Finite Group Databasebeta

← #10692231

Subgroups — #10692231

Order 768.

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

normal subgroups only

This group has 520 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×#3193C211yesmaximal, characteristic
2#82750C211yesmaximal, characteristic, complemented
3#71688C211yesmaximal, characteristic, complemented
4#71686C211yesmaximal, characteristic
5#71674C211yesmaximal, characteristic
6#67685C211yesmaximal, characteristic, complemented
7#67682C211yesmaximal, characteristic, complemented
8#9344—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C2×C3×#121C2211yescharacteristic
11C3×#484C412yescomplemented
12C3×#480—21no—
13C3×#468—22no—
14C3×#450C412yescomplemented
15C3×#448—21no—
16C3×#407C2211yescharacteristic
17#5229—21no—
18C2×#834—21no—
19C2×#834—21no—
20#5212C2211yescharacteristic
21#5211C2211yescharacteristic
22C4×#217C2211yescharacteristic
23#5207C2211yescharacteristic
24#4984—21no—
25#4980—21no—
26#4980—21no—
27#4951—21no—
28#4951—21no—
29#4538—22no—
30#4537—21no—
31#4537—21no—
32#3193D311yescharacteristic
33#2378—31no—
34#1964—31no—
35#1510—31no—
36#1508—31no—
37#1305—31no—
38#1284—31no—
39C22×C3×Q8D411yescharacteristic
40C22×C3×D4—21no—
41C22×C3×D4C2×C411yescharacteristic
42C23×C3×C4C2311yescharacteristic
43C2×C3×SD16—22no—
44C2×C3×M16—21no—
45C3×C22⋊Q8—21no—
46C3×C22⋊Q8—21no—
47C3×C22⋊Q8C2×C412yes—
48C3×C22⋊Q8D412yes—
49C3×#121—21no—
50C3×#121—21no—
51C3×#121—21no—
52C3×#121—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4C2×C411yescharacteristic
55C2×C3×C4⋊C4D411yescharacteristic
56C2×C3×C22⋊C4—21no—
57C2×C3×C22⋊C4—21no—
58C2×#217—21no—
59C2×#217—21no—
60C2×#217D411yescharacteristic
61C2×#217D411yescharacteristic
62C2×C4⋊Dic3—21no—
63C2×C4⋊Dic3—21no—
64C2×C4⋊Dic3—21no—
65C2×C4×Dic3—21no—
66C2×C4×Dic3—21no—
67C2×C4×Dic3—21no—
68C2×#208—21no—
69C3×#107—22no—
70C3×#103—22no—
71C3×#102—22no—
72C3×C22⋊C8—21no—
73C3×C22⋊C8C2×C412yes—
74#834—42no—
75#834—42no—
76C3⋊#95—21no—
77C3⋊#95—21no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80C3⋊#95—21no—
81C3⋊#95—21no—
82C3⋊#95—21no—
83C3⋊#95D411yescharacteristic
84C3⋊#95D411yescharacteristic
85#830—21no—
86#830—21no—
87C2×C22⋊Q8C2×D311yescharacteristic
88C2×#121C2×D311yescharacteristic
89#484Dic312yescomplemented
90#480—21no—
91#468—22no—
92#450Dic312yescomplemented
93#448—21no—
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⋊C4—31no—
107#328—62no—
108#327—61no—
109#327—61no—
110C24×C3—21no—
111C24×C3C2×D411yescharacteristic
112C2×C3×Q8—21no—
113C2×C3×Q8—81no—
114C2×C3×Q8C22⋊C412yes—
115C2×C3×D4—21no—
116C2×C3×D4—21no—
117C2×C3×D4—41no—
118C2×C3×D4—41no—
119C2×C3×D4—41no—
120C2×C3×D4—41no—
121C2×C3×D4—41no—
122C2×C3×D4—41no—
123C2×C3×D4—81no—
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—81no—
128C22×C3×C4C22×C411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4#4111yescharacteristic
133C22×C3×C4#4111yescharacteristic
134C22×Dic3—21no—
135C22×Dic3—21no—
136C22×Dic3—21no—
137C22×Dic3—21no—
138C22×Dic3—41no—
139C3×SD16—82no—
140C3×M16—81no—
141C2×C3×C8—22no—
142C3×C4⋊C4—22no—
143C3×C4⋊C4—22no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—21no—
147C3×C4⋊C4C22⋊C412yes—
148C3×C22⋊C4—22no—
149C3×C22⋊C4—22no—
150C3×C22⋊C4—41no—
151C3×C22⋊C4—41no—
152#217—22no—
153#217—41no—
154#217—41no—
155#217—41no—
156#217—81no—
157C4⋊Dic3—81no—
158C4⋊Dic3—81no—
159C4⋊Dic3—81no—
160C4×Dic3—81no—
161C4×Dic3—81no—
162C4×Dic3—81no—
163#208—81no—
164C2×C3⋊C8—41no—
165C22×Q8#6711yescharacteristic
166C22×D4—21no—
167C22×D4C2×Dic311yescharacteristic
168C23×C4C22×D311yescharacteristic
169C2×SD16—22no—
170C2×M16—21no—
171C2×M16—61no—
172C22⋊Q8—21no—
173C22⋊Q8—21no—
174C22⋊Q8C2×Dic312yes—
175C22⋊Q8#6712yes—
176#121—21no—
177#121—21no—
178#121—21no—
179#121—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—61no—
182C2×C4⋊C4—61no—
183C2×C4⋊C4—61no—
184C2×C4⋊C4C2×Dic311yescharacteristic
185C2×C4⋊C4#6711yescharacteristic
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—61no—
193C2×C42—61no—
194C2×C42—61no—
195#107—22no—
196#103—22no—
197#102—22no—
198#99—122no—
199#99—122no—
200C22⋊C8—21no—
201C22⋊C8—61no—
202C22⋊C8—61no—
203C22⋊C8C2×Dic312yes—
204#95—31no—
205#95—31no—
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—41no—
216C23×C3—41no—
217C23×C3—41no—
218C23×C3—41no—
219C23×C3—41no—
220C23×C3—81no—
221C23×C3—81no—
222C23×C3C4×D411yescharacteristic
223C23×C3#12011yescharacteristic
224C23×C3#12111yescharacteristic
225C3×Q8—42no—
226C3×Q8—81no—
227C3×D4—42no—
228C3×D4—81no—
229C3×D4—81no—
230C3×D4—81no—
231C3×D4—81no—
232C3×D4—81no—
233C2×C3×C4—22no—
234C2×C3×C4—22no—
235C2×C3×C4—41no—
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—41no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4C2×C22⋊C411yescharacteristic
249C2×C3×C4C4×D411yescharacteristic
250C2×C3×C4#12111yescharacteristic
251C2×C3×C4#12311yescharacteristic
252C2×Dic3—41no—
253C2×Dic3—41no—
254C2×Dic3—41no—
255C2×Dic3—41no—
256C2×Dic3—42no—
257C2×Dic3—41no—
258C2×Dic3—41no—
259C2×Dic3—41no—
260C2×Dic3—41no—
261C2×Dic3—41no—
262C2×Dic3—41no—
263C2×Dic3—41no—
264C2×Dic3—41no—
265C2×Dic3—81no—
266C2×Dic3—81no—
267C3×C8—42no—
268C3⋊C8—81no—
269C24—21no—
270C24D3×D411yescharacteristic
271C2×Q8—21no—
272C2×Q8—81no—
273C2×Q8#21712yes—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—81no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—61no—
287C22×C4—61no—
288C22×C4—61no—
289C22×C4—61no—
290C22×C4—81no—
291C22×C4—121no—
292C22×C4D3×D411yescharacteristic
293C22×C4#23711yescharacteristic
294C22×C4#23711yescharacteristic
295C22×C4C22×Dic311yescharacteristic
296C22×C4C2×#6711yescharacteristic
297C22×C4C2×#6711yescharacteristic
298SD16—82no—
299M16—81no—
300M16—241no—
301C2×C8—22no—
302C2×C8—121no—
303C4⋊C4—22no—
304C4⋊C4—21no—
305C4⋊C4—22no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—241no—
309C4⋊C4—241no—
310C4⋊C4—241no—
311C4⋊C4#21712yes—
312C22⋊C4—22no—
313C22⋊C4—22no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—62no—
317C22⋊C4—121no—
318C22⋊C4—121no—
319C22⋊C4—121no—
320C22⋊C4—241no—
321C42—241no—
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—81no—
335C22×C3—81no—
336C22×C3—81no—
337C22×C3—161no—
338C22×C3—161no—
339C22×C3#38611yescharacteristic
340C22×C3#45311yescharacteristic
341C22×C3#45411yescharacteristic
342C3×C4—21no—
343C3×C4—21no—
344C3×C4—21no—
345C3×C4—21no—
346C3×C4—42no—
347C3×C4—42no—
348C3×C4—81no—
349Dic3—81no—
350Dic3—81no—
351Dic3—81no—
352Dic3—81no—
353Dic3—82no—
354C23—21no—
355C23—21no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—81no—
362C23—81no—
363C23D4×Dic311yescharacteristic
364C23#93711yescharacteristic
365C23#93911yescharacteristic
366Q8—42no—
367Q8—81no—
368D4—42no—
369D4—81no—
370D4—81no—
371D4—81no—
372D4—81no—
373D4—81no—
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—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—121no—
390C2×C4—121no—
391C2×C4—122no—
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—241no—
403C2×C4—241no—
404C2×C4D4×Dic311yescharacteristic
405C2×C4#93511yescharacteristic
406C2×C4#93911yescharacteristic
407C2×C4C2×#21711yescharacteristic
408C8—42no—
409C8—241no—
410C2×C3—21no—
411C2×C3—21no—
412C2×C3—41no—
413C2×C3—41no—
414C2×C3—161no—
415C2×C3#188311yescharacteristic
416C2×C3#190511yescharacteristic
417C2×C3#191311yescharacteristic
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—81no—
427C22—81no—
428C22—81no—
429C22—81no—
430C22—81no—
431C22—161no—
432C22—161no—
433C22#519811yescharacteristic
434C22#520111yescharacteristic
435C22#522011yescharacteristic
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—42no—
441C4—42no—
442C4—81no—
443C4—241no—
444C4—241no—
445C4—241no—
446C4—241no—
447C4—242no—
448C3#934411yescharacteristic, complemented, Sylow
449C2—21no—
450C2—21no—
451C2—41no—
452C2—41no—
453C2—161no—
454C2#7184811yescharacteristic
455C2#7208811yescharacteristic
456C2#7210111yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#3193C2
socleC22×C3#386
prime radicalC2×C3×#121C22
cosocleC23×C3×C4C23
hypercenter#3193D3
hypocenterC3#9344
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3193

Complements

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

NH ≅ G/Ntype
2. #82750C2semidirect
3. #71688C2semidirect
6. #67685C2semidirect
7. #67682C2semidirect
11. C3×#484C4semidirect
14. C3×#450C4semidirect
89. #484Dic3semidirect
92. #450Dic3semidirect
448. C3#9344semidirect

Maximal subgroups

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

subgroupindex
C3×#31932
#827502
#716882
#716862
#716742
#676852
#676822
#93443