Finite Group Databasebeta

← #10693133

Subgroups — #10693133

Order 768.

684 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 95 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
1#84068C211yesmaximal, characteristic
2#78632C211yesmaximal, characteristic
3C3×#1784C211yesmaximal, characteristic, complemented
4#72133C211yesmaximal, characteristic, complemented
5#69968C211yesmaximal, characteristic
6#67685C211yesmaximal, characteristic, complemented
7#67684C211yesmaximal, characteristic, complemented
8#9505—31nomaximal, Sylow
9C23×C3×Q8C2211yescharacteristic
10C2×#946—21no—
11C2×C4×#67C2211yescharacteristic
12C2×C3×#101—22no—
13C3×#407C2211yescharacteristic
14C3×#407C2211yescharacteristic
15#5231—21no—
16#5230—21no—
17#5229—21no—
18C2×#836—22no—
19#5213C412yes—
20#5212C2211yescharacteristic
21#5210C412yes—
22#5207C2211yescharacteristic
23#5207C2211yescharacteristic
24#5127—22no—
25#4728—22no—
26D3×C22⋊C8—21no—
27#4719—21no—
28#4469—21no—
29#4468—21no—
30#2973—31no—
31#2409—31no—
32#1784—31no—
33#1784D311yescharacteristic, complemented
34#1465—31no—
35#1305—31no—
36#1305—31no—
37C22×C3×Q8—21no—
38C22×C3×Q8—21no—
39C22×C3×Q8—21no—
40C22×C3×Q8—21no—
41C22×C3×Q8—21no—
42C22×C3×Q8—21no—
43C22×C3×Q8—44no—
44C23×C3×C4C2×C412yes—
45C23×C3×C4C2311yescharacteristic
46C22×#67C2×C411yescharacteristic
47C22×C4×D3D411yescharacteristic
48C2×C3×M16—21no—
49C2×C3×M16D412yes—
50C2×#217C2×C411yescharacteristic
51C2×#217C2×C412yes—
52#946—44no—
53C4×#67—21no—
54C4×#67—21no—
55C4×#67—21no—
56C4×#67—21no—
57C2×(C2×D3)⋊C4—22no—
58C2×(C2×D3)⋊C4D411yescharacteristic
59C2×C4⋊Dic3—21no—
60C2×Dic3⋊C4—22no—
61C2×Dic3⋊C4D411yescharacteristic
62C2×C4×Dic3D411yescharacteristic
63C2×#208—21no—
64C2×#208D412yes—
65C2×D3×C8—22no—
66C3×#101—44no—
67C3×C22⋊C8—21no—
68C3×C22⋊C8—21no—
69C3×C22⋊C8—21no—
70C3×C22⋊C8—21no—
71#836—44no—
72C3⋊#95—21no—
73C3⋊#95—22no—
74C3⋊#95—22no—
75#830—21no—
76#830—21no—
77#830—21no—
78#830—21no—
79(C2×D3)⋊C8—22no—
80#815—22no—
81Dic3⋊C8—22no—
82C23×Q8C2×D311yescharacteristic
83C2×C22⋊Q8—61no—
84C2×C4×D4—31no—
85#435—62no—
86#435—62no—
87#432—61no—
88C2×#101—22no—
89C2×#101—62no—
90#407—31no—
91#407—31no—
92#407C2×D311yescharacteristic
93#407C2×D311yescharacteristic
94C2×C22⋊C8—61no—
95#395—61no—
96#393—61no—
97#391—61no—
98#386—32no—
99#385—32no—
100#380—31no—
101#324—61no—
102#323—61no—
103C24×C3C22×C411yescharacteristic
104C23×D3C22⋊C411yescharacteristic
105C2×C3×Q8—24no—
106C2×C3×Q8—24no—
107C2×C3×Q8—24no—
108C2×C3×Q8—42no—
109C2×C3×Q8—42no—
110C2×C3×Q8—42no—
111C2×C3×Q8—44no—
112C2×C3×Q8—44no—
113C2×C3×Q8—44no—
114C2×C3×Q8—41no—
115C2×C3×Q8—41no—
116C2×C3×Q8—41no—
117C2×C3×Q8—41no—
118C22×C3×C4—22no—
119C22×C3×C4—22no—
120C22×C3×C4—22no—
121C22×C3×C4—44no—
122C22×C3×C4—42no—
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4#4111yescharacteristic
128C22×C3×C4#4111yescharacteristic
129C2×#67—21no—
130C2×#67—21no—
131C2×#67—22no—
132C2×#67—41no—
133C22×Dic3—22no—
134C22×Dic3C22⋊C411yescharacteristic
135C22×Dic3C22⋊C411yescharacteristic
136C22×Dic3C22⋊C411yescharacteristic
137C2×C4×D3—21no—
138C2×C4×D3—21no—
139C2×C4×D3—42no—
140C3×M16—42no—
141C3×M16—42no—
142C2×C3×C8—22no—
143C2×C3×C8—41no—
144#217—21no—
145#217—21no—
146#217—42no—
147(C2×D3)⋊C4—21no—
148(C2×D3)⋊C4—21no—
149(C2×D3)⋊C4—44no—
150C4⋊Dic3—42no—
151Dic3⋊C4—21no—
152Dic3⋊C4—21no—
153Dic3⋊C4—44no—
154C4×Dic3—21no—
155C4×Dic3—21no—
156#208—42no—
157#208—42no—
158C2×C3⋊C8—22no—
159C2×C3⋊C8—41no—
160D3×C8—82no—
161C22×Q8—21no—
162C22×Q8—21no—
163C22×Q8—21no—
164C22×Q8—21no—
165C22×Q8—21no—
166C22×Q8—21no—
167C22×Q8—44no—
168C22×D4—31no—
169C23×C4—31no—
170C23×C4C4×D312yes—
171C23×C4C22×D311yescharacteristic
172C2×M16—21no—
173C2×M16—32no—
174C2×M16—61no—
175C2×M16#6712yes—
176C22×C8—62no—
177C22⋊Q8—124no—
178C4×D4—61no—
179C4×D4—61no—
180C4×D4—61no—
181C4×D4—61no—
182C2×C4⋊C4—31no—
183C2×C4⋊C4—62no—
184C2×C4⋊C4—61no—
185C2×C22⋊C4—31no—
186C2×C22⋊C4—31no—
187C2×C22⋊C4—32no—
188C2×C22⋊C4—62no—
189C2×C42—31no—
190C4⋊C8—62no—
191#101—44no—
192#101—124no—
193C22⋊C8—21no—
194C22⋊C8—21no—
195C22⋊C8—21no—
196C22⋊C8—21no—
197C22⋊C8—62no—
198C22⋊C8—61no—
199C22⋊C8—61no—
200C22⋊C8—61no—
201C22⋊C8—61no—
202#97—62no—
203#95—62no—
204#95—62no—
205#95—61no—
206C23×C3—21no—
207C23×C3—21no—
208C23×C3—41no—
209C23×C3—41no—
210C23×C3C2×C22⋊C411yescharacteristic
211C23×C3C4×D411yescharacteristic
212C23×C3C4×D411yescharacteristic
213C22×D3—21no—
214C22×D3—21no—
215C22×D3—41no—
216C22×D3—41no—
217C22×D3#10112yescomplemented
218C3×Q8—44no—
219C3×Q8—44no—
220C3×Q8—44no—
221C3×Q8—44no—
222C2×C3×C4—22no—
223C2×C3×C4—22no—
224C2×C3×C4—22no—
225C2×C3×C4—24no—
226C2×C3×C4—24no—
227C2×C3×C4—24no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—42no—
232C2×C3×C4—42no—
233C2×C3×C4—42no—
234C2×C3×C4—42no—
235C2×C3×C4#12011yescharacteristic
236C2×C3×C4#12111yescharacteristic
237C2×C3×C4#12111yescharacteristic
238C2×C3×C4#12311yescharacteristic
239#67—42no—
240#67—42no—
241C2×Dic3—21no—
242C2×Dic3—21no—
243C2×Dic3—21no—
244C2×Dic3—21no—
245C2×Dic3—21no—
246C2×Dic3—41no—
247C2×Dic3—41no—
248C2×Dic3—42no—
249C2×Dic3—42no—
250C2×Dic3—42no—
251C4×D3—42no—
252C4×D3—81no—
253C3×C8—42no—
254C3×C8—81no—
255C3⋊C8—42no—
256C3⋊C8—81no—
257C24—31no—
258C24C2×C4×D311yescharacteristic
259C2×Q8—24no—
260C2×Q8—24no—
261C2×Q8—24no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—44no—
266C2×Q8—44no—
267C2×Q8—44no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×D4—62no—
273C2×D4—61no—
274C2×D4—61no—
275C2×D4—121no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—31no—
280C22×C4—31no—
281C22×C4—31no—
282C22×C4—44no—
283C22×C4—42no—
284C22×C4—61no—
285C22×C4—61no—
286C22×C4—62no—
287C22×C4—122no—
288C22×C4#23511yescharacteristic
289C22×C4D3×D411yescharacteristic
290C22×C4D3×D411yescharacteristic
291C22×C4D3×D411yescharacteristic
292C22×C4#23711yescharacteristic
293C22×C4C2×#6711yescharacteristic
294M16—42no—
295M16—42no—
296M16—122no—
297M16—122no—
298C2×C8—22no—
299C2×C8—41no—
300C2×C8—62no—
301C2×C8—121no—
302C2×C8—242no—
303C4⋊C4—61no—
304C4⋊C4—61no—
305C4⋊C4—124no—
306C4⋊C4—122no—
307C22⋊C4—61no—
308C22⋊C4—61no—
309C22⋊C4—61no—
310C22⋊C4—61no—
311C22⋊C4—124no—
312C22⋊C4—122no—
313C42—61no—
314C42—61no—
315C22×C3—21no—
316C22×C3—21no—
317C22×C3—21no—
318C22×C3—21no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—42no—
324C22×C3—42no—
325C22×C3#38611yescharacteristic
326C22×C3C2×#10111yescharacteristic
327C22×C3#41311yescharacteristic
328C2×D3—22no—
329C2×D3—41no—
330C2×D3—41no—
331C2×D3—42no—
332C2×D3—81no—
333C3×C4—21no—
334C3×C4—21no—
335C3×C4—21no—
336C3×C4—44no—
337C3×C4B(4,𝔽2)12yes—
338Dic3—41no—
339Dic3—41no—
340Dic3—41no—
341Dic3—82no—
342C23—21no—
343C23—21no—
344C23—32no—
345C23—41no—
346C23—41no—
347C23—61no—
348C23—61no—
349C23—121no—
350C23—121no—
351C23D3×C22⋊C411yescharacteristic
352C23#67⋊C411yescharacteristic
353C23C4×#6711yescharacteristic
354Q8—44no—
355Q8—44no—
356Q8—44no—
357Q8—44no—
358D4—122no—
359D4—122no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—61no—
374C2×C4—61no—
375C2×C4—61no—
376C2×C4—61no—
377C2×C4—61no—
378C2×C4—122no—
379C2×C4—121no—
380C2×C4—121no—
381C2×C4—122no—
382C2×C4—122no—
383C2×C4—122no—
384C2×C4—241no—
385C2×C4#88311yescharacteristic
386C2×C4#88411yescharacteristic
387C2×C4#93711yescharacteristic
388C2×C4#93811yescharacteristic
389C8—42no—
390C8—81no—
391C8—122no—
392C8—241no—
393C2×C3—21no—
394C2×C3—21no—
395C2×C3—41no—
396C2×C3—41no—
397C2×C3#188911yescharacteristic
398C2×C3#190111yescharacteristic
399C2×C3#191511yescharacteristic
400D3—42no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—42no—
410C22—42no—
411C22—62no—
412C22—122no—
413C22—121no—
414C22—121no—
415C22—241no—
416C22D3×#10111yescharacteristic
417C22#475111yescharacteristic
418C22#495611yescharacteristic
419C4—21no—
420C4—21no—
421C4—21no—
422C4—44no—
423C4—121no—
424C4—121no—
425C4—121no—
426C4—242no—
427C4#503212yes—
428C3#950511yescharacteristic, complemented, Sylow
429C2—21no—
430C2—21no—
431C2—41no—
432C2—41no—
433C2—122no—
434C2#6949911yescharacteristic
435C2#7011511yescharacteristic
436C2#7012411yescharacteristic
subgroupG/H
centerC22#4956
commutatorC24×C3C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1784C2
socleC22×C3#386
prime radicalC22×#67C2×C4
cosocleC23×C3×C4C23
hypercenter#1784D3
hypocenterC3#9505
LCSG ▷ C24×C3 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1784

Complements

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

NH ≅ G/Ntype
3. C3×#1784C2semidirect
4. #72133C2semidirect
6. #67685C2semidirect
7. #67684C2semidirect
33. #1784D3semidirect
217. C22×D3#101semidirect
428. C3#9505semidirect

Maximal subgroups

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

subgroupindex
#840682
#786322
C3×#17842
#721332
#699682
#676852
#676842
#95053