Finite Group Databasebeta

← #10695583

Subgroups — #10695583

Order 768.

798 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 798 conjugacy classes of subgroups — too many to draw. Only its 85 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×#3007C211yesmaximal, characteristic
2#84090C211yesmaximal, characteristic
3#82785C211yesmaximal, characteristic, complemented
4#72138C211yesmaximal, characteristic
5#71689C211yesmaximal, characteristic, complemented
6#69971C211yesmaximal, characteristic, complemented
7#67682C211yesmaximal, characteristic, complemented
8#9943—31nomaximal, Sylow
9C22×C3×#41C2211yescharacteristic
10C2×C3×#137—21no—
11C2×C3×C22⋊Q8C2211yescharacteristic
12C2×#935—21no—
13C22×Dic3⋊C4C2211yescharacteristic
14C3×#452—22no—
15C3×#451—21no—
16C3×#450—21no—
17C3×#417—21no—
18C3×#407C2211yescharacteristic
19C2×#837—22no—
20#5230—21no—
21#5230—21no—
22#5222—21no—
23#5220—21no—
24#5211C2211yescharacteristic
25#5210—22no—
26C4×#217C2211yescharacteristic
27#5207C2211yescharacteristic
28#4983—21no—
29#4976—22no—
30#4946—21no—
31#4946—21no—
32#4538—22no—
33#4469—22no—
34#3007D311yescharacteristic
35#2583—31no—
36#2406—31no—
37#1789—31no—
38#1511—31no—
39#1463—31no—
40#1284—31no—
41C2×C3×#41—21no—
42C2×C3×#41—21no—
43C2×C3×#41—21no—
44C2×C3×#41—21no—
45C2×C3×#41—44no—
46C22×C3×Q8D411yescharacteristic
47C22×C3×Q8D411yescharacteristic
48C22×C3×D4—21no—
49C22×C3×D4D411yescharacteristic
50C23×C3×C4—21no—
51C23×C3×C4C2311yescharacteristic
52C23×Dic3D411yescharacteristic
53C23×Dic3D411yescharacteristic
54C3×#137—44no—
55C2×C3×Q16—22no—
56C2×C3×SD16—22no—
57C2×C3×M16—21no—
58C3×C22⋊Q8—21no—
59C3×C22⋊Q8—21no—
60C3×C22⋊Q8—21no—
61C3×C22⋊Q8—21no—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4D411yescharacteristic
64C2×C3×C22⋊C4—21no—
65C2×#217—21no—
66C2×#217—22no—
67C2×#217—22no—
68C2×#217—21no—
69#935—44no—
70C2×Dic3⋊C4—22no—
71C2×Dic3⋊C4—22no—
72C2×Dic3⋊C4—21no—
73C2×Dic3⋊C4—21no—
74C2×Dic3⋊C4—21no—
75C2×Dic3⋊C4—21no—
76C2×Dic3⋊C4—21no—
77C2×Dic3⋊C4—21no—
78C2×Dic3⋊C4—22no—
79C2×C4×Dic3D412yes—
80C2×#208—21no—
81C3×#103—22no—
82C3×#102—22no—
83C3×C22⋊C8—21no—
84C3×C22⋊C8D412yes—
85#837—44no—
86C3⋊#95—22no—
87C3⋊#95—22no—
88C3⋊#95—22no—
89C3⋊#95—21no—
90C3⋊#95—21no—
91C3⋊#95D412yes—
92#830—21no—
93#830—21no—
94C22×#41C2×D311yescharacteristic
95C2×#137—21no—
96C2×#123—61no—
97C2×C22⋊Q8C2×D311yescharacteristic
98C22×C4⋊C4—31no—
99#452—22no—
100#451—21no—
101#450—21no—
102C2×#104—62no—
103#417—21no—
104#407—31no—
105#407C2×D311yescharacteristic
106#398—61no—
107#395—61no—
108#395—61no—
109#394—61no—
110#389—62no—
111#386—61no—
112#385—62no—
113#385—61no—
114#385—61no—
115#381—31no—
116C4×C22⋊C4—31no—
117#328—62no—
118#324—62no—
119C24×C3—21no—
120C24×C3C2×D411yescharacteristic
121C3×#41—24no—
122C3×#41—24no—
123C3×#41—42no—
124C3×#41—42no—
125C3×#41—42no—
126C3×#41—42no—
127C3×#41—42no—
128C3×#41—42no—
129C2×C3×Q8—21no—
130C2×C3×Q8—21no—
131C2×C3×Q8—21no—
132C2×C3×Q8—21no—
133C2×C3×Q8—42no—
134C2×C3×Q8—42no—
135C2×C3×D4—24no—
136C2×C3×D4—21no—
137C2×C3×D4—21no—
138C2×C3×D4—42no—
139C2×C3×D4—42no—
140C2×C3×D4—41no—
141C2×C3×D4—41no—
142C22×C3×C4—21no—
143C22×C3×C4—21no—
144C22×C3×C4—21no—
145C22×C3×C4—42no—
146C22×C3×C4—42no—
147C22×C3×C4—42no—
148C22×C3×C4—41no—
149C22×C3×C4—41no—
150C22×C3×C4C2×D411yescharacteristic
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4#4111yescharacteristic
156C22×Dic3—22no—
157C22×Dic3—21no—
158C22×Dic3—21no—
159C22×Dic3—21no—
160C22×Dic3—22no—
161C22×Dic3—21no—
162C22×Dic3—21no—
163C22×Dic3—21no—
164C22×Dic3—42no—
165C22×Dic3—42no—
166C3×Q16—44no—
167C3×SD16—44no—
168C3×M16—42no—
169C2×C3×C8—22no—
170C3×C4⋊C4—21no—
171C3×C4⋊C4—21no—
172C3×C4⋊C4—41no—
173C3×C4⋊C4—41no—
174C3×C22⋊C4—41no—
175C3×C22⋊C4—41no—
176#217—24no—
177#217—44no—
178#217—42no—
179#217—44no—
180Dic3⋊C4—44no—
181Dic3⋊C4—44no—
182Dic3⋊C4—41no—
183Dic3⋊C4—41no—
184Dic3⋊C4—41no—
185Dic3⋊C4—41no—
186Dic3⋊C4—82no—
187C4×Dic3—42no—
188#208—42no—
189C2×C3⋊C8—41no—
190C2×#41—21no—
191C2×#41—21no—
192C2×#41—21no—
193C2×#41—21no—
194C2×#41—44no—
195C22×Q8#6711yescharacteristic
196C22×Q8#6711yescharacteristic
197C22×D4—21no—
198C22×D4#6711yescharacteristic
199C23×C4—21no—
200C23×C4—31no—
201C23×C4—31no—
202C23×C4C22×D311yescharacteristic
203#137—44no—
204C2×Q16—22no—
205C2×SD16—22no—
206C2×M16—21no—
207C2×M16—61no—
208#123—124no—
209C22⋊Q8—21no—
210C22⋊Q8—21no—
211C22⋊Q8—21no—
212C22⋊Q8—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—62no—
215C2×C4⋊C4—62no—
216C2×C4⋊C4—61no—
217C2×C4⋊C4—61no—
218C2×C4⋊C4—61no—
219C2×C4⋊C4—61no—
220C2×C4⋊C4—61no—
221C2×C4⋊C4—61no—
222C2×C4⋊C4—62no—
223C2×C4⋊C4#6711yescharacteristic
224C2×C22⋊C4—21no—
225C2×C22⋊C4—61no—
226C2×C22⋊C4—62no—
227C2×C22⋊C4—62no—
228C2×C22⋊C4—61no—
229C2×C42—32no—
230#104—124no—
231#103—22no—
232#102—22no—
233C22⋊C8—21no—
234C22⋊C8—61no—
235C22⋊C8—61no—
236C22⋊C8#6712yes—
237#95—32no—
238#95—62no—
239#95—62no—
240#95—62no—
241#95—61no—
242#95—61no—
243C23×C3—22no—
244C23×C3—22no—
245C23×C3—22no—
246C23×C3—41no—
247C23×C3—42no—
248C23×C3—41no—
249C23×C3—41no—
250C23×C3—42no—
251C23×C3#12011yescharacteristic
252C23×C3#12011yescharacteristic
253C23×C3#12111yescharacteristic
254C3×Q8—22no—
255C3×Q8—22no—
256C3×Q8—41no—
257C3×Q8—41no—
258C3×Q8—42no—
259C3×Q8—81no—
260C3×D4—22no—
261C3×D4—22no—
262C3×D4—41no—
263C3×D4—41no—
264C3×D4—44no—
265C3×D4—44no—
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4—22no—
269C2×C3×C4—41no—
270C2×C3×C4—41no—
271C2×C3×C4—41no—
272C2×C3×C4—41no—
273C2×C3×C4—44no—
274C2×C3×C4—44no—
275C2×C3×C4—41no—
276C2×C3×C4—41no—
277C2×C3×C4—42no—
278C2×C3×C4—41no—
279C2×C3×C4—41no—
280C2×C3×C4—42no—
281C2×C3×C4#12011yescharacteristic
282C2×C3×C4#12111yescharacteristic
283C2×C3×C4#12111yescharacteristic
284C2×C3×C4#12711yescharacteristic
285C2×Dic3—22no—
286C2×Dic3—24no—
287C2×Dic3—22no—
288C2×Dic3—22no—
289C2×Dic3—42no—
290C2×Dic3—41no—
291C2×Dic3—41no—
292C2×Dic3—41no—
293C2×Dic3—41no—
294C2×Dic3—42no—
295C2×Dic3—42no—
296C2×Dic3—42no—
297C2×Dic3—42no—
298C2×Dic3—41no—
299C2×Dic3—42no—
300C2×Dic3—42no—
301C2×Dic3—81no—
302C3×C8—42no—
303C3⋊C8—81no—
304C24—21no—
305C24D3×D411yescharacteristic
306#41—24no—
307#41—24no—
308#41—42no—
309#41—42no—
310#41—42no—
311#41—42no—
312#41—42no—
313#41—42no—
314C2×Q8—21no—
315C2×Q8—21no—
316C2×Q8—21no—
317C2×Q8—21no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×D4—24no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—41no—
326C2×D4—41no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—62no—
336C22×C4—61no—
337C22×C4—61no—
338C22×C4—61no—
339C22×C4—62no—
340C22×C4—61no—
341C22×C4—61no—
342C22×C4—61no—
343C22×C4—122no—
344C22×C4—122no—
345C22×C4D3×D411yescharacteristic
346C22×C4D3×D411yescharacteristic
347C22×C4#23711yescharacteristic
348C22×C4C2×#6711yescharacteristic
349C22×C4C2×#6711yescharacteristic
350C22×C4C2×#6711yescharacteristic
351Q16—44no—
352SD16—44no—
353M16—42no—
354M16—122no—
355C2×C8—22no—
356C2×C8—121no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—124no—
362C4⋊C4—124no—
363C4⋊C4—121no—
364C4⋊C4—121no—
365C4⋊C4—121no—
366C4⋊C4—121no—
367C4⋊C4—242no—
368C22⋊C4—41no—
369C22⋊C4—41no—
370C22⋊C4—64no—
371C22⋊C4—124no—
372C22⋊C4—122no—
373C22⋊C4—124no—
374C42—122no—
375C22×C3—21no—
376C22×C3—21no—
377C22×C3—21no—
378C22×C3—21no—
379C22×C3—21no—
380C22×C3—21no—
381C22×C3—21no—
382C22×C3—44no—
383C22×C3—44no—
384C22×C3—42no—
385C22×C3—82no—
386C22×C3—81no—
387C22×C3—81no—
388C22×C3#39211yescharacteristic
389C22×C3#45612yes—
390C22×C3B(4,𝔽2)12yes—
391C3×C4—21no—
392C3×C4—21no—
393C3×C4—21no—
394C3×C4—21no—
395C3×C4—42no—
396C3×C4—81no—
397C3×C4—81no—
398Dic3—44no—
399Dic3—42no—
400Dic3—82no—
401Dic3—81no—
402C23—22no—
403C23—22no—
404C23—22no—
405C23—41no—
406C23—42no—
407C23—41no—
408C23—42no—
409C23—41no—
410C23#93711yescharacteristic
411C23#93711yescharacteristic
412C23#93811yescharacteristic
413Q8—22no—
414Q8—22no—
415Q8—41no—
416Q8—41no—
417Q8—42no—
418Q8—81no—
419D4—22no—
420D4—22no—
421D4—41no—
422D4—41no—
423D4—44no—
424D4—44no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—64no—
441C2×C4—62no—
442C2×C4—62no—
443C2×C4—62no—
444C2×C4—122no—
445C2×C4—121no—
446C2×C4—121no—
447C2×C4—121no—
448C2×C4—121no—
449C2×C4—122no—
450C2×C4—122no—
451C2×C4—122no—
452C2×C4—122no—
453C2×C4—122no—
454C2×C4—121no—
455C2×C4—122no—
456C2×C4—241no—
457C2×C4#93911yescharacteristic
458C2×C4#93911yescharacteristic
459C2×C4#94011yescharacteristic
460C2×C4#95311yescharacteristic
461C8—42no—
462C8—241no—
463C2×C3—22no—
464C2×C3—22no—
465C2×C3—41no—
466C2×C3—82no—
467C2×C3#200511yescharacteristic
468C2×C3#201011yescharacteristic
469C2×C3#202111yescharacteristic
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—44no—
478C22—44no—
479C22—42no—
480C22—82no—
481C22—81no—
482C22—81no—
483C22#503212yes—
484C22#520512yes—
485C22#522311yescharacteristic
486C4—21no—
487C4—21no—
488C4—21no—
489C4—21no—
490C4—42no—
491C4—81no—
492C4—81no—
493C4—124no—
494C4—122no—
495C4—242no—
496C4—241no—
497C3#994311yescharacteristic, complemented, Sylow
498C2—22no—
499C2—22no—
500C2—41no—
501C2—82no—
502C2#7167111yescharacteristic
503C2#7210811yescharacteristic
504C2#7211811yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#3007C2
socleC22×C3#392
prime radicalC22×C3×#41C22
hypercenter#3007D3
hypocenterC3#9943
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3007

Complements

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

NH ≅ G/Ntype
3. #82785C2semidirect
5. #71689C2semidirect
6. #69971C2semidirect
7. #67682C2semidirect
497. C3#9943semidirect

Maximal subgroups

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

subgroupindex
C3×#30072
#840902
#827852
#721382
#716892
#699712
#676822
#99433