Finite Group Databasebeta

← #10695584

Subgroups — #10695584

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×#2583C211yesmaximal, characteristic, complemented
2#82735C211yesmaximal, characteristic
3#82192C211yesmaximal, characteristic
4#71689C211yesmaximal, characteristic, complemented
5#71552C211yesmaximal, characteristic, complemented
6#69454C211yesmaximal, characteristic
7#67616C211yesmaximal, characteristic, complemented
8#9943—31nomaximal, Sylow
9C22×#235C2211yescharacteristic
10C2×C3×C22⋊Q8C2211yescharacteristic
11C22×C3×C4⋊C4C2211yescharacteristic
12C2×#942—21no—
13C2×#929—21no—
14C3×#398—21no—
15C3×#395—21no—
16C3×#389—22no—
17C3×#385—21no—
18C3×#385—21no—
19C3×C4×C22⋊C4C2211yescharacteristic
20#5213—21no—
21#5211C2211yescharacteristic
22#5207C2211yescharacteristic
23#5207C2211yescharacteristic
24#5050—21no—
25#5047—22no—
26#5046—21no—
27#4991—21no—
28#4989—22no—
29#4979—21no—
30#4966—21no—
31C2×#805—22no—
32#4538—22no—
33#4528—22no—
34#3007—31no—
35#2583D311yescharacteristic, complemented
36#2406—31no—
37#1789—31no—
38#1511—31no—
39#1463—31no—
40#1284—31no—
41C22×C3×Q8D411yescharacteristic
42C23×C3×C4D411yescharacteristic
43C23×C3×C4D411yescharacteristic
44C23×C3×C4C2311yescharacteristic
45C22×#67—21no—
46C2×#235—21no—
47C2×#235—21no—
48C2×#235—21no—
49C2×#235—21no—
50C2×#235—44no—
51C22×D12D411yescharacteristic
52C22×C4×D3—21no—
53C22×Dic6D411yescharacteristic
54C3×C22⋊Q8—21no—
55C3×C22⋊Q8—21no—
56C3×C22⋊Q8—21no—
57C3×C22⋊Q8—21no—
58C2×C3×C4⋊C4—22no—
59C2×C3×C4⋊C4—21no—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—22no—
67C2×C3×C4⋊C4D411yescharacteristic
68C2×C3×C22⋊C4—21no—
69C2×C3×C22⋊C4—21no—
70C2×C3×C42D412yes—
71C2×#217—21no—
72C2×#216—22no—
73#942—44no—
74C2×#215—22no—
75#929—44no—
76C2×(C2×D3)⋊C4—22no—
77C2×(C2×D3)⋊C4—22no—
78C2×Dic3⋊C4—22no—
79C2×#208—21no—
80C2×#208—21no—
81C3×#95—22no—
82C3×#95—22no—
83C3×#95—21no—
84C3⋊#95—22no—
85C3⋊#95—21no—
86C3⋊#95D412yes—
87#830—21no—
88#830—21no—
89#830—21no—
90#830D412yes—
91#810—22no—
92#809—22no—
93#805—44no—
94C22×#41—31no—
95C2×#137—61no—
96C2×#123—61no—
97C2×C22⋊Q8C2×D311yescharacteristic
98C22×C4⋊C4C2×D311yescharacteristic
99#452—62no—
100#451—61no—
101#450—61no—
102C2×#104—62no—
103#417—61no—
104#407—31no—
105#407—31no—
106#398—21no—
107#395—21no—
108#395—61no—
109#394—61no—
110#389—22no—
111#386—61no—
112#385—21no—
113#385—21no—
114#385—62no—
115#381—31no—
116C4×C22⋊C4C2×D311yescharacteristic
117#328—62no—
118#324—62no—
119C24×C3C2×D411yescharacteristic
120C23×D3—21no—
121C2×C3×Q8—21no—
122C2×C3×Q8—21no—
123C2×C3×Q8—42no—
124C22×C3×C4—21no—
125C22×C3×C4—22no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—21no—
129C22×C3×C4—21no—
130C22×C3×C4—21no—
131C22×C3×C4—21no—
132C22×C3×C4—21no—
133C22×C3×C4—42no—
134C22×C3×C4—42no—
135C22×C3×C4—42no—
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4#4111yescharacteristic
142C2×#67—24no—
143C2×#67—42no—
144C2×#67—41no—
145C2×#67—41no—
146C22×Dic3—21no—
147C22×Dic3—22no—
148#235—24no—
149#235—24no—
150#235—42no—
151#235—42no—
152#235—42no—
153#235—42no—
154#235—42no—
155#235—42no—
156C2×D12—21no—
157C2×D12—21no—
158C2×D12—42no—
159C2×C4×D3—42no—
160C2×C4×D3—42no—
161C2×C4×D3—41no—
162C2×C4×D3—41no—
163C2×Dic6—21no—
164C2×Dic6—21no—
165C2×Dic6—42no—
166C3×C4⋊C4—21no—
167C3×C4⋊C4—21no—
168C3×C4⋊C4—44no—
169C3×C4⋊C4—41no—
170C3×C4⋊C4—41no—
171C3×C4⋊C4—41no—
172C3×C4⋊C4—41no—
173C3×C4⋊C4—41no—
174C3×C4⋊C4—41no—
175C3×C4⋊C4—82no—
176C3×C22⋊C4—24no—
177C3×C22⋊C4—41no—
178C3×C22⋊C4—41no—
179C3×C42—42no—
180#217—42no—
181#216—44no—
182#215—44no—
183(C2×D3)⋊C4—44no—
184(C2×D3)⋊C4—44no—
185Dic3⋊C4—44no—
186#208—42no—
187#208—42no—
188C2×C3⋊C8—22no—
189C2×C3⋊C8—41no—
190C2×#41—61no—
191C2×#41—61no—
192C2×#41—61no—
193C2×#41—61no—
194C2×#41—124no—
195C22×Q8—31no—
196C22×Q8#6711yescharacteristic
197C22×D4—31no—
198C22×D4—61no—
199C23×C4—61no—
200C23×C4#6711yescharacteristic
201C23×C4#6711yescharacteristic
202C23×C4C22×D311yescharacteristic
203#137—124no—
204C2×Q16—62no—
205C2×SD16—62no—
206C2×M16—61no—
207C2×M16—61no—
208#123—124no—
209C22⋊Q8—21no—
210C22⋊Q8—21no—
211C22⋊Q8—21no—
212C22⋊Q8—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—62no—
223C2×C4⋊C4#6711yescharacteristic
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—62no—
227C2×C22⋊C4—62no—
228C2×C22⋊C4—61no—
229C2×C42#6712yes—
230#104—124no—
231#103—62no—
232#102—62no—
233C22⋊C8—32no—
234C22⋊C8—61no—
235C22⋊C8—61no—
236C22⋊C8—61no—
237#95—22no—
238#95—22no—
239#95—21no—
240#95—32no—
241#95—62no—
242#95—61no—
243C23×C3—22no—
244C23×C3—22no—
245C23×C3—41no—
246C23×C3#12011yescharacteristic
247C23×C3#12011yescharacteristic
248C23×C3#12111yescharacteristic
249C22×D3—22no—
250C22×D3—42no—
251C22×D3—41no—
252C22×D3—41no—
253C22×D3—42no—
254C3×Q8—42no—
255C3×Q8—81no—
256C2×C3×C4—22no—
257C2×C3×C4—22no—
258C2×C3×C4—22no—
259C2×C3×C4—24no—
260C2×C3×C4—22no—
261C2×C3×C4—22no—
262C2×C3×C4—41no—
263C2×C3×C4—41no—
264C2×C3×C4—41no—
265C2×C3×C4—41no—
266C2×C3×C4—42no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—41no—
270C2×C3×C4—41no—
271C2×C3×C4—42no—
272C2×C3×C4—42no—
273C2×C3×C4—42no—
274C2×C3×C4—41no—
275C2×C3×C4—41no—
276C2×C3×C4—41no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4—81no—
280C2×C3×C4#12011yescharacteristic
281C2×C3×C4#12111yescharacteristic
282C2×C3×C4#12111yescharacteristic
283C2×C3×C4#12711yescharacteristic
284#67—44no—
285#67—44no—
286C2×Dic3—22no—
287C2×Dic3—41no—
288C2×Dic3—41no—
289C2×Dic3—42no—
290C2×Dic3—42no—
291C2×Dic3—42no—
292D12—22no—
293D12—22no—
294D12—41no—
295D12—41no—
296C4×D3—44no—
297C4×D3—44no—
298Dic6—22no—
299Dic6—22no—
300Dic6—41no—
301Dic6—41no—
302C3⋊C8—42no—
303C3⋊C8—81no—
304C24—61no—
305C24C2×#6711yescharacteristic
306#41—64no—
307#41—64no—
308#41—122no—
309#41—122no—
310#41—122no—
311#41—122no—
312#41—122no—
313#41—122no—
314C2×Q8—21no—
315C2×Q8—21no—
316C2×Q8—42no—
317C2×Q8—61no—
318C2×Q8—61no—
319C2×Q8—122no—
320C2×D4—64no—
321C2×D4—61no—
322C2×D4—61no—
323C2×D4—122no—
324C2×D4—122no—
325C2×D4—121no—
326C2×D4—121no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—61no—
340C22×C4—62no—
341C22×C4—122no—
342C22×C4—122no—
343C22×C4—121no—
344C22×C4—121no—
345C22×C4D3×D411yescharacteristic
346C22×C4D3×D411yescharacteristic
347C22×C4D3×D411yescharacteristic
348C22×C4#23711yescharacteristic
349C22×C4C2×#6711yescharacteristic
350C22×C4C2×#6711yescharacteristic
351Q16—124no—
352SD16—124no—
353M16—122no—
354M16—122no—
355C2×C8—62no—
356C2×C8—121no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—44no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—82no—
367C4⋊C4—124no—
368C22⋊C4—24no—
369C22⋊C4—41no—
370C22⋊C4—41no—
371C22⋊C4—124no—
372C22⋊C4—124no—
373C22⋊C4—122no—
374C42—42no—
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#39211yescharacteristic
384C22×C3#45612yes—
385C22×C3B(4,𝔽2)12yes—
386C2×D3—44no—
387C2×D3—42no—
388C2×D3—82no—
389C2×D3—81no—
390C2×D3—81no—
391C3×C4—21no—
392C3×C4—21no—
393C3×C4—21no—
394C3×C4—21no—
395C3×C4—44no—
396C3×C4—42no—
397C3×C4—81no—
398C3×C4—81no—
399C3×C4—81no—
400Dic3—42no—
401Dic3—82no—
402C23—22no—
403C23—22no—
404C23—41no—
405C23—62no—
406C23—122no—
407C23—121no—
408C23—121no—
409C23—122no—
410C23#93711yescharacteristic
411C23#93911yescharacteristic
412C23#95311yescharacteristic
413Q8—42no—
414Q8—62no—
415Q8—62no—
416Q8—81no—
417Q8—121no—
418Q8—121no—
419D4—62no—
420D4—62no—
421D4—121no—
422D4—121no—
423D4—124no—
424D4—124no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—24no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—42no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—62no—
449C2×C4—81no—
450C2×C4—124no—
451C2×C4—124no—
452C2×C4—121no—
453C2×C4—121no—
454C2×C4—122no—
455C2×C4—122no—
456C2×C4—122no—
457C2×C4#93711yescharacteristic
458C2×C4#93811yescharacteristic
459C2×C4#93911yescharacteristic
460C2×C4#94011yescharacteristic
461C8—122no—
462C8—241no—
463C2×C3—22no—
464C2×C3—22no—
465C2×C3—41no—
466C2×C3#200511yescharacteristic
467C2×C3#201011yescharacteristic
468C2×C3#202111yescharacteristic
469D3—82no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—44no—
478C22—124no—
479C22—122no—
480C22—242no—
481C22—241no—
482C22—241no—
483C22#509512yes—
484C22#522311yescharacteristic
485C22#524412yes—
486C4—21no—
487C4—21no—
488C4—21no—
489C4—21no—
490C4—44no—
491C4—42no—
492C4—81no—
493C4—81no—
494C4—81no—
495C4—122no—
496C4—242no—
497C3#994311yescharacteristic, complemented, Sylow
498C2—22no—
499C2—22no—
500C2—41no—
501C2—242no—
502C2#7171511yescharacteristic
503C2#7179511yescharacteristic
504C2#7185111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#2583C2
socleC22×C3#392
prime radicalC22×#235C22
hypercenter#2583D3
hypocenterC3#9943
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2583

Complements

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

NH ≅ G/Ntype
1. C3×#2583C2semidirect
4. #71689C2semidirect
5. #71552C2semidirect
7. #67616C2semidirect
35. #2583D3semidirect
497. C3#9943semidirect

Maximal subgroups

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

subgroupindex
C3×#25832
#827352
#821922
#716892
#715522
#694542
#676162
#99433