Finite Group Databasebeta

← #10692227

Subgroups — #10692227

Order 768.

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

normal subgroups only

This group has 520 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#82750C211yesmaximal, characteristic, complemented
2#82738C211yesmaximal, characteristic
3C3×#1508C211yesmaximal, characteristic, complemented
4#69978C211yesmaximal, characteristic
5#69456C211yesmaximal, characteristic
6#67684C211yesmaximal, characteristic, complemented
7#67616C211yesmaximal, characteristic, complemented
8#9344—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C2×#930C2211yescharacteristic
11C3×#407C2211yescharacteristic
12C3×C4×C22⋊C4C2211yescharacteristic
13#5229—21no—
14#5227—21no—
15#5212C2211yescharacteristic
16#5211C2211yescharacteristic
17#5207C2211yescharacteristic
18#5049—22no—
19#5046C412yescomplemented
20#5044C412yescomplemented
21#4984—21no—
22#4980—21no—
23#4980—21no—
24#4967—21no—
25C2×#806—21no—
26C2×#806—21no—
27#4951—21no—
28#4951—21no—
29C3×#328—22no—
30#4470—21no—
31#4470—21no—
32#3193—31no—
33#2378—31no—
34#1964—31no—
35#1510—31no—
36#1508D311yescharacteristic, complemented
37#1305—31no—
38#1284—31no—
39C22×C3×Q8D411yescharacteristic
40C23×C3×C4C2311yescharacteristic
41C22×#67—21no—
42C22×D12C2×C411yescharacteristic
43C2×C3×M16—21no—
44C3×C22⋊Q8—21no—
45C3×C22⋊Q8—21no—
46C3×C22⋊Q8C2×C412yes—
47C3×C22⋊Q8D412yes—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4D411yescharacteristic
50C2×C3×C22⋊C4—21no—
51C2×C3×C22⋊C4—21no—
52C2×C3×C42—21no—
53C2×#217—21no—
54C2×#217D411yescharacteristic
55C2×#217D411yescharacteristic
56C2×#215—22no—
57#930—21no—
58#930—21no—
59#930—21no—
60#930—21no—
61C2×(C2×D3)⋊C4—21no—
62C2×C4⋊Dic3—21no—
63C2×C4⋊Dic3—21no—
64C2×C4⋊Dic3—21no—
65C2×C4⋊Dic3C2×C411yescharacteristic
66C2×C4×Dic3—21no—
67C2×C4×Dic3—21no—
68C2×#208—21no—
69C3×C22⋊C8—21no—
70C3×C22⋊C8—21no—
71C3×#95—21no—
72#835—22no—
73C3⋊#95—21no—
74C3⋊#95—21no—
75C3⋊#95—21no—
76C3⋊#95—21no—
77C3⋊#95—21no—
78C3⋊#95—21no—
79C3⋊#95D411yescharacteristic
80C3⋊#95D411yescharacteristic
81#830—21no—
82#830C2×C412yes—
83#809—22no—
84#807—22no—
85#806—42no—
86#806—42no—
87C2×C22⋊Q8C2×D311yescharacteristic
88C2×#121—31no—
89#484—32no—
90#480—61no—
91#468—62no—
92#450—32no—
93#448—61no—
94C2×#99—61no—
95C2×#99—61no—
96#407—31no—
97#407C2×D311yescharacteristic
98#391—61no—
99#389—61no—
100#388—61no—
101#388—61no—
102#387—61no—
103#387—61no—
104#381—31no—
105#380—31no—
106C4×C22⋊C4C2×D311yescharacteristic
107#328—22no—
108#327—61no—
109#327—61no—
110C24×C3C2×D411yescharacteristic
111C23×D3—21no—
112C2×C3×Q8—21no—
113C2×C3×Q8—81no—
114C2×C3×Q8C22⋊C412yes—
115C22×C3×C4—21no—
116C22×C3×C4—21no—
117C22×C3×C4—41no—
118C22×C3×C4—81no—
119C22×C3×C4C22×C411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4#4111yescharacteristic
124C22×C3×C4#4111yescharacteristic
125C2×#67—41no—
126C2×#67—41no—
127C2×#67—41no—
128C2×#67—41no—
129C2×#67—41no—
130C2×#67—41no—
131C22×Dic3—21no—
132C22×Dic3—21no—
133C22×Dic3—21no—
134C22×Dic3—21no—
135C22×Dic3—21no—
136C2×D12—21no—
137C2×D12—21no—
138C2×D12—81no—
139C3×M16—81no—
140C2×C3×C8—41no—
141C3×C4⋊C4—22no—
142C3×C4⋊C4—22no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4C22⋊C412yes—
145C3×C22⋊C4—22no—
146C3×C22⋊C4—22no—
147C3×C22⋊C4—22no—
148C3×C22⋊C4—41no—
149C3×C42—81no—
150#217—41no—
151#217—41no—
152#217—81no—
153#215—82no—
154(C2×D3)⋊C4—41no—
155(C2×D3)⋊C4—41no—
156C4⋊Dic3—21no—
157C4⋊Dic3—21no—
158C4⋊Dic3—81no—
159C4⋊Dic3—81no—
160C4⋊Dic3—81no—
161C4×Dic3—81no—
162C4×Dic3—81no—
163#208—81no—
164C2×C3⋊C8—22no—
165C22×Q8D1211yescharacteristic
166C22×D4—31no—
167C22×D4—61no—
168C23×C4C22×D311yescharacteristic
169C2×SD16—62no—
170C2×M16—21no—
171C2×M16—61no—
172C22⋊Q8—21no—
173C22⋊Q8—21no—
174C22⋊Q8C4×D312yes—
175C22⋊Q8#6712yes—
176#121—61no—
177#121—61no—
178#121—61no—
179#121—61no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—31no—
182C2×C4⋊C4—61no—
183C2×C4⋊C4—61no—
184C2×C4⋊C4—61no—
185C2×C4⋊C4D1211yescharacteristic
186C2×C22⋊C4—21no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—31no—
189C2×C22⋊C4—31no—
190C2×C22⋊C4—61no—
191C2×C22⋊C4—61no—
192C2×C42—21no—
193C2×C42—61no—
194C2×C42—61no—
195#107—62no—
196#103—62no—
197#102—62no—
198#99—122no—
199#99—122no—
200C22⋊C8—21no—
201C22⋊C8—21no—
202C22⋊C8—32no—
203C22⋊C8—61no—
204#95—21no—
205#95—31no—
206#95—31no—
207#95—61no—
208#95—61no—
209#95—61no—
210#95—61no—
211#95—61no—
212#95—61no—
213C23×C3—21no—
214C23×C3—21no—
215C23×C3—41no—
216C23×C3—41no—
217C23×C3C4×D411yescharacteristic
218C23×C3#12011yescharacteristic
219C23×C3#12111yescharacteristic
220C22×D3—41no—
221C22×D3—41no—
222C22×D3—41no—
223C22×D3—81no—
224C22×D3—81no—
225C3×Q8—42no—
226C3×Q8—81no—
227C2×C3×C4—22no—
228C2×C3×C4—22no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—42no—
240C2×C3×C4—81no—
241C2×C3×C4—81no—
242C2×C3×C4C2×C22⋊C411yescharacteristic
243C2×C3×C4C4×D411yescharacteristic
244C2×C3×C4#12111yescharacteristic
245C2×C3×C4#12311yescharacteristic
246#67—81no—
247#67—81no—
248#67—81no—
249#67—81no—
250C2×Dic3—41no—
251C2×Dic3—41no—
252C2×Dic3—41no—
253C2×Dic3—41no—
254C2×Dic3—41no—
255C2×Dic3—41no—
256C2×Dic3—41no—
257C2×Dic3—41no—
258C2×Dic3—41no—
259C2×Dic3—41no—
260C2×Dic3—41no—
261C2×Dic3—41no—
262C2×Dic3—41no—
263C2×Dic3—41no—
264C2×Dic3—41no—
265D12—42no—
266D12—81no—
267C3×C8—81no—
268C3⋊C8—42no—
269C24—61no—
270C24D3×D411yescharacteristic
271C2×Q8—21no—
272C2×Q8—81no—
273C2×Q8(C2×D3)⋊C412yes—
274C2×D4—61no—
275C2×D4—61no—
276C2×D4—121no—
277C2×D4—121no—
278C2×D4—121no—
279C2×D4—121no—
280C2×D4—121no—
281C2×D4—121no—
282C2×D4—241no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—41no—
286C22×C4—61no—
287C22×C4—61no—
288C22×C4—61no—
289C22×C4—61no—
290C22×C4—61no—
291C22×C4—81no—
292C22×C4C2×C4×D311yescharacteristic
293C22×C4C2×D1211yescharacteristic
294C22×C4D3×D411yescharacteristic
295C22×C4#23711yescharacteristic
296C22×C4#23711yescharacteristic
297C22×C4C2×#6711yescharacteristic
298SD16—242no—
299M16—81no—
300M16—241no—
301C2×C8—41no—
302C2×C8—62no—
303C4⋊C4—22no—
304C4⋊C4—21no—
305C4⋊C4—22no—
306C4⋊C4—61no—
307C4⋊C4—61no—
308C4⋊C4—241no—
309C4⋊C4—241no—
310C4⋊C4—241no—
311C4⋊C4(C2×D3)⋊C412yes—
312C22⋊C4—22no—
313C22⋊C4—22no—
314C22⋊C4—22no—
315C22⋊C4—41no—
316C22⋊C4—121no—
317C22⋊C4—121no—
318C22⋊C4—121no—
319C22⋊C4—121no—
320C22⋊C4—241no—
321C42—81no—
322C42—241no—
323C42—241no—
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—21no—
327C22×C3—21no—
328C22×C3—21no—
329C22×C3—21no—
330C22×C3—21no—
331C22×C3—21no—
332C22×C3—81no—
333C22×C3—81no—
334C22×C3#38611yescharacteristic
335C22×C3#45311yescharacteristic
336C22×C3#45411yescharacteristic
337C2×D3—81no—
338C2×D3—81no—
339C2×D3—81no—
340C2×D3—161no—
341C2×D3—161no—
342C3×C4—21no—
343C3×C4—21no—
344C3×C4—21no—
345C3×C4—21no—
346C3×C4—42no—
347C3×C4—42no—
348C3×C4—82no—
349Dic3—81no—
350Dic3—81no—
351Dic3—81no—
352Dic3—81no—
353Dic3—81no—
354C23—21no—
355C23—21no—
356C23—41no—
357C23—41no—
358C23—121no—
359C23—121no—
360C23—121no—
361C23—241no—
362C23—241no—
363C23#67⋊C411yescharacteristic
364C23#88211yescharacteristic
365C23#93911yescharacteristic
366Q8—42no—
367Q8—81no—
368D4—122no—
369D4—241no—
370D4—241no—
371D4—241no—
372D4—241no—
373D4—241no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—121no—
390C2×C4—121no—
391C2×C4—121no—
392C2×C4—121no—
393C2×C4—121no—
394C2×C4—121no—
395C2×C4—121no—
396C2×C4—121no—
397C2×C4—121no—
398C2×C4—121no—
399C2×C4—121no—
400C2×C4—121no—
401C2×C4—121no—
402C2×C4—121no—
403C2×C4—121no—
404C2×C4#67⋊C411yescharacteristic
405C2×C4#88611yescharacteristic
406C2×C4C2×(C2×D3)⋊C411yescharacteristic
407C2×C4#93911yescharacteristic
408C8—81no—
409C8—122no—
410C2×C3—21no—
411C2×C3—21no—
412C2×C3—41no—
413C2×C3—41no—
414C2×C3#188311yescharacteristic
415C2×C3#190511yescharacteristic
416C2×C3#191311yescharacteristic
417D3—161no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—81no—
427C22—81no—
428C22—241no—
429C22—241no—
430C22—241no—
431C22—481no—
432C22—481no—
433C22#482211yescharacteristic
434C22#482711yescharacteristic
435C22#495811yescharacteristic
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—42no—
441C4—42no—
442C4—82no—
443C4—241no—
444C4—241no—
445C4—241no—
446C4—241no—
447C4—241no—
448C3#934411yescharacteristic, complemented, Sylow
449C2—21no—
450C2—21no—
451C2—41no—
452C2—41no—
453C2—481no—
454C2#7002411yescharacteristic
455C2#7033411yescharacteristic
456C2#7036611yescharacteristic
subgroupG/H
centerC22#4958
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1508C2
socleC22×C3#386
prime radicalC2×#930C22
cosocleC23×C3×C4C23
hypercenter#1508D3
hypocenterC3#9344
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1508

Complements

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

NH ≅ G/Ntype
1. #82750C2semidirect
3. C3×#1508C2semidirect
6. #67684C2semidirect
7. #67616C2semidirect
19. #5046C4semidirect
20. #5044C4semidirect
36. #1508D3semidirect
448. C3#9344semidirect

Maximal subgroups

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

subgroupindex
#827502
#827382
C3×#15082
#699782
#694562
#676842
#676162
#93443