Finite Group Databasebeta

← #10691436

Subgroups — #10691436

Order 768.

686 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×D4×Dic3C211yesmaximal, characteristic
2#82314C211yesmaximal, characteristic
3C3×#1486C211yesmaximal, characteristic
4#70792C211yesmaximal, characteristic, complemented
5#70514C211yesmaximal, characteristic, complemented
6#67736C211yesmaximal, characteristic, complemented
7#67715C211yesmaximal, characteristic, complemented
8#9231—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10C2×D4×Dic3—21no—
11C22×C4×Dic3—21no—
12#5470C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×#423C2211yescharacteristic
15#5218—21no—
16C4×#217—21no—
17#5019C412yescomplemented
18C4×#214—22no—
19#5008C412yescomplemented
20Dic3×C4⋊C4—21no—
21#4971—21no—
22Dic3×C22⋊C4—22no—
23C4×C4⋊Dic3C2211yescharacteristic
24C42×Dic3C2211yescharacteristic
25#4925C2211yescharacteristic
26C3×D4⋊C8—22no—
27#4555—21no—
28#4555—21no—
29#4467—22no—
30#3101—31no—
31C42×D4—31no—
32#1994—31no—
33#1503—31no—
34#1486D311yescharacteristic
35#1303—31no—
36#1276—31no—
37C22×C3×D4D411yescharacteristic
38C23×C3×C4—21no—
39C23×Dic3—22no—
40C2×C3×M16—21no—
41C3×C4×D4—21no—
42C3×C4×D4—21no—
43C3×C4×D4C2×C412yes—
44C3×C4×D4D412yes—
45C2×C3×C4⋊C4D411yescharacteristic
46C2×C3×C22⋊C4—21no—
47C2×C3×C42C2311yescharacteristic
48C2×#217—22no—
49D4×Dic3—28no—
50C2×#214—22no—
51#926—21no—
52C2×C4⋊Dic3—21no—
53#924—21no—
54#924—21no—
55C2×C4×Dic3—21no—
56C2×C4×Dic3—22no—
57C2×C4×Dic3—21no—
58C2×C4×Dic3—21no—
59C2×C4×Dic3—42no—
60C2×C4×Dic3—42no—
61C2×C4×Dic3D411yescharacteristic
62C2×C4×Dic3D411yescharacteristic
63#870—21no—
64#869C2×C411yescharacteristic, complemented
65#869C2×C411yescharacteristic, complemented
66C4×Dic6—21no—
67C4×Dic6—21no—
68C3×C4⋊C8—21no—
69C3×C4⋊C8—21no—
70C3×C4⋊C8—21no—
71C3×C22⋊C8—21no—
72C3×C4×C8—21no—
73#832—22no—
74C3⋊#95—22no—
75C3⋊#95D412yes—
76#830—21no—
77#810—22no—
78#808—22no—
79C4⋊(C3⋊C8)C2×C412yescomplemented
80C4×C3⋊C8—21no—
81#533—31no—
82C2×C4×D4—61no—
83C2×C4×D4C2×D311yescharacteristic
84C22×C42—61no—
85#483—61no—
86#476—32no—
87#461—32no—
88#450—61no—
89C4×SD16—62no—
90#431—31no—
91#431C2×D311yescharacteristic
92#423C2×D311yescharacteristic
93C4×C4⋊C4—31no—
94C4×C4⋊C4—61no—
95C4×C22⋊C4—62no—
96C4×C22⋊C4—61no—
97C43—31no—
98#339—61no—
99#339—61no—
100#328—62no—
101D4⋊C8—22no—
102C24×C3—21no—
103C2×C3×D4—24no—
104C2×C3×D4—21no—
105C2×C3×D4C22⋊C412yes—
106C22×C3×C4—21no—
107C22×C3×C4—22no—
108C22×C3×C4—22no—
109C22×C3×C4—41no—
110C22×C3×C4—41no—
111C22×C3×C4—41no—
112C22×C3×C4—41no—
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×Dic3—21no—
117C22×Dic3—28no—
118C22×Dic3—21no—
119C22×Dic3—22no—
120C22×Dic3—44no—
121C22×Dic3—44no—
122C2×Dic6—21no—
123C2×Dic6—21no—
124C3×M16—41no—
125C3×M16—41no—
126C2×C3×C8—41no—
127C2×C3×C8—41no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4C22⋊C412yes—
130C3×C22⋊C4—22no—
131C3×C22⋊C4—22no—
132C3×C42C22×C411yescharacteristic
133C3×C42C2×D411yescharacteristic
134C3×C42#4111yescharacteristic
135C3×C42#4111yescharacteristic
136#217—28no—
137#217—41no—
138#214—44no—
139C4⋊Dic3—24no—
140C4⋊Dic3—21no—
141C4⋊Dic3—21no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144Dic3⋊C4—41no—
145Dic3⋊C4—41no—
146C4×Dic3—24no—
147C4×Dic3—21no—
148C4×Dic3—21no—
149C4×Dic3—21no—
150C4×Dic3—42no—
151C4×Dic3—44no—
152C4×Dic3—41no—
153C4×Dic3—41no—
154C4×Dic3—41no—
155C4×Dic3SD1612yescomplemented
156C2×C3⋊C8—22no—
157C22×D4D1211yescharacteristic
158C23×C4—21no—
159C23×C4—62no—
160C2×SD16—62no—
161C2×M16—21no—
162C4⋊Q8—31no—
163C4⋊Q8—31no—
164#125—61no—
165C22⋊Q8—61no—
166C22⋊Q8—61no—
167C4×Q8—61no—
168C4×Q8—61no—
169C4×D4—21no—
170C4×D4—21no—
171C4×D4—68no—
172C4×D4C4×D312yes—
173C4×D4#6712yes—
174#117—61no—
175C2×C4⋊C4—61no—
176C2×C4⋊C4D1211yescharacteristic
177C2×C22⋊C4—21no—
178C2×C22⋊C4—62no—
179C2×C42—31no—
180C2×C42—31no—
181C2×C42—61no—
182C2×C42—62no—
183C2×C42—61no—
184C2×C42—61no—
185C2×C42—122no—
186C2×C42—122no—
187C2×C42C22×D311yescharacteristic
188#106—62no—
189C4⋊C8—21no—
190C4⋊C8—21no—
191C4⋊C8—21no—
192C4⋊C8—32no—
193#103—62no—
194#102—62no—
195C22⋊C8—21no—
196C22⋊C8—61no—
197C4×C8—21no—
198C4×C8—61no—
199#95—32no—
200#95—62no—
201C23×C3—24no—
202C23×C3—22no—
203C23×C3—42no—
204C23×C3—41no—
205C23×C3—41no—
206C23×C3#12011yescharacteristic
207C3×D4—24no—
208C3×D4—22no—
209C3×D4#10414yes—
210C2×C3×C4—22no—
211C2×C3×C4—22no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4—42no—
216C2×C3×C4—42no—
217C2×C3×C4—42no—
218C2×C3×C4—42no—
219C2×C3×C4C2×C22⋊C411yescharacteristic
220C2×C3×C4C4×D411yescharacteristic
221C2×C3×C4C4×D411yescharacteristic
222C2×C3×C4#12111yescharacteristic
223C2×C3×C4#12111yescharacteristic
224C2×C3×C4#12311yescharacteristic
225C2×C3×C4C2×SD1611yescharacteristic
226C2×C3×C4#13511yescharacteristic
227C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
228C2×C3×C4#13711yescharacteristic
229C2×Dic3—22no—
230C2×Dic3—22no—
231C2×Dic3—22no—
232C2×Dic3—22no—
233C2×Dic3—24no—
234C2×Dic3—416no—
235C2×Dic3—41no—
236C2×Dic3—42no—
237C2×Dic3—42no—
238C2×Dic3—41no—
239C2×Dic3—41no—
240C2×Dic3—41no—
241Dic6—42no—
242Dic6—81no—
243C3×C8—81no—
244C3×C8—81no—
245C3⋊C8—42no—
246C24—21no—
247C2×Q8—61no—
248C2×Q8—61no—
249C2×D4—24no—
250C2×D4—21no—
251C2×D4(C2×D3)⋊C412yes—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—61no—
260C22×C4—68no—
261C22×C4—62no—
262C22×C4—61no—
263C22×C4—124no—
264C22×C4—124no—
265C22×C4C2×D1211yescharacteristic
266C22×C4D3×D411yescharacteristic
267C22×C4D3×D411yescharacteristic
268SD16—124no—
269M16—41no—
270M16—41no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—62no—
274C4⋊C4—21no—
275C4⋊C4—64no—
276C4⋊C4—61no—
277C4⋊C4—61no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C4⋊C4—121no—
281C4⋊C4—121no—
282C4⋊C4(C2×D3)⋊C412yes—
283C22⋊C4—22no—
284C22⋊C4—22no—
285C22⋊C4—68no—
286C22⋊C4—121no—
287C42—32no—
288C42—64no—
289C42—61no—
290C42—61no—
291C42—61no—
292C42—124no—
293C42—122no—
294C42—121no—
295C42—121no—
296C42—121no—
297C42C2×C4×D311yescharacteristic
298C42#23711yescharacteristic
299C42#23711yescharacteristic
300C42C2×#6711yescharacteristic
301C22×C3—21no—
302C22×C3—21no—
303C22×C3—24no—
304C22×C3—44no—
305C22×C3—44no—
306C22×C3—42no—
307C22×C3—42no—
308C22×C3#38611yescharacteristic
309C22×C3#45011yescharacteristic
310C22×C3#45211yescharacteristic
311C3×C4—21no—
312C3×C4—21no—
313C3×C4—21no—
314C3×C4—42no—
315C3×C4C2×#10412yes—
316C3×C4C4×SD1611yescharacteristic
317C3×C4#44011yescharacteristic
318C3×C4#46111yescharacteristic
319C3×C4#46311yescharacteristic
320Dic3—24no—
321Dic3—42no—
322Dic3—44no—
323Dic3—81no—
324Dic3—81no—
325C23—24no—
326C23—22no—
327C23—42no—
328C23—41no—
329C23—41no—
330C23#88211yescharacteristic
331Q8—122no—
332Q8—241no—
333D4—24no—
334D4—22no—
335D4#82514yes—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—62no—
346C2×C4—62no—
347C2×C4—62no—
348C2×C4—62no—
349C2×C4—64no—
350C2×C4—1216no—
351C2×C4—121no—
352C2×C4—122no—
353C2×C4—122no—
354C2×C4—121no—
355C2×C4—121no—
356C2×C4—121no—
357C2×C4#67⋊C411yescharacteristic
358C2×C4#67⋊C411yescharacteristic
359C2×C4#88611yescharacteristic
360C2×C4#91111yescharacteristic
361C2×C4#91211yescharacteristic
362C2×C4D3×SD1611yescharacteristic
363C2×C4#91511yescharacteristic
364C2×C4C2×(C2×D3)⋊C411yescharacteristic
365C2×C4#93911yescharacteristic
366C2×C4#93911yescharacteristic
367C8—81no—
368C8—81no—
369C8—122no—
370C2×C3—21no—
371C2×C3—21no—
372C2×C3—44no—
373C2×C3#187711yescharacteristic
374C2×C3#189611yescharacteristic
375C2×C3#190611yescharacteristic
376C22—21no—
377C22—21no—
378C22—24no—
379C22—44no—
380C22—44no—
381C22—42no—
382C22—42no—
383C22#477611yescharacteristic
384C22#478311yescharacteristic
385C22#495811yescharacteristic
386C4—21no—
387C4—21no—
388C4—21no—
389C4—42no—
390C4—64no—
391C4—124no—
392C4—122no—
393C4—241no—
394C4—241no—
395C4#475711yescharacteristic
396C4#475811yescharacteristic
397C4C2×#82512yes—
398C4#515311yescharacteristic
399C4#516211yescharacteristic
400C3#923111yescharacteristic, complemented, Sylow
401C2—21no—
402C2—21no—
403C2—44no—
404C2#6996511yescharacteristic
405C2#7017111yescharacteristic
406C2#7039511yescharacteristic
subgroupG/H
centerC2×C4#939
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1486C2
socleC22×C3#386
prime radicalC22×C3×D4D4
cosocleC2×C3×C42C23
hypercenter#1486D3
hypocenterC3#9231
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1486

Complements

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

NH ≅ G/Ntype
4. #70792C2semidirect
5. #70514C2semidirect
6. #67736C2semidirect
7. #67715C2semidirect
17. #5019C4semidirect
19. #5008C4semidirect
64. #869C2×C4semidirect
65. #869C2×C4semidirect
79. C4⋊(C3⋊C8)C2×C4semidirect
155. C4×Dic3SD16semidirect
400. C3#9231semidirect

Maximal subgroups

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

subgroupindex
C4×D4×Dic32
#823142
C3×#14862
#707922
#705142
#677362
#677152
#92313