C3×C8—42no—
250C3⋊C8—42no—
251C3⋊C8—42no—
252C3⋊C8—81no—
253C24—121no—
254C2×Q8—21no—
255C2×Q8—21no—
256C2×Q8—21no—
257C2×Q8—21no—
258C2×Q8—21no—
259C2×Q8—22no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8#21312yes—
266C2×D4—62no—
267C2×D4—62no—
268C2×D4—62no—
269C2×D4—122no—
270C2×D4—122no—
271C2×D4—121no—
272C2×D4—121no—
273C2×D4—121no—
274C2×D4—121no—
275C2×D4—121no—
276C2×D4—121no—
277C2×D4—121no—
278C2×D4—242no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—41no—
282C22×C4D3×D411yescharacteristic
283C22×C4#23711yescharacteristic
284C22×C4C2×#6711yescharacteristic
285Q16—48no—
286SD16—128no—
287C2×C8—22no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—62no—
291C2×C8—62no—
292C2×C8—62no—
293C2×C8—62no—
294C2×C8—121no—
295C2×C8—121no—
296C2×C8—122no—
297C2×C8—121no—
298C4⋊C4—22no—
299C4⋊C4—21no—
300C4⋊C4—22no—
301C4⋊C4—21no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C42—41no—
307C42—41no—
308C42D3×D411yescharacteristic
309C42D3×D411yescharacteristic
310C42C2×#6711yescharacteristic
311C42C2×#6711yescharacteristic
312C22×C3#39211yescharacteristic
313C22×C3#46412yes—
314C22×C3#46611yescharacteristic
315C22×C3#46911yescharacteristic
316C22×C3#47112yes—
317C2×D3—82no—
318C2×D3—82no—
319C2×D3—82no—
320C2×D3—162no—
321C2×D3—162no—
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—42no—
325C3×C4—82no—
326C3×C4—81no—
327C3×C4#44711yescharacteristic
328C3×C4#44711yescharacteristic
329C3×C4#44911yescharacteristic
330C3×C4#44911yescharacteristic
331C3×C4#45212yes—
332C3×C4#49511yescharacteristic
333C3×C4#49711yescharacteristic
334C23—122no—
335C23—242no—
336C23—242no—
337C23—241no—
338C23—241no—
339C23#93911yescharacteristic
340Q8—22no—
341Q8—22no—
342Q8—41no—
343Q8—41no—
344Q8—42no—
345Q8—42no—
346Q8—81no—
347Q8—81no—
348D4—124no—
349D4—122no—
350D4—122no—
351D4—242no—
352D4—242no—
353D4—241no—
354D4—241no—
355D4—241no—
356D4—241no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4#91812yes—
367C2×C4#91912yes—
368C2×C4C2×#21311yescharacteristic
369C2×C4#93211yescharacteristic
370C2×C4#93711yescharacteristic
371C2×C4#93711yescharacteristic
372C2×C4#93811yescharacteristic
373C2×C4#93911yescharacteristic
374C2×C4#94011yescharacteristic
375C2×C4#94911yescharacteristic
376C2×C4#95011yescharacteristic
377C2×C4#95311yescharacteristic
378C8—42no—
379C8—122no—
380C8—122no—
381C8—241no—
382C2×C3#168212yes—
383C2×C3#170212yes—
384C2×C3#199911yescharacteristic
385C2×C3#201511yescharacteristic
386C2×C3#201911yescharacteristic
387D3—162no—
388C22—242no—
389C22—242no—
390C22—242no—
391C22—482no—
392C22—482no—
393C22#503811yescharacteristic
394C22#503911yescharacteristic
395C22#518512yes—
396C22#520312yes—
397C22#522311yescharacteristic
398C4—21no—
399C4—21no—
400C4—42no—
401C4—82no—
402C4—81no—
403C4#503611yescharacteristic
404C4#503711yescharacteristic
405C4#518712yes—
406C4#519011yescharacteristic
407C4#519111yescharacteristic
408C4#521711yescharacteristic
409C4#523811yescharacteristic
410C3#1026311yescharacteristic, complemented, Sylow
411C2—482no—
412C2#7105512yes—
413C2#7105912yes—
414C2#7168411yescharacteristic
415C2#7207211yescharacteristic
416C2#7210711yescharacteristic
subgroupG/H
centerC23#939
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#462C2
socleC23×C3#121
prime radicalC2×#874C22
hypercenterC2×#462D3
hypocenterC3#10263
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#462

Complements

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

NH ≅ G/Ntype
1. C2×C3×#462C2semidirect
4. C2×#4547C2semidirect
5. #71634C2semidirect
7. #67703C2semidirect
31. C2×#462D3semidirect
410. C3#10263semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4622
C2×#50882
C2×#50272
C2×#45472
#716342
C2×#44852
#677032
#102633