Finite Group Databasebeta

← #10692098

Subgroups — #10692098

Order 768.

650 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 650 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3109C211yesmaximal, characteristic
2#82317C211yesmaximal, characteristic, complemented
3#71818C211yesmaximal, characteristic, complemented
4#71812C211yesmaximal, characteristic, complemented
5#71599C211yesmaximal, characteristic, complemented
6#67750C211yesmaximal, characteristic, complemented
7#67748C211yesmaximal, characteristic, complemented
8#9325—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C3×C4×D4C2211yescharacteristic
11C2×D4×Dic3—21no—
12C3×#481—21no—
13C3×#476C412yescomplemented
14C3×#459C412yescomplemented
15C3×#447—21no—
16C3×#442—22no—
17C3×#432C2211yescharacteristic
18C2×#837—22no—
19C4×#217—21no—
20C2×C3⋊#95—21no—
21Dic3×C4⋊C4—21no—
22Dic3×C22⋊C4—22no—
23C4×C4⋊Dic3C2211yescharacteristic
24#4932C2211yescharacteristic
25#4926C2211yescharacteristic
26C4×#208C2211yescharacteristic
27#4537—21no—
28#4537—21no—
29D4⋊(C3⋊C8)—22no—
30#3109D311yescharacteristic
31#2275—31no—
32#1980—31no—
33#1505—31no—
34#1490—31no—
35#1300—31no—
36#1285—31no—
37C2×C3×#41—21no—
38C22×C3×D4D411yescharacteristic
39C23×C3×C4—21no—
40C23×Dic3—22no—
41C2×C3×D8—22no—
42C3×C4⋊Q8C2×C411yescharacteristic, complemented
43C3×#127C2×C411yescharacteristic, complemented
44C3×#124—21no—
45C3×#121—21no—
46C3×#121—21no—
47C3×C4×D4—21no—
48C3×C4×D4—21no—
49C3×C4×D4—21no—
50C3×C4×D4—21no—
51C3×C4×D4C2×C412yes—
52C3×C4×D4D412yes—
53C2×C3×C4⋊C4D411yescharacteristic
54C2×C3×C22⋊C4—21no—
55C2×C3×C42C2311yescharacteristic
56C2×#217—22no—
57D4×Dic3—28no—
58C2×C4⋊Dic3—21no—
59C2×C4×Dic3—21no—
60C2×C4×Dic3—22no—
61C2×C4×Dic3D412yes—
62C2×#208—21no—
63C3×#106—22no—
64C3×C4⋊C8C2×C412yescomplemented
65C3×#102—22no—
66C3×#102—22no—
67C3×C22⋊C8—21no—
68C3×#97—21no—
69#837—44no—
70C3⋊#95—22no—
71C3⋊#95—21no—
72C3⋊#95—21no—
73C3⋊#95D411yescharacteristic
74C3⋊#95D411yescharacteristic
75#830—21no—
76C4⋊(C3⋊C8)—21no—
77#803—21no—
78#803—21no—
79C4×C3⋊C8—21no—
80#532C2×D311yescharacteristic
81C2×C4×D4—61no—
82C2×C4×D4C2×D311yescharacteristic
83#481—21no—
84#476Dic312yescomplemented
85#459Dic312yescomplemented
86#447—21no—
87#442—22no—
88#432—31no—
89#432C2×D311yescharacteristic
90C2×#104—62no—
91C4×M16—31no—
92C4×C4⋊C4—31no—
93C4×C4⋊C4—61no—
94C4×C22⋊C4—61no—
95C4×C22⋊C4—62no—
96#376—31no—
97C2×#95—61no—
98#327—61no—
99#327—61no—
100D4⋊C8—62no—
101C24×C3—21no—
102C3×#41—42no—
103C3×#41—42no—
104C2×C3×Q8—21no—
105C2×C3×D4—24no—
106C2×C3×D4—21no—
107C2×C3×D4—21no—
108C2×C3×D4—21no—
109C2×C3×D4—21no—
110C2×C3×D4—41no—
111C2×C3×D4C22⋊C412yes—
112C22×C3×C4—21no—
113C22×C3×C4—22no—
114C22×C3×C4—22no—
115C22×C3×C4—41no—
116C22×C3×C4—82no—
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×Dic3—21no—
121C22×Dic3—22no—
122C22×Dic3—21no—
123C22×Dic3—28no—
124C22×Dic3—82no—
125C22×Dic3—82no—
126C3×D8—44no—
127C2×C3×C8—22no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4C22⋊C412yes—
132C3×C22⋊C4—22no—
133C3×C22⋊C4—22no—
134C3×C22⋊C4—41no—
135C3×C22⋊C4—41no—
136C3×C42C22×C411yescharacteristic
137C3×C42C2×D411yescharacteristic
138C3×C42#4111yescharacteristic
139C3×C42#4111yescharacteristic
140#217—28no—
141C4⋊Dic3—24no—
142C4×Dic3—24no—
143C4×Dic3—42no—
144C4×Dic3—82no—
145#208—22no—
146#208—41no—
147C2×C3⋊C8—41no—
148C2×C3⋊C8—41no—
149C2×#41—21no—
150C22×D4#6711yescharacteristic
151C23×C4—21no—
152C23×C4—62no—
153C2×D8—22no—
154C2×M16—61no—
155C4⋊Q8C2×Dic311yescharacteristic, complemented
156#127C2×Dic311yescharacteristic, complemented
157#124—21no—
158#121—21no—
159#121—21no—
160C4×D4—21no—
161C4×D4—21no—
162C4×D4—21no—
163C4×D4—21no—
164C4×D4—68no—
165C4×D4C2×Dic312yes—
166C4×D4#6712yes—
167C2×C4⋊C4—61no—
168C2×C4⋊C4#6711yescharacteristic
169C2×C22⋊C4—21no—
170C2×C22⋊C4—62no—
171C2×C42—32no—
172C2×C42—61no—
173C2×C42—62no—
174C2×C42C22×D311yescharacteristic
175#106—22no—
176C4⋊C8—61no—
177C4⋊C8C2×Dic312yescomplemented
178#104—124no—
179#102—22no—
180#102—22no—
181C22⋊C8—21no—
182C22⋊C8—61no—
183#97—21no—
184#97—61no—
185#97—61no—
186C4×C8—61no—
187#95—31no—
188#95—31no—
189#95—62no—
190#95—61no—
191#95—61no—
192C23×C3—24no—
193C23×C3—22no—
194C23×C3—42no—
195C23×C3—41no—
196C23×C3—41no—
197C23×C3—41no—
198C23×C3#12011yescharacteristic
199C3×Q8—42no—
200C3×D4—24no—
201C3×D4—22no—
202C3×D4—42no—
203C3×D4—42no—
204C3×D4—81no—
205C3×D4—81no—
206C3×D4—81no—
207C3×D4#10414yescomplemented
208C2×C3×C4—21no—
209C2×C3×C4—21no—
210C2×C3×C4—22no—
211C2×C3×C4—22no—
212C2×C3×C4—41no—
213C2×C3×C4—42no—
214C2×C3×C4—41no—
215C2×C3×C4—82no—
216C2×C3×C4—82no—
217C2×C3×C4—82no—
218C2×C3×C4C2×C22⋊C411yescharacteristic
219C2×C3×C4C4×D411yescharacteristic
220C2×C3×C4C4×D411yescharacteristic
221C2×C3×C4#12111yescharacteristic
222C2×C3×C4#12111yescharacteristic
223C2×C3×C4#12311yescharacteristic
224C2×Dic3—22no—
225C2×Dic3—22no—
226C2×Dic3—24no—
227C2×Dic3—41no—
228C2×Dic3—42no—
229C2×Dic3—41no—
230C2×Dic3—42no—
231C2×Dic3—41no—
232C2×Dic3—41no—
233C2×Dic3—88no—
234C3×C8—42no—
235C3⋊C8—42no—
236C3⋊C8—81no—
237C24—21no—
238#41—42no—
239#41—42no—
240C2×Q8—21no—
241C2×D4—24no—
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—41no—
247C2×D4#21712yes—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—41no—
252C22×C4—61no—
253C22×C4—61no—
254C22×C4—62no—
255C22×C4—68no—
256C22×C4—82no—
257C22×C4—242no—
258C22×C4—242no—
259C22×C4D3×D411yescharacteristic
260C22×C4D3×D411yescharacteristic
261C22×C4C2×#6711yescharacteristic
262D8—44no—
263M16—62no—
264M16—121no—
265C2×C8—22no—
266C2×C8—121no—
267C2×C8—121no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—64no—
272C4⋊C4#21712yes—
273C22⋊C4—22no—
274C22⋊C4—22no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—68no—
278C42—64no—
279C42—122no—
280C42—242no—
281C42#23711yescharacteristic
282C42#23711yescharacteristic
283C42C22×Dic311yescharacteristic
284C42C2×#6711yescharacteristic
285C22×C3—21no—
286C22×C3—21no—
287C22×C3—24no—
288C22×C3—44no—
289C22×C3—44no—
290C22×C3—42no—
291C22×C3—42no—
292C22×C3—81no—
293C22×C3—81no—
294C22×C3—81no—
295C22×C3#38611yescharacteristic
296C22×C3#45311yescharacteristic
297C22×C3#45411yescharacteristic
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—21no—
302C3×C4—21no—
303C3×C4—42no—
304C3×C4—81no—
305C3×C4C2×#10412yes—
306Dic3—42no—
307Dic3—44no—
308Dic3—42no—
309C23—24no—
310C23—22no—
311C23—42no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23#93711yescharacteristic
316Q8—42no—
317D4—24no—
318D4—22no—
319D4—42no—
320D4—42no—
321D4—81no—
322D4—81no—
323D4—81no—
324D4#83714yescomplemented
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—62no—
333C2×C4—62no—
334C2×C4—64no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—121no—
339C2×C4—122no—
340C2×C4—121no—
341C2×C4—122no—
342C2×C4—121no—
343C2×C4—121no—
344C2×C4—248no—
345C2×C4D4×Dic311yescharacteristic
346C2×C4D4×Dic311yescharacteristic
347C2×C4#93511yescharacteristic
348C2×C4#93911yescharacteristic
349C2×C4#93911yescharacteristic
350C2×C4C2×#21711yescharacteristic
351C8—42no—
352C8—122no—
353C8—241no—
354C2×C3—21no—
355C2×C3—21no—
356C2×C3—44no—
357C2×C3—161no—
358C2×C3#188311yescharacteristic
359C2×C3#189611yescharacteristic
360C2×C3#190411yescharacteristic
361C22—21no—
362C22—21no—
363C22—24no—
364C22—44no—
365C22—44no—
366C22—42no—
367C22—42no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22#519811yescharacteristic
372C22#520111yescharacteristic
373C22#522011yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—42no—
380C4—81no—
381C4—122no—
382C4—124no—
383C4—122no—
384C4C2×#83712yes—
385C3#932511yescharacteristic, complemented, Sylow
386C2—21no—
387C2—21no—
388C2—44no—
389C2—161no—
390C2#7171111yescharacteristic
391C2#7208811yescharacteristic
392C2#7208911yescharacteristic
subgroupG/H
centerC22#5220
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#3109C2
socleC22×C3#386
cosocleC2×C3×C42C23
hypercenter#3109D3
hypocenterC3#9325
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3109

Complements

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

NH ≅ G/Ntype
2. #82317C2semidirect
3. #71818C2semidirect
4. #71812C2semidirect
5. #71599C2semidirect
6. #67750C2semidirect
7. #67748C2semidirect
13. C3×#476C4semidirect
14. C3×#459C4semidirect
42. C3×C4⋊Q8C2×C4semidirect
43. C3×#127C2×C4semidirect
64. C3×C4⋊C8C2×C4semidirect
84. #476Dic3semidirect
85. #459Dic3semidirect
155. C4⋊Q8C2×Dic3semidirect
156. #127C2×Dic3semidirect
177. C4⋊C8C2×Dic3semidirect
207. C3×D4#104semidirect
324. D4#837semidirect
385. C3#9325semidirect

Maximal subgroups

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

subgroupindex
C3×#31092
#823172
#718182
#718122
#715992
#677502
#677482
#93253