Finite Group Databasebeta

← #10686708

Subgroups — #10686708

Order 768.

726 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 135 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×#3491C211yesmaximal, characteristic
2#71752C212yesmaximal, complemented
3#71748C212yesmaximal, complemented
4#71582C211yesmaximal, characteristic, complemented
5#71572C211yesmaximal, characteristic, complemented
6#8348—31nomaximal, Sylow
7C3×#552—22no—
8C3×#552—22no—
9C3×#551—22no—
10C3×#551—22no—
11C3×#543C411yescharacteristic
12C3×#539—21no—
13C3×#539C412yes—
14C3×#534C411yescharacteristic
15C3×#529—21no—
16C3×#529C2212yes—
17C2×C3×#123—21no—
18C3×#518—21no—
19C2×C3×#117C2211yescharacteristic
20C2×#830—21no—
21#4926C2212yescomplemented
22C2×#803C2211yescharacteristic, complemented
23C2×C4×C3⋊C8C2211yescharacteristic, complemented
24#4543—22no—
25#4540—22no—
26#4535—22no—
27D4⋊(C3⋊C8)—22no—
28#4528—21no—
29#4525—21no—
30#4524—21no—
31#3491D311yescharacteristic
32#1750—31no—
33#1532—32no—
34#1494—32no—
35#1464—31no—
36C22×C3×D4—21no—
37C23×C3×C4D411yescharacteristic
38C3×#126—22no—
39C3×#126—22no—
40C3×#126—22no—
41C3×#126—22no—
42C3×#125—21no—
43C3×#125—22no—
44C3×#125—22no—
45C3×#125D412yes—
46C3×#124—21no—
47C3×#124D412yes—
48C3×#123—22no—
49C3×#123—22no—
50C3×#123—22no—
51C3×#123—22no—
52C3×#123—22no—
53C3×#123—22no—
54C3×#123—22no—
55C3×C22⋊Q8—22no—
56C3×C22⋊Q8—22no—
57C3×C22⋊Q8—22no—
58C3×C22⋊Q8—21no—
59C3×C22⋊Q8—22no—
60C3×C22⋊Q8C2×C412yes—
61C3×#121—21no—
62C3×#121C2×C412yes—
63C3×#120—22no—
64C3×C4×Q8—21no—
65C3×C4×Q8D412yes—
66C3×C4×D4—22no—
67C3×C4×D4—22no—
68C3×C4×D4—21no—
69C3×C4×D4—22no—
70C3×C4×D4—22no—
71C3×C4×D4D412yes—
72C3×#117—22no—
73C3×#117—22no—
74C3×#117—22no—
75C3×#117—22no—
76C3×#117—22no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4C2×C411yescharacteristic
80C2×C3×C4⋊C4D411yescharacteristic
81C2×C3×C22⋊C4—22no—
82C2×C3×C22⋊C4—21no—
83C2×C3×C22⋊C4—21no—
84C2×C3×C22⋊C4—21no—
85C2×C3×C22⋊C4C2×C411yescharacteristic
86C2×C3×C22⋊C4D411yescharacteristic
87C2×C3×C42D411yescharacteristic
88C2×C3×C42C2311yescharacteristic
89C22×C3⋊C8—21no—
90C22×C3⋊C8—21no—
91#830—24no—
92#830—22no—
93C4⋊(C3⋊C8)—22no—
94#803—22no—
95#803—22no—
96C4×C3⋊C8—22no—
97#552—22no—
98#552—22no—
99#551—22no—
100#551—22no—
101#543Dic311yescharacteristic
102#539—21no—
103#539Dic312yes—
104#534Dic311yescharacteristic
105#529—21no—
106#529C2×D312yes—
107C2×#123—21no—
108#518—21no—
109C2×#117C2×D311yescharacteristic
110#432—32no—
111C2×C22⋊C8—61no—
112C2×#97—31no—
113C2×C4×C8—31no—
114#336—61no—
115#330—62no—
116#329—62no—
117Q8⋊C8—62no—
118D4⋊C8—62no—
119#324—61no—
120#323—61no—
121C24×C3—21no—
122C24×C3C22⋊C411yescharacteristic
123C2×C3×Q8—21no—
124C2×C3×Q8C22⋊C412yes—
125C2×C3×D4—22no—
126C2×C3×D4—22no—
127C2×C3×D4—21no—
128C2×C3×D4—42no—
129C2×C3×D4—42no—
130C2×C3×D4C22⋊C412yes—
131C22×C3×C4—21no—
132C22×C3×C4—22no—
133C22×C3×C4—21no—
134C22×C3×C4—21no—
135C22×C3×C4—22no—
136C22×C3×C4—21no—
137C22×C3×C4—21no—
138C22×C3×C4—22no—
139C22×C3×C4—41no—
140C22×C3×C4C22⋊C411yescharacteristic
141C22×C3×C4C22⋊C411yescharacteristic
142C22×C3×C4C22⋊C411yescharacteristic
143C22×C3×C4C22×C411yescharacteristic
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4C2×D411yescharacteristic
146C3×C4⋊C4—22no—
147C3×C4⋊C4—22no—
148C3×C4⋊C4—22no—
149C3×C4⋊C4—22no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4—21no—
152C3×C4⋊C4—22no—
153C3×C4⋊C4—22no—
154C3×C4⋊C4—22no—
155C3×C4⋊C4—22no—
156C3×C4⋊C4—22no—
157C3×C4⋊C4—22no—
158C3×C4⋊C4C22⋊C412yes—
159C3×C4⋊C4C22⋊C412yes—
160C3×C22⋊C4—22no—
161C3×C22⋊C4—22no—
162C3×C22⋊C4—22no—
163C3×C22⋊C4—22no—
164C3×C22⋊C4—22no—
165C3×C22⋊C4—22no—
166C3×C22⋊C4—22no—
167C3×C22⋊C4—22no—
168C3×C22⋊C4—22no—
169C3×C22⋊C4—22no—
170C3×C22⋊C4—22no—
171C3×C22⋊C4—22no—
172C3×C42—22no—
173C3×C42—21no—
174C3×C42—21no—
175C3×C42—22no—
176C3×C42C2×D412yes—
177C3×C42C2×D412yes—
178C2×C3⋊C8—22no—
179C2×C3⋊C8—22no—
180C2×C3⋊C8—41no—
181C2×C3⋊C8—41no—
182C2×C3⋊C8—41no—
183C2×C3⋊C8—41no—
184C2×C3⋊C8—42no—
185C22×D4—21no—
186C23×C4#6711yescharacteristic
187C22×C8—61no—
188C22×C8—61no—
189#126—22no—
190#126—22no—
191#126—22no—
192#126—22no—
193#125—21no—
194#125—22no—
195#125—22no—
196#125#6712yes—
197#124—21no—
198#124#6712yes—
199#123—22no—
200#123—22no—
201#123—22no—
202#123—22no—
203#123—22no—
204#123—22no—
205#123—22no—
206C22⋊Q8—22no—
207C22⋊Q8—22no—
208C22⋊Q8—22no—
209C22⋊Q8—21no—
210C22⋊Q8—22no—
211C22⋊Q8C2×Dic312yes—
212#121—21no—
213#121C2×Dic312yes—
214#120—22no—
215C4×Q8—21no—
216C4×Q8#6712yes—
217C4×D4—22no—
218C4×D4—22no—
219C4×D4—21no—
220C4×D4—22no—
221C4×D4—22no—
222C4×D4#6712yes—
223#117—22no—
224#117—22no—
225#117—22no—
226#117—22no—
227#117—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4C2×Dic311yescharacteristic
231C2×C4⋊C4#6711yescharacteristic
232C2×C22⋊C4—22no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4C2×Dic311yescharacteristic
237C2×C22⋊C4#6711yescharacteristic
238C2×C42#6711yescharacteristic
239C2×C42C22×D311yescharacteristic
240C4⋊C8—62no—
241C22⋊C8—64no—
242C22⋊C8—62no—
243#97—62no—
244#97—62no—
245C4×C8—62no—
246C23×C3—21no—
247C23×C3—21no—
248C23×C3—22no—
249C23×C3—41no—
250C23×C3—41no—
251C23×C3—42no—
252C23×C3—81no—
253C23×C3—81no—
254C23×C3#10412yes—
255C23×C3C2×C22⋊C411yescharacteristic
256C3×Q8—42no—
257C3×Q8—81no—
258C3×D4—42no—
259C3×D4—44no—
260C3×D4—42no—
261C3×D4—42no—
262C3×D4—81no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—21no—
267C2×C3×C4—22no—
268C2×C3×C4—22no—
269C2×C3×C4—22no—
270C2×C3×C4—21no—
271C2×C3×C4—21no—
272C2×C3×C4—21no—
273C2×C3×C4—41no—
274C2×C3×C4—41no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—41no—
279C2×C3×C4—41no—
280C2×C3×C4—82no—
281C2×C3×C4—81no—
282C2×C3×C4—82no—
283C2×C3×C4—81no—
284C2×C3×C4C2×C22⋊C412yes—
285C2×C3×C4#12012yes—
286C2×C3×C4#12011yescharacteristic
287C2×C3×C4#12011yescharacteristic
288C3⋊C8—42no—
289C3⋊C8—42no—
290C3⋊C8—82no—
291C24—21no—
292C24#21711yescharacteristic
293C2×Q8—21no—
294C2×Q8#21712yes—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—21no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4#21712yes—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—41no—
310C22×C4#21711yescharacteristic
311C22×C4#21711yescharacteristic
312C22×C4#21711yescharacteristic
313C22×C4C22×Dic311yescharacteristic
314C22×C4C2×#6711yescharacteristic
315C22×C4C2×#6711yescharacteristic
316C2×C8—62no—
317C2×C8—62no—
318C2×C8—121no—
319C2×C8—121no—
320C2×C8—121no—
321C2×C8—121no—
322C2×C8—122no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4#21712yes—
336C4⋊C4#21712yes—
337C22⋊C4—22no—
338C22⋊C4—22no—
339C22⋊C4—22no—
340C22⋊C4—22no—
341C22⋊C4—22no—
342C22⋊C4—22no—
343C22⋊C4—22no—
344C22⋊C4—22no—
345C22⋊C4—22no—
346C22⋊C4—22no—
347C22⋊C4—22no—
348C22⋊C4—22no—
349C42—22no—
350C42—21no—
351C42—21no—
352C42—22no—
353C42C2×#6712yes—
354C42C2×#6712yes—
355C22×C3—21no—
356C22×C3—21no—
357C22×C3—22no—
358C22×C3—42no—
359C22×C3—41no—
360C22×C3—41no—
361C22×C3—42no—
362C22×C3—42no—
363C22×C3—42no—
364C22×C3—81no—
365C22×C3—162no—
366C22×C3#37911yescharacteristic
367C22×C3C2×#10411yescharacteristic
368C22×C3#42111yescharacteristic
369C3×C4—22no—
370C3×C4—21no—
371C3×C4—21no—
372C3×C4—22no—
373C3×C4—41no—
374C3×C4—42no—
375C3×C4—41no—
376C3×C4—42no—
377C3×C4—41no—
378C3×C4—42no—
379C23—21no—
380C23—21no—
381C23—22no—
382C23—41no—
383C23—41no—
384C23—42no—
385C23—81no—
386C23—81no—
387C23#83712yes—
388C23C2×#21711yescharacteristic
389Q8—42no—
390Q8—81no—
391D4—42no—
392D4—44no—
393D4—42no—
394D4—42no—
395D4—81no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—21no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4C2×#21712yes—
418C2×C4#95311yescharacteristic
419C2×C4#95311yescharacteristic
420C2×C4#95312yes—
421C8—122no—
422C8—122no—
423C8—242no—
424C2×C3—21no—
425C2×C3—21no—
426C2×C3—42no—
427C2×C3—82no—
428C2×C3#178911yescharacteristic
429C2×C3#179111yescharacteristic
430C2×C3#179411yescharacteristic
431C22—21no—
432C22—21no—
433C22—22no—
434C22—42no—
435C22—41no—
436C22—41no—
437C22—42no—
438C22—42no—
439C22—42no—
440C22—81no—
441C22—162no—
442C22C2×#83711yescharacteristic
443C22#523611yescharacteristic
444C22#524711yescharacteristic
445C4—21no—
446C4—21no—
447C4—22no—
448C4—22no—
449C4—41no—
450C4—42no—
451C4—41no—
452C4—42no—
453C4—41no—
454C4—42no—
455C3#834811yescharacteristic, complemented, Sylow
456C2—21no—
457C2—21no—
458C2—42no—
459C2—82no—
460C2#7213811yescharacteristic
461C2#7214511yescharacteristic
462C2#7214611yescharacteristic
subgroupG/H
centerC22#5247
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3491C2
socleC22×C3#379
prime radicalC3×#534C4
cosocleC2×C3×C42C23
hypercenter#3491D3
hypocenterC3#8348
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3491

Complements

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

NH ≅ G/Ntype
2. #71752C2semidirect
3. #71748C2semidirect
4. #71582C2semidirect
5. #71572C2semidirect
21. #4926C22semidirect
22. C2×#803C22semidirect
23. C2×C4×C3⋊C8C22semidirect
455. C3#8348semidirect

Maximal subgroups

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

subgroupindex
C3×#34912
#717522
#717482
#715822
#715722
#83483