Finite Group Databasebeta

← #10711963

Subgroups — #10711963

Order 768.

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

normal subgroups only

This group has 504 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
1#82840C211yesmaximal, characteristic
2C2×C3×#352C211yesmaximal, characteristic
3#71704C211yesmaximal, characteristic
4#70068C211yesmaximal, characteristic
5#70021C211yesmaximal, characteristic
6#67698C211yesmaximal, characteristic
7#67695C211yesmaximal, characteristic
8#12961—31nomaximal, Sylow
9C2×C3×#123C2211yescharacteristic
10#5798C2211yescharacteristic
11C2×C3×#100C2211yescharacteristic
12C2×C3×#99C2211yescharacteristic
13#5221C2211yescharacteristic
14#5220C2211yescharacteristic
15C2×#834C2211yescharacteristic
16#5212—21no—
17#5211—21no—
18#5121—21no—
19#4984—21no—
20#4980—21no—
21#4951—21no—
22#4951—21no—
23#4950—21no—
24#4950—21no—
25C3×#352C414yescomplemented
26#4555—21no—
27#4527—21no—
28#4527—21no—
29#2376—31no—
30C2×#352D311yescharacteristic
31#1890—31no—
32#1867—31no—
33#1856—31no—
34#1393—31no—
35#1390—31no—
36C22×C3×D4C2311yescharacteristic
37C23×C3×C4D411yescharacteristic
38C23×Dic3D411yescharacteristic
39C2×C3×M16—21no—
40C3×#123—21no—
41C3×#123—21no—
42C3×#123C2×C412yes—
43C3×#123Q812yes—
44C2×C3×C4⋊C4—21no—
45C2×C3×C22⋊C4—21no—
46C2×C3×C22⋊C4—21no—
47C2×C3×C22⋊C4D411yescharacteristic
48C2×#217—21no—
49C2×#217—21no—
50C2×#217—21no—
51C2×#217—21no—
52C2×#217—21no—
53C2×#217—21no—
54C2×#217D411yescharacteristic
55D4×Dic3—41no—
56D4×Dic3—41no—
57#926—21no—
58C2×C4⋊Dic3—21no—
59C2×C4⋊Dic3—21no—
60C2×C4⋊Dic3—21no—
61C2×C4⋊Dic3—41no—
62C2×C4⋊Dic3D411yescharacteristic
63C2×C4×Dic3—21no—
64C2×C4×Dic3D411yescharacteristic
65C3×#100—21no—
66C3×#100C2×C412yes—
67C3×#99—21no—
68C3×#99C2×C412yes—
69#834—21no—
70#834—21no—
71C3⋊#95—21no—
72C3⋊#95—21no—
73C3⋊#95—21no—
74C3⋊#95—21no—
75C3⋊#95—21no—
76C3⋊#95—41no—
77C3⋊#95—41no—
78C3⋊#95—41no—
79#818—41no—
80#817—41no—
81C2×#123C2×D311yescharacteristic
82#518—31no—
83#428—61no—
84C2×#100C2×D311yescharacteristic
85C2×#99—31no—
86C2×#99C2×D311yescharacteristic
87#390—31no—
88#389—61no—
89#388—61no—
90#388—61no—
91#387—61no—
92#386—31no—
93#385—61no—
94#385—61no—
95#381—61no—
96#380—61no—
97#352Dic314yescomplemented
98#342—61no—
99#342—61no—
100#339—61no—
101C24×C3C2×D411yescharacteristic
102C24×C3C2×D411yescharacteristic
103C2×C3×D4—41no—
104C2×C3×D4—41no—
105C2×C3×D4C22×C411yescharacteristic
106C2×C3×D4C2×Q811yescharacteristic
107C2×C3×D4#4111yescharacteristic
108C2×C3×D4#4111yescharacteristic
109C22×C3×C4—21no—
110C22×C3×C4—21no—
111C22×C3×C4—21no—
112C22×C3×C4—41no—
113C22×C3×C4—41no—
114C22×C3×C4—41no—
115C22×C3×C4C4⋊C412yescomplemented
116C22×C3×C4C2×D411yescharacteristic
117C22×Dic3—21no—
118C22×Dic3—21no—
119C22×Dic3—21no—
120C22×Dic3—21no—
121C22×Dic3—21no—
122C22×Dic3—21no—
123C22×Dic3—21no—
124C22×Dic3—21no—
125C22×Dic3—41no—
126C22×Dic3—41no—
127C22×Dic3—81no—
128C3×M16—22no—
129C3×M16—41no—
130C2×C3×C8—41no—
131C3×C4⋊C4—22no—
132C3×C4⋊C4—22no—
133C3×C22⋊C4—22no—
134C3×C22⋊C4—22no—
135C3×C22⋊C4—21no—
136C3×C22⋊C4—22no—
137C3×C22⋊C4—41no—
138C3×C22⋊C4C4⋊C412yescomplemented
139#217—41no—
140#217—41no—
141#217—41no—
142#217—41no—
143#217—41no—
144#217—41no—
145#217—81no—
146#217—81no—
147#217—81no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150C4⋊Dic3—41no—
151C4⋊Dic3—81no—
152C4⋊Dic3—81no—
153C4⋊Dic3—81no—
154C4⋊Dic3—82no—
155C4×Dic3—41no—
156C4×Dic3—41no—
157C4×Dic3—81no—
158C22×D4C22×D311yescharacteristic
159C23×C4—31no—
160C23×C4D1211yescharacteristic
161C2×M16—21no—
162#123—21no—
163#123—21no—
164#123Dic612yes—
165#123C2×Dic312yes—
166C4×D4—121no—
167C4×D4—121no—
168#117—61no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—31no—
171C2×C4⋊C4—61no—
172C2×C4⋊C4—61no—
173C2×C4⋊C4—61no—
174C2×C4⋊C4—121no—
175C2×C22⋊C4—21no—
176C2×C22⋊C4—21no—
177C2×C22⋊C4—31no—
178C2×C22⋊C4—61no—
179C2×C22⋊C4—61no—
180C2×C22⋊C4—61no—
181C2×C22⋊C4—61no—
182C2×C22⋊C4—61no—
183C2×C22⋊C4—61no—
184C2×C22⋊C4D1211yescharacteristic
185C2×C42—31no—
186C2×C42—61no—
187#107—121no—
188#106—121no—
189#100—21no—
190#100C2×Dic312yes—
191#99—21no—
192#99—61no—
193#99—61no—
194#99C2×Dic312yes—
195#95—61no—
196#95—61no—
197#95—61no—
198#95—61no—
199#95—61no—
200#95—121no—
201#95—121no—
202#95—121no—
203C23×C3—21no—
204C23×C3—21no—
205C23×C3—21no—
206C23×C3—21no—
207C23×C3—41no—
208C23×C3—41no—
209C23×C3—41no—
210C23×C3—41no—
211C23×C3C2×C4⋊C411yescharacteristic
212C23×C3C4×D411yescharacteristic
213C23×C3#12011yescharacteristic
214C23×C3C22⋊Q811yescharacteristic
215C23×C3#12311yescharacteristic
216C3×D4—41no—
217C3×D4—41no—
218C3×D4—41no—
219C3×D4—41no—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—22no—
223C2×C3×C4—22no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—42no—
229C2×C3×C4—42no—
230C2×C3×C4—81no—
231C2×C3×C4—81no—
232C2×C3×C4—81no—
233C2×C3×C4C4×D411yescharacteristic
234C2×C3×C4C22⋊Q811yescharacteristic
235C2×Dic3—41no—
236C2×Dic3—41no—
237C2×Dic3—41no—
238C2×Dic3—41no—
239C2×Dic3—41no—
240C2×Dic3—41no—
241C2×Dic3—41no—
242C2×Dic3—41no—
243C2×Dic3—42no—
244C2×Dic3—41no—
245C2×Dic3—41no—
246C2×Dic3—41no—
247C2×Dic3—41no—
248C2×Dic3—41no—
249C2×Dic3—41no—
250C2×Dic3—81no—
251C2×Dic3—81no—
252C2×Dic3—81no—
253C2×Dic3—81no—
254C2×Dic3—81no—
255C2×Dic3—81no—
256C2×Dic3—81no—
257C2×Dic3—81no—
258C3×C8—42no—
259C24C2×D1211yescharacteristic
260C24D3×D411yescharacteristic
261C2×D4—41no—
262C2×D4—41no—
263C2×D4C2×Dic611yescharacteristic
264C2×D4#23711yescharacteristic
265C2×D4#23711yescharacteristic
266C2×D4C22×Dic311yescharacteristic
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—61no—
274C22×C4—61no—
275C22×C4—61no—
276C22×C4—61no—
277C22×C4—61no—
278C22×C4—61no—
279C22×C4—61no—
280C22×C4—61no—
281C22×C4—121no—
282C22×C4—121no—
283C22×C4—241no—
284C22×C4C4⋊Dic312yescomplemented
285C22×C4D3×D411yescharacteristic
286M16—22no—
287M16—41no—
288C2×C8—41no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—61no—
292C4⋊C4—61no—
293C4⋊C4—121no—
294C4⋊C4—241no—
295C4⋊C4—241no—
296C4⋊C4—241no—
297C4⋊C4—242no—
298C22⋊C4—22no—
299C22⋊C4—21no—
300C22⋊C4—22no—
301C22⋊C4—22no—
302C22⋊C4—41no—
303C22⋊C4—121no—
304C22⋊C4—121no—
305C22⋊C4—121no—
306C22⋊C4—121no—
307C22⋊C4—121no—
308C22⋊C4—121no—
309C22⋊C4—241no—
310C22⋊C4—241no—
311C22⋊C4—241no—
312C22⋊C4C4⋊Dic312yescomplemented
313C42—121no—
314C42—121no—
315C42—241no—
316C22×C3—21no—
317C22×C3—21no—
318C22×C3—21no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—21no—
322C22×C3—41no—
323C22×C3—41no—
324C22×C3—41no—
325C22×C3—41no—
326C22×C3—41no—
327C22×C3—41no—
328C22×C3—41no—
329C22×C3—41no—
330C22×C3—81no—
331C22×C3—81no—
332C22×C3#38511yescharacteristic
333C22×C3B(4,𝔽2)11yescharacteristic
334C22×C3#45811yescharacteristic
335C3×C4—21no—
336C3×C4—21no—
337C3×C4—42no—
338C3×C4—42no—
339C3×C4—82no—
340Dic3—81no—
341Dic3—81no—
342Dic3—81no—
343Dic3—81no—
344Dic3—81no—
345Dic3—81no—
346Dic3—161no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23#87811yescharacteristic
356C23#88211yescharacteristic
357C23#88611yescharacteristic
358C23C2×C4⋊Dic311yescharacteristic
359C23D4×Dic311yescharacteristic
360D4—41no—
361D4—41no—
362D4—41no—
363D4—41no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—121no—
378C2×C4—121no—
379C2×C4—121no—
380C2×C4—121no—
381C2×C4—121no—
382C2×C4—121no—
383C2×C4—121no—
384C2×C4—121no—
385C2×C4—122no—
386C2×C4—121no—
387C2×C4—121no—
388C2×C4—121no—
389C2×C4—121no—
390C2×C4—121no—
391C2×C4—121no—
392C2×C4—241no—
393C2×C4—241no—
394C2×C4—241no—
395C2×C4—241no—
396C2×C4—241no—
397C2×C4—241no—
398C2×C4—241no—
399C2×C4—241no—
400C2×C4#87811yescharacteristic
401C2×C4D4×Dic311yescharacteristic
402C8—42no—
403C2×C3—21no—
404C2×C3—21no—
405C2×C3—41no—
406C2×C3—41no—
407C2×C3—41no—
408C2×C3—41no—
409C2×C3#186711yescharacteristic
410C2×C3#220011yescharacteristic
411C2×C3#220111yescharacteristic
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—81no—
427C22—81no—
428C22#474111yescharacteristic
429C22#474211yescharacteristic
430C22#495011yescharacteristic
431C4—21no—
432C4—21no—
433C4—42no—
434C4—42no—
435C4—82no—
436C4—241no—
437C4—241no—
438C4—241no—
439C4—241no—
440C4—241no—
441C4—241no—
442C4—481no—
443C3#1296111yescharacteristic, complemented, Sylow
444C2—21no—
445C2—21no—
446C2—41no—
447C2—41no—
448C2—41no—
449C2—41no—
450C2#6913911yescharacteristic
451C2#6914111yescharacteristic
452C2#7002111yescharacteristic
subgroupG/H
centerC22#4741
commutatorC2×C3×D4C22×C4
FrattiniC22×D4C22×D3
FittingC2×C3×#352C2
socleC22×C3B(4,𝔽2)
prime radical, cosocleC22×C3×D4C23
hypercenterC2×#352D3
hypocenterC3#12961
LCSG ▷ C2×C3×D4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×#352

Complements

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

NH ≅ G/Ntype
25. C3×#352C4semidirect
97. #352Dic3semidirect
115. C22×C3×C4C4⋊C4semidirect
138. C3×C22⋊C4C4⋊C4semidirect
284. C22×C4C4⋊Dic3semidirect
312. C22⋊C4C4⋊Dic3semidirect
443. C3#12961semidirect

Maximal subgroups

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

subgroupindex
#828402
C2×C3×#3522
#717042
#700682
#700212
#676982
#676952
#129613