Finite Group Databasebeta

← #10795146

Subgroups — #10795146

Order 768.

1262 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Q8×Dic3C211yesmaximal, characteristic
2#78619C212yesmaximal
3C3×#1784C211yesmaximal, characteristic
4#72133C211yesmaximal, characteristic
5#70097C214yesmaximal, complemented
6Dic3×#101C214yesmaximal, complemented
7#69973C212yesmaximal
8#20727—31nomaximal, Sylow
9C23×C3×Q8C2211yescharacteristic
10C2×Q8×Dic3—22no—
11C2×Q8×Dic3—22no—
12C2×Q8×Dic3—22no—
13C2×Q8×Dic3—22no—
14C2×Q8×Dic3C412yes—
15C2×Q8×Dic3C412yes—
16C2×Q8×Dic3C2212yes—
17C2×Q8×Dic3C2212yes—
18C22×C4⋊Dic3C412yes—
19C22×C4⋊Dic3C2211yescharacteristic
20C22×C4×Dic3C412yes—
21C22×C4×Dic3C2211yescharacteristic
22C2×C3×#101C2214yes—
23C3×#407C2212yes—
24C2×#836C2214yes—
25#5207C2212yes—
26#5118C2214yes—
27Dic3×M16C2214yes—
28#4719—22no—
29#4719—22no—
30#4718—22no—
31#4718—22no—
32#4554C2218yes—
33C22×C4×Q8—31no—
34#2920—32no—
35#1821—32no—
36#1801—34no—
37#1784—31no—
38#1784D311yescharacteristic
39C4×#101—34no—
40C22×C3×Q8—28no—
41C22×C3×Q8—22no—
42C22×C3×Q8—22no—
43C22×C3×Q8C2×C412yes—
44C22×C3×Q8C2312yes—
45C23×C3×C4C2×C412yes—
46C23×C3×C4C2311yescharacteristic
47C23×Dic3C2×C411yescharacteristic
48C23×Dic3C2×C412yes—
49C23×Dic3C2×C411yescharacteristic
50C2×C3×M16C2314yes—
51Q8×Dic3—24no—
52Q8×Dic3—24no—
53Q8×Dic3—24no—
54Q8×Dic3—24no—
55Q8×Dic3—24no—
56Q8×Dic3—24no—
57Q8×Dic3—24no—
58Q8×Dic3—24no—
59C2×C4⋊Dic3—22no—
60C2×C4⋊Dic3—22no—
61C2×C4⋊Dic3—24no—
62C2×C4⋊Dic3—24no—
63C2×C4⋊Dic3C2×C414yes—
64C2×C4⋊Dic3C2×C414yes—
65C2×C4⋊Dic3C2×C412yes—
66C2×C4⋊Dic3C2312yes—
67C2×C4×Dic3—22no—
68C2×C4×Dic3—22no—
69C2×C4×Dic3—22no—
70C2×C4×Dic3—22no—
71C2×C4×Dic3C2×C414yes—
72C2×C4×Dic3C2×C414yes—
73C2×C4×Dic3C2×C411yescharacteristic
74C2×C4×Dic3C2×C411yescharacteristic
75C2×C4×Dic3D412yes—
76C2×C4×Dic3D412yes—
77C2×C4×Dic3D412yes—
78C2×C4×Dic3D412yes—
79C2×C4×Dic3C2311yescharacteristic
80C2×C4×Dic3C2311yescharacteristic
81C2×#208C2314yes—
82C3×#101—28no—
83C3×C22⋊C8—22no—
84C3×C22⋊C8—22no—
85#836—28no—
86#830—22no—
87#830—22no—
88#815—24no—
89Dic3⋊C8—24no—
90Dic3⋊C8—24no—
91C8×Dic3—24no—
92C23×Q8C2×D311yescharacteristic
93C2×C4×Q8—32no—
94C2×C4×Q8—32no—
95C2×C4×Q8—32no—
96C2×C4×Q8—32no—
97C2×C4×Q8—62no—
98C2×C4×Q8—62no—
99C2×C4×Q8—62no—
100C2×C4×Q8—62no—
101C22×C4⋊C4—32no—
102C22×C4⋊C4—31no—
103C22×C42—31no—
104C22×C42—32no—
105#432—62no—
106#432—62no—
107#431—62no—
108#431—62no—
109#423—34no—
110C2×#101—34no—
111C2×#101C2×D314yes—
112#407—32no—
113#407C2×D312yes—
114C4×M16—34no—
115#343—38no—
116C24×C3C22×C411yescharacteristic
117C2×C3×Q8—216no—
118C2×C3×Q8—216no—
119C2×C3×Q8—24no—
120C2×C3×Q8—28no—
121C2×C3×Q8—24no—
122C2×C3×Q8—24no—
123C2×C3×Q8#4118yes—
124C22×C3×C4—22no—
125C22×C3×C4—22no—
126C22×C3×C4—28no—
127C22×C3×C4—24no—
128C22×C3×C4C22×C411yescharacteristic
129C22×C3×C4C22×C414yes—
130C22×C3×C4C2×D412yes—
131C22×C3×C4C2×D412yes—
132C22×C3×C4C2411yescharacteristic
133C22×Dic3—24no—
134C22×Dic3—22no—
135C22×Dic3—22no—
136C22×Dic3—42no—
137C22×Dic3—44no—
138C22×Dic3C22⋊C411yescharacteristic
139C22×Dic3C22⋊C411yescharacteristic
140C22×Dic3C22⋊C411yescharacteristic
141C22×Dic3C22⋊C412yes—
142C22×Dic3C22⋊C412yes—
143C22×Dic3C22⋊C411yescharacteristic
144C22×Dic3C22×C411yescharacteristic
145C22×Dic3C22×C411yescharacteristic
146C22×Dic3C22×C414yes—
147C22×Dic3C22×C411yescharacteristic
148C22×Dic3C22×C411yescharacteristic
149C3×M16—28no—
150C2×C3×C8—24no—
151C4⋊Dic3—24no—
152C4⋊Dic3—24no—
153C4⋊Dic3—28no—
154C4⋊Dic3—28no—
155C4×Dic3—24no—
156C4×Dic3—24no—
157C4×Dic3—24no—
158C4×Dic3—24no—
159C4×Dic3C22⋊C412yes—
160C4×Dic3C22⋊C412yes—
161C4×Dic3C22⋊C412yes—
162C4×Dic3C22⋊C412yes—
163C4×Dic3C2×D412yes—
164C4×Dic3C2×D412yes—
165C4×Dic3C2×D412yes—
166C4×Dic3C2×D412yes—
167#208—28no—
168C2×C3⋊C8—24no—
169C22×Q8—28no—
170C22×Q8—22no—
171C22×Q8—22no—
172C22×Q8C4×D312yes—
173C22×Q8C22×D312yes—
174C23×C4—31no—
175C23×C4—31no—
176C23×C4—32no—
177C23×C4C4×D312yes—
178C23×C4C22×D311yescharacteristic
179C2×M16—34no—
180C2×M16C22×D314yes—
181C4×Q8—64no—
182C4×Q8—64no—
183C4×Q8—64no—
184C4×Q8—64no—
185C4×Q8—64no—
186C4×Q8—64no—
187C4×Q8—64no—
188C4×Q8—64no—
189C2×C4⋊C4—34no—
190C2×C4⋊C4—34no—
191C2×C4⋊C4—32no—
192C2×C4⋊C4—32no—
193C2×C4⋊C4—64no—
194C2×C4⋊C4—64no—
195C2×C4⋊C4—62no—
196C2×C4⋊C4—62no—
197C2×C42—31no—
198C2×C42—31no—
199C2×C42—32no—
200C2×C42—32no—
201C2×C42—32no—
202C2×C42—32no—
203C2×C42—31no—
204C2×C42—31no—
205C2×C42—34no—
206C2×C42—34no—
207C2×C42—62no—
208C2×C42—62no—
209C2×C42—62no—
210C2×C42—62no—
211C4⋊C8—64no—
212C4⋊C8—64no—
213#101—28no—
214#101—68no—
215C22⋊C8—22no—
216C22⋊C8—22no—
217C22⋊C8—62no—
218C22⋊C8—62no—
219#97—64no—
220C4×C8—64no—
221C23×C3—22no—
222C23×C3—22no—
223C23×C3—22no—
224C23×C3C2×C22⋊C411yescharacteristic
225C23×C3C2×C22⋊C411yescharacteristic
226C23×C3C23×C411yescharacteristic
227C3×Q8—216no—
228C3×Q8—216no—
229C2×C3×C4—28no—
230C2×C3×C4—28no—
231C2×C3×C4—24no—
232C2×C3×C4—22no—
233C2×C3×C4—24no—
234C2×C3×C4—24no—
235C2×C3×C4—24no—
236C2×C3×C4C2×C22⋊C412yes—
237C2×C3×C4#11718yes—
238C2×C3×C4C22×D412yes—
239C2×C3×C4C2×#4114yes—
240C2×Dic3—22no—
241C2×Dic3—24no—
242C2×Dic3—22no—
243C2×Dic3—22no—
244C2×Dic3—22no—
245C2×Dic3—22no—
246C2×Dic3—24no—
247C2×Dic3—42no—
248C2×Dic3—41no—
249C2×Dic3—41no—
250C2×Dic3—44no—
251C2×Dic3—44no—
252C2×Dic3—44no—
253C2×Dic3#10114yescomplemented
254C2×Dic3C2×C22⋊C411yescharacteristic
255C2×Dic3C2×C22⋊C411yescharacteristic
256C2×Dic3C2×C22⋊C412yes—
257C2×Dic3C2×C22⋊C412yes—
258C2×Dic3C2×C22⋊C411yescharacteristic
259C2×Dic3C2×C22⋊C411yescharacteristic
260C3×C8—44no—
261C3⋊C8—44no—
262C24C2×C4×D311yescharacteristic
263C2×Q8—216no—
264C2×Q8—216no—
265C2×Q8—28no—
266C2×Q8—24no—
267C2×Q8—24no—
268C2×Q8—24no—
269C2×Q8#23718yes—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—28no—
273C22×C4—24no—
274C22×C4—31no—
275C22×C4—31no—
276C22×C4—31no—
277C22×C4—31no—
278C22×C4—32no—
279C22×C4—31no—
280C22×C4—31no—
281C22×C4—32no—
282C22×C4—31no—
283C22×C4—31no—
284C22×C4—34no—
285C22×C4—64no—
286C22×C4—62no—
287C22×C4—62no—
288C22×C4—122no—
289C22×C4—124no—
290C22×C4C2×C4×D314yes—
291C22×C4C2×C4×D311yescharacteristic
292C22×C4D3×D412yes—
293C22×C4D3×D412yes—
294C22×C4C23×D311yescharacteristic
295M16—28no—
296M16—68no—
297C2×C8—24no—
298C2×C8—64no—
299C4⋊C4—68no—
300C4⋊C4—68no—
301C4⋊C4—64no—
302C4⋊C4—64no—
303C42—32no—
304C42—32no—
305C42—32no—
306C42—32no—
307C42—32no—
308C42—32no—
309C42—32no—
310C42—32no—
311C42—64no—
312C42—64no—
313C42—64no—
314C42—64no—
315C22×C3—28no—
316C22×C3—21no—
317C22×C3—21no—
318C22×C3—21no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—21no—
322C22×C3C2×#10112yes—
323C22×C3#41312yes—
324C22×C3C22×C22⋊C411yescharacteristic
325C22×C3C2×#11712yes—
326C3×C4—22no—
327C3×C4—28no—
328C3×C4#52514yes—
329Dic3—21no—
330Dic3—21no—
331Dic3—22no—
332Dic3—22no—
333Dic3—44no—
334Dic3C2×#10112yes—
335Dic3C2×#10112yes—
336C23—22no—
337C23—22no—
338C23—22no—
339C23D3×C22⋊C411yescharacteristic
340C23D3×C22⋊C411yescharacteristic
341C23C22×C4×D311yescharacteristic
342Q8—216no—
343Q8—216no—
344C2×C4—28no—
345C2×C4—28no—
346C2×C4—24no—
347C2×C4—22no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—31no—
352C2×C4—31no—
353C2×C4—34no—
354C2×C4—31no—
355C2×C4—31no—
356C2×C4—32no—
357C2×C4—32no—
358C2×C4—62no—
359C2×C4—62no—
360C2×C4—62no—
361C2×C4—62no—
362C2×C4—62no—
363C2×C4—64no—
364C2×C4—64no—
365C2×C4—122no—
366C2×C4—121no—
367C2×C4—121no—
368C2×C4—124no—
369C2×C4—124no—
370C2×C4—124no—
371C2×C4#87718yes—
372C2×C4D3×C22⋊C412yes—
373C2×C4C2×D3×D412yes—
374C2×C4C2×#23714yes—
375C8—44no—
376C8—124no—
377C2×C3—22no—
378C2×C3—22no—
379C2×C3—22no—
380C2×C3#228011yescharacteristic
381C2×C3C22×#10111yescharacteristic
382C2×C3C2×#41311yescharacteristic
383C22—28no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22D3×#10112yes—
391C22#475112yes—
392C22C2×#87712yes—
393C22C2×D3×C22⋊C411yescharacteristic
394C4—28no—
395C4—22no—
396C4—32no—
397C4—32no—
398C4—61no—
399C4—61no—
400C4—62no—
401C4—62no—
402C4—124no—
403C4#558614yes—
404C3#2072711yescharacteristic, complemented, Sylow
405C2—22no—
406C2—22no—
407C2—22no—
408C2#7381711yescharacteristic
409C2C2×D3×#10111yescharacteristic
410C2C2×#475111yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC23×C3C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1784C2
socleC22×C3C22×C22⋊C4
prime radicalC24×C3C22×C4
cosocleC22×C3×C4C24
hypercenter#1784D3
hypocenterC3#20727
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1784

Complements

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

NH ≅ G/Ntype
5. #70097C2semidirect
6. Dic3×#101C2semidirect
253. C2×Dic3#101semidirect
404. C3#20727semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×Dic32
#786192
C3×#17842
#721332
#700972
Dic3×#1012
#699732
#207273