Finite Group Databasebeta

← #10691237

Subgroups — #10691237

Order 768.

612 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 105 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#82822C211yesmaximal, characteristic
2#74004C211yesmaximal, characteristic
3C3×#1520C211yesmaximal, characteristic, complemented
4#71690C211yesmaximal, characteristic, complemented
5C2×#4468C211yesmaximal, characteristic
6#67676C211yesmaximal, characteristic, complemented
7#67625C211yesmaximal, characteristic, complemented
8#9201—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C22×(C2×D3)⋊C4C2211yescharacteristic
11C2×C3×C22⋊C8C2211yescharacteristic
12C3×#380C2211yescharacteristic
13#5231C412yescomplemented
14#5212C2211yescharacteristic
15#5212C2211yescharacteristic
16C2×#830C2211yescharacteristic
17#4994—22no—
18#4993C412yescomplemented
19#4990—22no—
20#4955—21no—
21#4955—21no—
22#4946—21no—
23Dic3×C22⋊C4—21no—
24#4670—22no—
25C3×#328—22no—
26#4538—22no—
27#4468—22no—
28#2626—31no—
29#2312—31no—
30#1520—31no—
31#1520D311yescharacteristic, complemented
32C2×#323—31no—
33#1289—31no—
34#1289—31no—
35C24×D3D411yescharacteristic
36C22×C3×Q8C2×C411yescharacteristic
37C23×C3×C4C2311yescharacteristic
38C23×Dic3D411yescharacteristic
39C22×C3×C8—21no—
40C3×C22⋊Q8—21no—
41C3×C22⋊Q8—21no—
42C3×C22⋊Q8—21no—
43C3×C22⋊Q8—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4C2×C411yescharacteristic
47C2×C3×C22⋊C4—21no—
48C2×C3×C22⋊C4—21no—
49C2×#217—21no—
50C2×#217—21no—
51C2×(C2×D3)⋊C4—21no—
52C2×(C2×D3)⋊C4—21no—
53C2×(C2×D3)⋊C4—21no—
54C2×(C2×D3)⋊C4—21no—
55C2×(C2×D3)⋊C4—22no—
56C2×(C2×D3)⋊C4—22no—
57C2×(C2×D3)⋊C4—22no—
58C2×(C2×D3)⋊C4—22no—
59C2×(C2×D3)⋊C4C2×C412yes—
60C2×(C2×D3)⋊C4D412yes—
61C2×C4⋊Dic3—21no—
62C2×C4⋊Dic3—21no—
63C2×Dic3⋊C4—22no—
64C2×C4×Dic3—22no—
65C22×C3⋊C8—21no—
66C3×C22⋊C8—21no—
67C3×C22⋊C8—21no—
68C3×#95D412yes—
69C3⋊#95—22no—
70C3⋊#95—22no—
71C3⋊#95C2×C412yes—
72C3⋊#95D412yes—
73#830—21no—
74#830—21no—
75C2×C22⋊Q8C2×D311yescharacteristic
76C22×C22⋊C4—31no—
77C2×C22⋊C8—31no—
78C2×C22⋊C8C2×D311yescharacteristic
79#399—32no—
80#396—62no—
81#393—32no—
82#388—62no—
83#388—62no—
84#385—61no—
85#380—31no—
86#380—31no—
87#380C2×D311yescharacteristic
88#379—61no—
89#379—61no—
90C4×C22⋊C4—61no—
91#328—22no—
92#328—62no—
93#323—62no—
94C24×C3C2×D411yescharacteristic
95C23×D3—21no—
96C23×D3—21no—
97C23×D3—21no—
98C23×D3—21no—
99C23×D3—21no—
100C23×D3—21no—
101C23×D3—81no—
102C23×D3—81no—
103C23×D3C22⋊C412yes—
104C2×C3×Q8—21no—
105C2×C3×Q8—21no—
106C2×C3×Q8—81no—
107C22×C3×C4—21no—
108C22×C3×C4—21no—
109C22×C3×C4—22no—
110C22×C3×C4—81no—
111C22×C3×C4C22×C411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4#4111yescharacteristic
116C22×C3×C4#4111yescharacteristic
117C22×Dic3—21no—
118C22×Dic3—21no—
119C22×Dic3—22no—
120C22×Dic3—22no—
121C22×Dic3—81no—
122C22×Dic3C22⋊C412yes—
123C2×C3×C8—41no—
124C2×C3×C8—81no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—41no—
128C3×C4⋊C4—41no—
129C3×C4⋊C4—81no—
130C3×C22⋊C4—41no—
131C3×C22⋊C4—41no—
132C3×C22⋊C4—41no—
133C3×C22⋊C4—41no—
134#217—41no—
135#217—41no—
136#217—41no—
137#217—41no—
138(C2×D3)⋊C4—22no—
139(C2×D3)⋊C4—22no—
140(C2×D3)⋊C4—22no—
141(C2×D3)⋊C4—22no—
142(C2×D3)⋊C4—82no—
143(C2×D3)⋊C4—82no—
144(C2×D3)⋊C4—82no—
145(C2×D3)⋊C4—82no—
146C4⋊Dic3—81no—
147C4⋊Dic3—81no—
148Dic3⋊C4—82no—
149C4×Dic3—82no—
150C2×C3⋊C8—41no—
151C2×C3⋊C8—81no—
152C25—31no—
153C22×Q8C4×D311yescharacteristic
154C23×C4—31no—
155C23×C4C22×D311yescharacteristic
156C22×C8—21no—
157C22×C8—61no—
158C22⋊Q8—21no—
159C22⋊Q8—21no—
160C22⋊Q8—21no—
161C22⋊Q8—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—62no—
165C2×C4⋊C4—61no—
166C2×C4⋊C4—61no—
167C2×C4⋊C4C4×D311yescharacteristic
168C2×C22⋊C4—21no—
169C2×C22⋊C4—21no—
170C2×C22⋊C4—32no—
171C2×C22⋊C4—32no—
172C2×C22⋊C4—61no—
173C2×C22⋊C4—61no—
174C2×C22⋊C4—61no—
175C2×C22⋊C4—61no—
176C2×C22⋊C4—61no—
177C2×C22⋊C4—61no—
178C2×C22⋊C4—62no—
179C2×C22⋊C4—62no—
180C2×C22⋊C4—62no—
181C2×C22⋊C4—62no—
182C2×C42—62no—
183C22⋊C8—21no—
184C22⋊C8—21no—
185C22⋊C8—61no—
186C22⋊C8—61no—
187#95—32no—
188#95—32no—
189#95—62no—
190#95—62no—
191#95#6712yes—
192C23×C3—41no—
193C23×C3—41no—
194C23×C3C2×C22⋊C411yescharacteristic
195C23×C3#12011yescharacteristic
196C23×C3#12311yescharacteristic
197C23×C3#13511yescharacteristic
198C23×C3#13511yescharacteristic
199C23×C3#13711yescharacteristic
200C23×C3#13711yescharacteristic
201C22×D3—22no—
202C22×D3—22no—
203C22×D3—22no—
204C22×D3—42no—
205C22×D3—42no—
206C22×D3—42no—
207C22×D3—42no—
208C22×D3—41no—
209C22×D3—41no—
210C22×D3—41no—
211C22×D3—41no—
212C22×D3—81no—
213C22×D3—81no—
214C22×D3—82no—
215C22×D3—82no—
216C22×D3—82no—
217C22×D3—82no—
218C3×Q8—81no—
219C3×Q8—81no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—42no—
227C2×C3×C4—42no—
228C2×C3×C4—42no—
229C2×C3×C4—42no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—42no—
233C2×C3×C4C4×D411yescharacteristic
234C2×C3×C4C4×D411yescharacteristic
235C2×C3×C4#12111yescharacteristic
236C2×C3×C4#12111yescharacteristic
237C2×Dic3—22no—
238C2×Dic3—42no—
239C2×Dic3—42no—
240C2×Dic3—42no—
241C2×Dic3—42no—
242C2×Dic3—42no—
243C2×Dic3—42no—
244C2×Dic3—42no—
245C2×Dic3—42no—
246C2×Dic3—81no—
247C3×C8—81no—
248C3⋊C8—81no—
249C24—32no—
250C24—61no—
251C24—61no—
252C24—61no—
253C24—61no—
254C24—61no—
255C24—61no—
256C24—241no—
257C24—241no—
258C24D3×D411yescharacteristic
259C2×Q8—21no—
260C2×Q8—21no—
261C2×Q8—81no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—32no—
266C22×C4—61no—
267C22×C4—61no—
268C22×C4—62no—
269C22×C4—62no—
270C22×C4—81no—
271C22×C4—241no—
272C22×C4C2×C4×D311yescharacteristic
273C22×C4#23511yescharacteristic
274C22×C4D3×D411yescharacteristic
275C22×C4D3×D411yescharacteristic
276C22×C4#23711yescharacteristic
277C22×C4C2×#6711yescharacteristic
278C2×C8—41no—
279C2×C8—81no—
280C2×C8—121no—
281C2×C8—241no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—81no—
287C4⋊C4—242no—
288C4⋊C4—241no—
289C4⋊C4—241no—
290C22⋊C4—41no—
291C22⋊C4—41no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—62no—
295C22⋊C4—62no—
296C22⋊C4—62no—
297C22⋊C4—62no—
298C22⋊C4—121no—
299C22⋊C4—121no—
300C22⋊C4—121no—
301C22⋊C4—121no—
302C22⋊C4—242no—
303C22⋊C4—242no—
304C22⋊C4—242no—
305C22⋊C4—242no—
306C42—242no—
307C22×C3—21no—
308C22×C3—21no—
309C22×C3—21no—
310C22×C3—21no—
311C22×C3—21no—
312C22×C3—21no—
313C22×C3—41no—
314C22×C3—41no—
315C22×C3—41no—
316C22×C3—41no—
317C22×C3#38611yescharacteristic
318C22×C3#41611yescharacteristic
319C22×C3#41911yescharacteristic
320C22×C3#45211yescharacteristic
321C22×C3#45211yescharacteristic
322C22×C3#48511yescharacteristic
323C22×C3#48511yescharacteristic
324C2×D3—44no—
325C2×D3—42no—
326C2×D3—42no—
327C2×D3—42no—
328C2×D3—82no—
329C2×D3—82no—
330C2×D3—82no—
331C2×D3—82no—
332C2×D3—82no—
333C3×C4—21no—
334C3×C4—21no—
335C3×C4—21no—
336C3×C4—21no—
337C3×C4—82no—
338C3×C4—81no—
339C3×C4—81no—
340Dic3—42no—
341Dic3—82no—
342Dic3—82no—
343C23—41no—
344C23—41no—
345C23—62no—
346C23—62no—
347C23—62no—
348C23—122no—
349C23—122no—
350C23—122no—
351C23—122no—
352C23—121no—
353C23—121no—
354C23—121no—
355C23—121no—
356C23—241no—
357C23—241no—
358C23—242no—
359C23—242no—
360C23—242no—
361C23—242no—
362C23D3×C22⋊C411yescharacteristic
363C23#88311yescharacteristic
364C23#91511yescharacteristic
365C23#91611yescharacteristic
366C23#91811yescharacteristic
367C23#91911yescharacteristic
368C23#93711yescharacteristic
369Q8—81no—
370Q8—81no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—62no—
385C2×C4—122no—
386C2×C4—122no—
387C2×C4—122no—
388C2×C4—122no—
389C2×C4—122no—
390C2×C4—122no—
391C2×C4—122no—
392C2×C4—122no—
393C2×C4—241no—
394C2×C4#67⋊C411yescharacteristic
395C2×C4#88411yescharacteristic
396C2×C4C4×#6711yescharacteristic
397C2×C4#93811yescharacteristic
398C8—81no—
399C8—241no—
400C2×C3—21no—
401C2×C3—21no—
402C2×C3—21no—
403C2×C3—21no—
404C2×C3—21no—
405C2×C3—21no—
406C2×C3#187311yescharacteristic
407C2×C3#191111yescharacteristic
408C2×C3#191311yescharacteristic
409D3—82no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—124no—
421C22—122no—
422C22—122no—
423C22—122no—
424C22—242no—
425C22—242no—
426C22—242no—
427C22—242no—
428C22—242no—
429C22#480211yescharacteristic
430C22#480411yescharacteristic
431C22#481411yescharacteristic
432C22#481511yescharacteristic
433C22#495611yescharacteristic
434C22#517311yescharacteristic
435C22#518711yescharacteristic
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—82no—
441C4—81no—
442C4—81no—
443C4—122no—
444C4—242no—
445C4—242no—
446C3#920111yescharacteristic, complemented, Sylow
447C2—21no—
448C2—21no—
449C2—21no—
450C2—21no—
451C2—21no—
452C2—21no—
453C2—242no—
454C2#7003411yescharacteristic
455C2#7026911yescharacteristic
456C2#7040811yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1520C2
socleC22×C3#386
prime radicalC24×D3D4
cosocleC23×C3×C4C23
hypercenter#1520D3
hypocenterC3#9201
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1520

Complements

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

NH ≅ G/Ntype
3. C3×#1520C2semidirect
4. #71690C2semidirect
6. #67676C2semidirect
7. #67625C2semidirect
13. #5231C4semidirect
18. #4993C4semidirect
31. #1520D3semidirect
446. C3#9201semidirect

Maximal subgroups

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

subgroupindex
#828222
#740042
C3×#15202
#716902
C2×#44682
#676762
#676252
#92013