Finite Group Databasebeta

← #10693250

Subgroups — #10693250

Order 768.

538 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 103 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×#3094C211yesmaximal, characteristic
2#82635C211yesmaximal, characteristic
3#71674C211yesmaximal, characteristic
4#71594C211yesmaximal, characteristic, complemented
5#71590C211yesmaximal, characteristic, complemented
6#67686C211yesmaximal, characteristic, complemented
7#67682C211yesmaximal, characteristic, complemented
8#9525—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C2×C3×#117C2211yescharacteristic
11C3×#482—22no—
12C3×#481C412yescomplemented
13C3×#480C412yescomplemented
14C3×#428—21no—
15C3×#418—21no—
16C3×#407C2211yescharacteristic
17#5221—21no—
18#5220—21no—
19#5220—21no—
20#5211C2211yescharacteristic
21#5211C2211yescharacteristic
22C4×#217C2211yescharacteristic
23#5207C2211yescharacteristic
24#4980—21no—
25#4951—21no—
26Dic3×C22⋊C4—21no—
27#4537—22no—
28#4532—21no—
29#4532—21no—
30#4526—21no—
31#4526—21no—
32#3094D311yescharacteristic
33#2333—31no—
34#1826—31no—
35#1820—31no—
36#1510—31no—
37#1306—31no—
38#1284—31no—
39C22×C3×D4—21no—
40C22×C3×D4D411yescharacteristic
41C23×C3×C4C2311yescharacteristic
42C23×Dic3—21no—
43C2×C3×M16—21no—
44C3×#121—21no—
45C3×#121—21no—
46C3×#121C2×C412yes—
47C3×#121D412yes—
48C3×#117—21no—
49C3×#117—21no—
50C3×#117—21no—
51C3×#117—21no—
52C2×C3×C4⋊C4C2×C411yescharacteristic, complemented
53C2×C3×C4⋊C4C2×C411yescharacteristic, complemented
54C2×C3×C4⋊C4D411yescharacteristic
55C2×C3×C22⋊C4—21no—
56C2×C3×C22⋊C4—21no—
57C2×C3×C42—21no—
58C2×#217—21no—
59C2×#217—21no—
60C2×#217—21no—
61C2×#217—21no—
62C2×#217—21no—
63C2×#217D411yescharacteristic
64C2×#217D411yescharacteristic
65C2×C4⋊Dic3—21no—
66C2×C4×Dic3—21no—
67C2×C4×Dic3D411yescharacteristic
68C2×C4×Dic3D411yescharacteristic
69C2×#208—21no—
70C3×#107—22no—
71C3×#106—22no—
72C3×#102—22no—
73C3×#102—22no—
74C3×C22⋊C8—21no—
75C3×C22⋊C8C2×C412yescomplemented
76C3⋊#95—21no—
77C3⋊#95—21no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80C3⋊#95—21no—
81C3⋊#95—21no—
82C3⋊#95—21no—
83#830—21no—
84#830—21no—
85C2×#121C2×D311yescharacteristic
86C2×#117C2×D311yescharacteristic
87#482—22no—
88#481Dic312yescomplemented
89#480Dic312yescomplemented
90#428—21no—
91#418—21no—
92#407—31no—
93#407C2×D311yescharacteristic
94#390—61no—
95#388—61no—
96#387—61no—
97#386—61no—
98#386—61no—
99#381—31no—
100#381—31no—
101C4×C22⋊C4—31no—
102C4×C22⋊C4—61no—
103#342—61no—
104#342—61no—
105#339—61no—
106#339—61no—
107#327—62no—
108C24×C3—21no—
109C24×C3C2×D411yescharacteristic
110C2×C3×D4—22no—
111C2×C3×D4—22no—
112C2×C3×D4—21no—
113C2×C3×D4—41no—
114C2×C3×D4—41no—
115C2×C3×D4—81no—
116C2×C3×D4—81no—
117C2×C3×D4C22⋊C412yes—
118C22×C3×C4—21no—
119C22×C3×C4—21no—
120C22×C3×C4—21no—
121C22×C3×C4—41no—
122C22×C3×C4—41no—
123C22×C3×C4C22×C411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4#4111yescharacteristic
128C22×C3×C4#4111yescharacteristic
129C22×Dic3—21no—
130C22×Dic3—21no—
131C22×Dic3—21no—
132C22×Dic3—21no—
133C22×Dic3—42no—
134C22×Dic3—41no—
135C22×Dic3—81no—
136C22×Dic3—81no—
137C3×M16—41no—
138C3×M16—41no—
139C2×C3×C8—22no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4C22⋊C412yes—
146C3×C22⋊C4—22no—
147C3×C22⋊C4—22no—
148C3×C22⋊C4—41no—
149C3×C22⋊C4—41no—
150C3×C42—41no—
151C3×C42—41no—
152#217—42no—
153#217—42no—
154#217—41no—
155#217—41no—
156#217—41no—
157#217—41no—
158#217—81no—
159#217—81no—
160C4⋊Dic3—81no—
161C4×Dic3—41no—
162C4×Dic3—41no—
163C4×Dic3—81no—
164#208—41no—
165#208—41no—
166C2×C3⋊C8—41no—
167C22×D4—21no—
168C22×D4#6711yescharacteristic
169C23×C4—61no—
170C23×C4C22×D311yescharacteristic
171C2×M16—21no—
172C2×M16—61no—
173#121—21no—
174#121—21no—
175#121C2×Dic312yes—
176#121#6712yes—
177#117—21no—
178#117—21no—
179#117—21no—
180#117—21no—
181C2×C4⋊C4—61no—
182C2×C4⋊C4C2×Dic311yescharacteristic, complemented
183C2×C4⋊C4C2×Dic311yescharacteristic, complemented
184C2×C4⋊C4#6711yescharacteristic
185C2×C22⋊C4—21no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—31no—
188C2×C22⋊C4—31no—
189C2×C22⋊C4—61no—
190C2×C22⋊C4—61no—
191C2×C22⋊C4—61no—
192C2×C22⋊C4—61no—
193C2×C22⋊C4—61no—
194C2×C42—21no—
195C2×C42—31no—
196C2×C42—31no—
197C2×C42—61no—
198#107—22no—
199#106—22no—
200#102—22no—
201#102—22no—
202C22⋊C8—21no—
203C22⋊C8—61no—
204C22⋊C8—61no—
205C22⋊C8C2×Dic312yescomplemented
206#95—61no—
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—22no—
216C23×C3—41no—
217C23×C3—41no—
218C23×C3—41no—
219C23×C3—41no—
220C23×C3—41no—
221C23×C3—41no—
222C23×C3—41no—
223C23×C3—41no—
224C23×C3C4×D411yescharacteristic
225C23×C3#12011yescharacteristic
226C23×C3#12111yescharacteristic
227C3×D4—42no—
228C3×D4—81no—
229C3×D4—82no—
230C3×D4—82no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—22no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—81no—
245C2×C3×C4—81no—
246C2×C3×C4C2×C22⋊C411yescharacteristic
247C2×C3×C4C4×D411yescharacteristic
248C2×C3×C4#12111yescharacteristic
249C2×C3×C4#12311yescharacteristic
250C2×Dic3—22no—
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—84no—
263C2×Dic3—81no—
264C2×Dic3—81no—
265C2×Dic3—81no—
266C3×C8—42no—
267C3⋊C8—81no—
268C24—21no—
269C24D3×D411yescharacteristic
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—21no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—81no—
276C2×D4—81no—
277C2×D4#21712yes—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—61no—
284C22×C4—61no—
285C22×C4—61no—
286C22×C4—61no—
287C22×C4—122no—
288C22×C4—121no—
289C22×C4—241no—
290C22×C4—241no—
291C22×C4D3×D411yescharacteristic
292C22×C4#23711yescharacteristic
293C22×C4#23711yescharacteristic
294C22×C4C22×Dic311yescharacteristic
295C22×C4C2×#6711yescharacteristic
296C22×C4C2×#6711yescharacteristic
297M16—41no—
298M16—41no—
299M16—121no—
300M16—121no—
301C2×C8—22no—
302C2×C8—121no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—241no—
309C4⋊C4#21712yes—
310C22⋊C4—22no—
311C22⋊C4—22no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—122no—
315C22⋊C4—122no—
316C22⋊C4—121no—
317C22⋊C4—121no—
318C22⋊C4—121no—
319C22⋊C4—121no—
320C22⋊C4—241no—
321C22⋊C4—241no—
322C42—41no—
323C42—41no—
324C42—121no—
325C42—121no—
326C42—241no—
327C22×C3—21no—
328C22×C3—21no—
329C22×C3—21no—
330C22×C3—21no—
331C22×C3—21no—
332C22×C3—21no—
333C22×C3—21no—
334C22×C3—21no—
335C22×C3—42no—
336C22×C3—42no—
337C22×C3—42no—
338C22×C3—81no—
339C22×C3—81no—
340C22×C3—81no—
341C22×C3—81no—
342C22×C3—81no—
343C22×C3—81no—
344C22×C3#38611yescharacteristic
345C22×C3#45811yescharacteristic
346C22×C3#45811yescharacteristic
347C3×C4—21no—
348C3×C4—21no—
349C3×C4—21no—
350C3×C4—21no—
351C3×C4—42no—
352C3×C4—81no—
353C3×C4—81no—
354Dic3—42no—
355Dic3—81no—
356Dic3—81no—
357Dic3—81no—
358Dic3—161no—
359C23—21no—
360C23—21no—
361C23—22no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23D4×Dic311yescharacteristic
371C23#93711yescharacteristic
372C23#93911yescharacteristic
373D4—42no—
374D4—81no—
375D4—82no—
376D4—82no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—22no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—62no—
391C2×C4—81no—
392C2×C4—81no—
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—244no—
405C2×C4—241no—
406C2×C4—241no—
407C2×C4—241no—
408C2×C4D4×Dic311yescharacteristic
409C2×C4#93511yescharacteristic
410C2×C4#93911yescharacteristic
411C2×C4C2×#21711yescharacteristic
412C8—42no—
413C8—241no—
414C2×C3—21no—
415C2×C3—21no—
416C2×C3—41no—
417C2×C3—41no—
418C2×C3—82no—
419C2×C3#189011yescharacteristic
420C2×C3#190411yescharacteristic
421C2×C3#190611yescharacteristic
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—42no—
431C22—42no—
432C22—42no—
433C22—81no—
434C22—81no—
435C22—81no—
436C22—81no—
437C22—81no—
438C22—81no—
439C22#505311yescharacteristic
440C22#505311yescharacteristic
441C22#522011yescharacteristic
442C4—21no—
443C4—21no—
444C4—21no—
445C4—21no—
446C4—42no—
447C4—81no—
448C4—81no—
449C4—122no—
450C4—241no—
451C4—241no—
452C4—241no—
453C4—481no—
454C3#952511yescharacteristic, complemented, Sylow
455C2—21no—
456C2—21no—
457C2—41no—
458C2—41no—
459C2—82no—
460C2#7170411yescharacteristic
461C2#7208911yescharacteristic
462C2#7211411yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#3094C2
socleC22×C3#386
prime radicalC2×C3×#121C22
cosocleC23×C3×C4C23
hypercenter#3094D3
hypocenterC3#9525
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3094

Complements

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

NH ≅ G/Ntype
4. #71594C2semidirect
5. #71590C2semidirect
6. #67686C2semidirect
7. #67682C2semidirect
12. C3×#481C4semidirect
13. C3×#480C4semidirect
52. C2×C3×C4⋊C4C2×C4semidirect
53. C2×C3×C4⋊C4C2×C4semidirect
75. C3×C22⋊C8C2×C4semidirect
88. #481Dic3semidirect
89. #480Dic3semidirect
182. C2×C4⋊C4C2×Dic3semidirect
183. C2×C4⋊C4C2×Dic3semidirect
205. C22⋊C8C2×Dic3semidirect
454. C3#9525semidirect

Maximal subgroups

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

subgroupindex
C3×#30942
#826352
#716742
#715942
#715902
#676862
#676822
#95253