Finite Group Databasebeta

← #10697713

Subgroups — #10697713

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×#4849C211yesmaximal, characteristic, complemented
2C2×#4848C211yesmaximal, characteristic
3C2×#4694C211yesmaximal, characteristic
4C2×#4482C211yesmaximal, characteristic, complemented
5C2×#4485C211yesmaximal, characteristic
6#69788C211yesmaximal, characteristic, complemented
7C3×#1277C211yesmaximal, characteristic, complemented
8#10263—31nomaximal, Sylow
9C22×Dic12—21no—
10C22×#204—21no—
11C2×#874C2211yescharacteristic
12C2×#869C2211yescharacteristic
13C2×C4×Dic6C2211yescharacteristic
14C2×C3×C4⋊C8C2211yescharacteristic
15C2×C3×C4⋊C8C2211yescharacteristic
16C2×C3×C4×C8C2211yescharacteristic
17C2×#821—22no—
18C2×#821—21no—
19C2×#816—21no—
20C2×C4⋊(C3⋊C8)C2211yescharacteristic
21#4849—22no—
22#4849—22no—
23#4848—22no—
24#4848—22no—
25#4694—22no—
26#4694—22no—
27#4550—22no—
28#4485—22no—
29#4482—22no—
30C2×#486—31no—
31C2×#462—31no—
32C2×#460—31no—
33#1984—31no—
34C2×#332—31no—
35C2×#331—31no—
36#1277D311yescharacteristic, complemented
37C22×D12—21no—
38C22×D12—21no—
39C22×D12—21no—
40C22×Dic6—21no—
41C22×Dic6D411yescharacteristic
42C22×C3×C8—21no—
43C22×C3×C8—21no—
44C22×C3×C8D412yes—
45C2×C3×C42C2311yescharacteristic
46C2×C4⋊Dic3—21no—
47C2×C4⋊Dic3—21no—
48C2×C4⋊Dic3D411yescharacteristic
49C2×Dic3⋊C4—21no—
50C2×C4×Dic3—21no—
51C22×C3⋊C8—21no—
52C2×Dic12—24no—
53C2×Dic12—42no—
54C2×Dic12—42no—
55C2×#204—24no—
56C2×#204—42no—
57C2×#204—42no—
58#874—21no—
59#874—21no—
60#874—21no—
61#874D412yes—
62#869—21no—
63#869—21no—
64#869—21no—
65#869D412yes—
66C4×Dic6—21no—
67C4×Dic6—21no—
68C4×Dic6—21no—
69C4×Dic6—21no—
70C3×C4⋊C8—21no—
71C3×C4⋊C8—21no—
72C3×C4⋊C8D412yes—
73C3×C4⋊C8D412yes—
74C3×C4×C8—21no—
75C3×C4×C8—21no—
76#821—28no—
77#821—24no—
78#816—24no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)—21no—
81C22×Q16—61no—
82C22×SD16—61no—
83C2×C4⋊Q8—31no—
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—61no—
97C2×#102—62no—
98C2×C4×C8C2×D311yescharacteristic
99#336—62no—
100#332—62no—
101#331—62no—
102C23×D3—41no—
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4#4111yescharacteristic
106C22×Dic3—21no—
107C22×Dic3—21no—
108C22×Dic3—41no—
109C2×D12—22no—
110C2×D12—22no—
111C2×D12—22no—
112C2×D12—41no—
113C2×D12—42no—
114C2×D12—41no—
115C2×D12—41no—
116C2×D12—41no—
117C2×D12—42no—
118C2×D12—41no—
119C2×D12—41no—
120C2×D12—41no—
121C2×D12—82no—
122C2×Dic6—21no—
123C2×Dic6—21no—
124C2×Dic6—21no—
125C2×Dic6—21no—
126C2×Dic6—21no—
127C2×Dic6—22no—
128C2×Dic6—41no—
129C2×Dic6—41no—
130C2×Dic6—41no—
131C2×Dic6—41no—
132C2×Dic6—41no—
133C2×Dic6D812yes—
134C2×C3×C8—22no—
135C2×C3×C8—22no—
136C2×C3×C8—22no—
137C2×C3×C8—22no—
138C2×C3×C8—22no—
139C2×C3×C8—41no—
140C2×C3×C8—41no—
141C2×C3×C8—41no—
142C2×C3×C8—41no—
143C3×C42C2×D411yescharacteristic
144C3×C42C2×D411yescharacteristic
145C3×C42C2×D411yescharacteristic
146C3×C42C2×D411yescharacteristic
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149C4⋊Dic3—22no—
150C4⋊Dic3—22no—
151C4⋊Dic3—41no—
152C4⋊Dic3—41no—
153Dic3⋊C4—41no—
154Dic3⋊C4—41no—
155C4×Dic3—41no—
156C4×Dic3—41no—
157C2×C3⋊C8—41no—
158C2×C3⋊C8—42no—
159Dic12—48no—
160#204—48no—
161C22×Q8—31no—
162C22×Q8—61no—
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—61no—
175C22×C8D1212yes—
176C4⋊Q8—32no—
177C4⋊Q8—61no—
178C4⋊Q8—61no—
179C4⋊Q8—61no—
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—31no—
189C2×C4⋊C4—61no—
190C2×C4⋊C4—61no—
191C2×C4⋊C4—61no—
192C2×C42—61no—
193C2×C42C22×D311yescharacteristic
194C4⋊C8—21no—
195C4⋊C8—21no—
196C4⋊C8—61no—
197C4⋊C8—61no—
198C4⋊C8D1212yes—
199C4⋊C8D1212yes—
200#103—64no—
201#102—64no—
202#102—68no—
203C4×C8—21no—
204C4×C8—21no—
205C23×C3#12111yescharacteristic
206C22×D3—42no—
207C22×D3—82no—
208C22×D3—81no—
209C22×D3—81no—
210C22×D3—82no—
211C2×C3×C4—22no—
212C2×C3×C4#12011yescharacteristic
213C2×C3×C4#12011yescharacteristic
214C2×C3×C4#12011yescharacteristic
215C2×C3×C4#12111yescharacteristic
216C2×C3×C4#12111yescharacteristic
217C2×C3×C4#12711yescharacteristic
218C2×C3×C4C2×D811yescharacteristic
219C2×C3×C4#13512yes—
220C2×C3×C4#13511yescharacteristic
221C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
222C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
223C2×C3×C4#13712yes—
224C2×Dic3—22no—
225C2×Dic3—41no—
226C2×Dic3—41no—
227C2×Dic3—41no—
228C2×Dic3—42no—
229C2×Dic3—42no—
230C2×Dic3—81no—
231C2×Dic3—81no—
232D12—42no—
233D12—42no—
234D12—44no—
235D12—82no—
236D12—82no—
237D12—81no—
238D12—81no—
239D12—81no—
240D12—81no—
241Dic6—22no—
242Dic6—22no—
243Dic6—41no—
244Dic6—41no—
245Dic6—42no—
246Dic6—42no—
247Dic6—81no—
248Dic6—81no—
249C3×C8—42no—
250C3×C8—42no—
251C3×C8—42no—
252C3⋊C8—81no—
253C24—121no—
254C2×Q8—32no—
255C2×Q8—61no—
256C2×Q8—61no—
257C2×Q8—61no—
258C2×Q8—62no—
259C2×Q8—61no—
260C2×Q8—61no—
261C2×Q8—121no—
262C2×Q8—121no—
263C2×Q8—121no—
264C2×Q8—121no—
265C2×Q8—121no—
266C2×D4—62no—
267C2×D4—62no—
268C2×D4—62no—
269C2×D4—121no—
270C2×D4—121no—
271C2×D4—121no—
272C2×D4—122no—
273C2×D4—122no—
274C2×D4—121no—
275C2×D4—121no—
276C2×D4—121no—
277C2×D4—121no—
278C2×D4—242no—
279C22×C4—61no—
280C22×C4—61no—
281C22×C4—121no—
282C22×C4C2×D1211yescharacteristic
283C22×C4D3×D411yescharacteristic
284C22×C4#23911yescharacteristic
285Q16—128no—
286SD16—128no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—121no—
297C2×C8—122no—
298C4⋊C4—61no—
299C4⋊C4—62no—
300C4⋊C4—61no—
301C4⋊C4—62no—
302C4⋊C4—121no—
303C4⋊C4—121no—
304C4⋊C4—121no—
305C4⋊C4—121no—
306C42—121no—
307C42—121no—
308C42C2×D1211yescharacteristic
309C42C2×D1211yescharacteristic
310C42D3×D411yescharacteristic
311C42D3×D411yescharacteristic
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#44711yescharacteristic
325C3×C4#44711yescharacteristic
326C3×C4#44911yescharacteristic
327C3×C4#44911yescharacteristic
328C3×C4#45212yes—
329C3×C4#49511yescharacteristic
330C3×C4#49711yescharacteristic
331Dic3—42no—
332Dic3—82no—
333Dic3—81no—
334C23—122no—
335C23—241no—
336C23—241no—
337C23—242no—
338C23—242no—
339C23#89511yescharacteristic
340Q8—62no—
341Q8—62no—
342Q8—121no—
343Q8—121no—
344Q8—122no—
345Q8—122no—
346Q8—241no—
347Q8—241no—
348D4—124no—
349D4—122no—
350D4—122no—
351D4—242no—
352D4—242no—
353D4—241no—
354D4—241no—
355D4—241no—
356D4—241no—
357C2×C4—22no—
358C2×C4—62no—
359C2×C4—121no—
360C2×C4—121no—
361C2×C4—122no—
362C2×C4—121no—
363C2×C4—122no—
364C2×C4—241no—
365C2×C4—241no—
366C2×C4#87411yescharacteristic
367C2×C4#88211yescharacteristic
368C2×C4#88211yescharacteristic
369C2×C4#88211yescharacteristic
370C2×C4#89511yescharacteristic
371C2×C4#89511yescharacteristic
372C2×C4#90412yes—
373C2×C4#90912yes—
374C2×C4D3×D811yescharacteristic
375C2×C4#91411yescharacteristic
376C2×C4#91411yescharacteristic
377C2×C4#91911yescharacteristic
378C8—42no—
379C8—42no—
380C8—42no—
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#467111yescharacteristic
394C22#484512yes—
395C22#486511yescharacteristic
396C22#488311yescharacteristic
397C22#489812yes—
398C4—21no—
399C4—21no—
400C4—122no—
401C4—242no—
402C4—241no—
403C4#469611yescharacteristic
404C4#471511yescharacteristic
405C4#473312yes—
406C4#477311yescharacteristic
407C4#477311yescharacteristic
408C4#481011yescharacteristic
409C4#481011yescharacteristic
410C3#1026311yescharacteristic, complemented, Sylow
411C2—482no—
412C2#6855412yes—
413C2#6859012yes—
414C2#6867711yescharacteristic
415C2#6888611yescharacteristic
416C2#6894311yescharacteristic
subgroupG/H
centerC23#895
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1277C2
socleC23×C3#121
prime radicalC2×#874C22
hypercenter#1277D3
hypocenterC3#10263
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1277

Complements

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

NH ≅ G/Ntype
1. C2×#4849C2semidirect
4. C2×#4482C2semidirect
6. #69788C2semidirect
7. C3×#1277C2semidirect
36. #1277D3semidirect
410. C3#10263semidirect

Maximal subgroups

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

subgroupindex
C2×#48492
C2×#48482
C2×#46942
C2×#44822
C2×#44852
#697882
C3×#12772
#102633