Finite Group Databasebeta

← #10697709

Subgroups — #10697709

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×#5090C211yesmaximal, characteristic, complemented
2C2×#5088C211yesmaximal, characteristic
3C2×#4848C211yesmaximal, characteristic
4C2×C3×#332C211yesmaximal, characteristic, complemented
5#70472C211yesmaximal, characteristic, complemented
6C2×#4485C211yesmaximal, characteristic
7#67703C211yesmaximal, characteristic, complemented
8#10263—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C22×#216—21no—
11C22×#204—21no—
12C2×#874C2211yescharacteristic
13C2×C4×Dic6C2211yescharacteristic
14C2×C3×C4⋊C8C2211yescharacteristic
15C2×C3×C4⋊C8C2211yescharacteristic
16C2×#835—21no—
17C2×#821—21no—
18#5090—22no—
19#5090—22no—
20#5088—22no—
21#5088—22no—
22C2×#809—22no—
23C2×C4⋊(C3⋊C8)C2211yescharacteristic
24C2×C4×C3⋊C8C2211yescharacteristic
25#4848—22no—
26#4848—22no—
27C3×#332—22no—
28#4550—22no—
29#4485—22no—
30C2×#486—31no—
31C2×#462—31no—
32C2×#460—31no—
33#1984—31no—
34C2×#332D311yescharacteristic, complemented
35C2×#331—31no—
36#1277—31no—
37C22×C3×Q8—21no—
38C22×D12—21no—
39C22×D12—21no—
40C22×D12—21no—
41C22×Dic6D411yescharacteristic
42C22×C3×C8—21no—
43C22×C3×C8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—21no—
47C3×C4⋊Q8D412yes—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C42C2311yescharacteristic
51C2×#216—24no—
52C2×#216—42no—
53C2×#216—42no—
54C2×C4⋊Dic3D411yescharacteristic
55C2×Dic3⋊C4—21no—
56C2×C4×Dic3—21no—
57C22×C3⋊C8—21no—
58C22×C3⋊C8D412yes—
59C2×#204—24no—
60C2×#204—42no—
61C2×#204—42no—
62#874—21no—
63#874—21no—
64#874—21no—
65#874D412yes—
66C4×Dic6—21no—
67C4×Dic6—21no—
68C4×Dic6—21no—
69C4×Dic6—21no—
70C3×C4⋊C8—21no—
71C3×C4⋊C8—21no—
72C3×C4⋊C8—21no—
73C3×C4⋊C8D412yes—
74#835—24no—
75#821—24no—
76#809—28no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)D412yes—
79C4×C3⋊C8—21no—
80C4×C3⋊C8—21no—
81C22×Q16—61no—
82C22×SD16—61no—
83C2×C4⋊Q8C2×D311yescharacteristic
84C2×#127—31no—
85C2×C4×Q8—31no—
86#486—62no—
87#486—62no—
88#462—62no—
89#462—62no—
90#460—62no—
91#460—62no—
92C2×C4⋊C8—31no—
93C2×C4⋊C8C2×D311yescharacteristic
94C2×C4⋊C8C2×D311yescharacteristic
95C2×#103—61no—
96C2×#102—62no—
97C2×#102—61no—
98C2×C4×C8—31no—
99#336—62no—
100#332—22no—
101#331—62no—
102C23×D3—41no—
103C2×C3×Q8—22no—
104C2×C3×Q8—41no—
105C2×C3×Q8—41no—
106C2×C3×Q8—41no—
107C2×C3×Q8—41no—
108C2×C3×Q8—41no—
109C22×C3×C4—41no—
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4#4111yescharacteristic
113C22×Dic3—21no—
114C22×Dic3—21no—
115C2×D12—22no—
116C2×D12—22no—
117C2×D12—22no—
118C2×D12—41no—
119C2×D12—41no—
120C2×D12—42no—
121C2×D12—41no—
122C2×D12—41no—
123C2×D12—42no—
124C2×D12—41no—
125C2×D12—41no—
126C2×D12—41no—
127C2×D12—82no—
128C2×Dic6—21no—
129C2×Dic6—21no—
130C2×Dic6—21no—
131C2×Dic6—21no—
132C2×Dic6—21no—
133C2×Dic6D812yes—
134C2×C3×C8—22no—
135C2×C3×C8—41no—
136C2×C3×C8—41no—
137C2×C3×C8—41no—
138C2×C3×C8—42no—
139C3×C4⋊C4—22no—
140C3×C4⋊C4—22no—
141C3×C4⋊C4—41no—
142C3×C4⋊C4—41no—
143C3×C42C2×D411yescharacteristic
144C3×C42C2×D411yescharacteristic
145C3×C42C2×D411yescharacteristic
146C3×C42C2×D411yescharacteristic
147#216—48no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150Dic3⋊C4—41no—
151Dic3⋊C4—41no—
152C4×Dic3—41no—
153C4×Dic3—41no—
154C2×C3⋊C8—22no—
155C2×C3⋊C8—22no—
156C2×C3⋊C8—22no—
157C2×C3⋊C8—22no—
158C2×C3⋊C8—41no—
159C2×C3⋊C8—41no—
160#204—48no—
161C22×Q8—21no—
162C22×Q8—31no—
163C22×D4—61no—
164C22×D4—61no—
165C22×D4—61no—
166C2×Q16—64no—
167C2×Q16—122no—
168C2×Q16—122no—
169C2×SD16—64no—
170C2×SD16—122no—
171C2×SD16—122no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—32no—
175C22×C8—61no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8D1212yes—
180#127—32no—
181#127—61no—
182#127—61no—
183#127—61no—
184C4×Q8—61no—
185C4×Q8—61no—
186C4×Q8—61no—
187C4×Q8—61no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—31no—
191C2×C4⋊C4—61no—
192C2×C42—61no—
193C2×C42C22×D311yescharacteristic
194C4⋊C8—21no—
195C4⋊C8—21no—
196C4⋊C8—21no—
197C4⋊C8—32no—
198C4⋊C8—61no—
199C4⋊C8#6712yes—
200#103—64no—
201#102—68no—
202#102—64no—
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—42no—
212C3×Q8—42no—
213C3×Q8—81no—
214C3×Q8—81no—
215C2×C3×C4—22no—
216C2×C3×C4—42no—
217C2×C3×C4—81no—
218C2×C3×C4—81no—
219C2×C3×C4#12011yescharacteristic
220C2×C3×C4#12011yescharacteristic
221C2×C3×C4#12011yescharacteristic
222C2×C3×C4#12111yescharacteristic
223C2×C3×C4#12111yescharacteristic
224C2×C3×C4#12711yescharacteristic
225C2×C3×C4C2×D811yescharacteristic
226C2×C3×C4#13511yescharacteristic
227C2×C3×C4#13512yes—
228C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
229C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
230C2×C3×C4#13712yes—
231C2×Dic3—22no—
232C2×Dic3—41no—
233C2×Dic3—41no—
234C2×Dic3—41no—
235C2×Dic3—42no—
236D12—42no—
237D12—42no—
238D12—44no—
239D12—82no—
240D12—82no—
241D12—81no—
242D12—81no—
243D12—81no—
244D12—81no—
245Dic6—22no—
246Dic6—22no—
247Dic6—41no—
248Dic6—41no—
249C3×C8—42no—
250C3×C8—81no—
251C3⋊C8—42no—
252C3⋊C8—42no—
253C24—121no—
254C2×Q8—22no—
255C2×Q8—32no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—61no—
262C2×Q8—61no—
263C2×Q8—61no—
264C2×Q8—61no—
265C2×Q8—61no—
266C2×D4—62no—
267C2×D4—62no—
268C2×D4—62no—
269C2×D4—121no—
270C2×D4—121no—
271C2×D4—121no—
272C2×D4—121no—
273C2×D4—122no—
274C2×D4—121no—
275C2×D4—122no—
276C2×D4—121no—
277C2×D4—121no—
278C2×D4—242no—
279C22×C4—41no—
280C22×C4—61no—
281C22×C4—61no—
282C22×C4#23511yescharacteristic
283C22×C4D3×D411yescharacteristic
284C22×C4D3×D411yescharacteristic
285Q16—128no—
286SD16—128no—
287C2×C8—22no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—42no—
291C2×C8—41no—
292C2×C8—62no—
293C2×C8—62no—
294C2×C8—62no—
295C2×C8—62no—
296C2×C8—121no—
297C2×C8—121no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—61no—
303C4⋊C4—61no—
304C4⋊C4—121no—
305C4⋊C4—121no—
306C42—121no—
307C42—121no—
308C42C2×D1211yescharacteristic
309C42D3×D411yescharacteristic
310C42D3×D411yescharacteristic
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—82no—
325C3×C4#44711yescharacteristic
326C3×C4#44711yescharacteristic
327C3×C4#44911yescharacteristic
328C3×C4#44911yescharacteristic
329C3×C4#45212yes—
330C3×C4#49511yescharacteristic
331C3×C4#49711yescharacteristic
332Dic3—42no—
333Dic3—81no—
334C23—122no—
335C23—242no—
336C23—241no—
337C23—241no—
338C23—242no—
339C23#88411yescharacteristic
340Q8—42no—
341Q8—42no—
342Q8—62no—
343Q8—62no—
344Q8—81no—
345Q8—81no—
346Q8—121no—
347Q8—121no—
348D4—122no—
349D4—122no—
350D4—124no—
351D4—242no—
352D4—242no—
353D4—241no—
354D4—241no—
355D4—241no—
356D4—241no—
357C2×C4—22no—
358C2×C4—42no—
359C2×C4—62no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—121no—
363C2×C4—121no—
364C2×C4—121no—
365C2×C4—122no—
366C2×C4#88211yescharacteristic
367C2×C4#88211yescharacteristic
368C2×C4#88411yescharacteristic
369C2×C4D3×D811yescharacteristic
370C2×C4#91111yescharacteristic
371C2×C4#91411yescharacteristic
372C2×C4#91612yes—
373C2×C4#91611yescharacteristic
374C2×C4#91812yes—
375C2×C4#93011yescharacteristic
376C2×C4#93711yescharacteristic
377C2×C4#94011yescharacteristic
378C8—42no—
379C8—81no—
380C8—122no—
381C8—122no—
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#475412yes—
394C22#477511yescharacteristic
395C22#478211yescharacteristic
396C22#479812yes—
397C22#496011yescharacteristic
398C4—21no—
399C4—21no—
400C4—82no—
401C4—122no—
402C4—241no—
403C4#477311yescharacteristic
404C4#478111yescharacteristic
405C4#480812yes—
406C4#515611yescharacteristic
407C4#516811yescharacteristic
408C4#517211yescharacteristic
409C4#519011yescharacteristic
410C3#1026311yescharacteristic, complemented, Sylow
411C2—482no—
412C2#6847012yes—
413C2#6847812yes—
414C2#7011711yescharacteristic
415C2#7020411yescharacteristic
416C2#7027811yescharacteristic
subgroupG/H
centerC23#884
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#332C2
socleC23×C3#121
prime radicalC2×#874C22
hypercenterC2×#332D3
hypocenterC3#10263
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#332

Complements

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

NH ≅ G/Ntype
1. C2×#5090C2semidirect
4. C2×C3×#332C2semidirect
5. #70472C2semidirect
7. #67703C2semidirect
34. C2×#332D3semidirect
410. C3#10263semidirect

Maximal subgroups

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

subgroupindex
C2×#50902
C2×#50882
C2×#48482
C2×C3×#3322
#704722
C2×#44852
#677032
#102633