Finite Group Databasebeta

← #10691157

Subgroups — #10691157

Order 768.

582 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 91 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#82711C211yesmaximal, characteristic
2#78632C211yesmaximal, characteristic, complemented
3C3×#1507C211yesmaximal, characteristic, complemented
4#71673C211yesmaximal, characteristic, complemented
5C4×#806C211yesmaximal, characteristic, complemented
6#67682C211yesmaximal, characteristic, complemented
7#67616C211yesmaximal, characteristic, complemented
8#9185—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C2×C4×#67C2211yescharacteristic
11C3×#407C2211yescharacteristic
12C3×C4×C22⋊C4C2211yescharacteristic
13#5223—21no—
14#5223C411yescharacteristic, complemented
15#5222—21no—
16#5222C411yescharacteristic, complemented
17#5211C2211yescharacteristic
18C4×#217C2211yescharacteristic
19#5207C2211yescharacteristic
20#5127—22no—
21#4993C411yescharacteristic, complemented
22#4985C411yescharacteristic, complemented
23#4956—21no—
24C2×#806—21no—
25#4944—21no—
26#4728—22no—
27D3×C22⋊C8—21no—
28#4719—21no—
29C3×#327—22no—
30#4537—22no—
31#4527—21no—
32#2973—31no—
33#2635—31no—
34C4×#99—31no—
35#1507—31no—
36#1507D311yescharacteristic, complemented
37#1284—31no—
38#1284—31no—
39C22×C3×D4—21no—
40C22×C3×D4C2×C411yescharacteristic
41C23×C3×C4C2311yescharacteristic
42C22×#67—21no—
43C22×#67D411yescharacteristic
44C22×C4×D3D411yescharacteristic
45C2×C3×M16—21no—
46C3×#121—21no—
47C3×#121—21no—
48C3×#121—21no—
49C3×#121—21no—
50C2×C3×C4⋊C4C2×C411yescharacteristic
51C2×C3×C22⋊C4—21no—
52C2×C3×C22⋊C4—21no—
53C2×C3×C42—21no—
54C2×#217—21no—
55C2×#217—21no—
56C2×#217—21no—
57C2×#217—21no—
58C2×#217—21no—
59C2×#217D411yescharacteristic
60C4×#67—21no—
61C4×#67—21no—
62C4×#67—21no—
63C4×#67—21no—
64C2×(C2×D3)⋊C4—21no—
65C2×(C2×D3)⋊C4C2×C411yescharacteristic
66C2×Dic3⋊C4—21no—
67C2×Dic3⋊C4C2×C411yescharacteristic
68C2×C4×Dic3—21no—
69C2×C4×Dic3D411yescharacteristic
70C2×#208—21no—
71C2×D3×C8—22no—
72C3×C22⋊C8—21no—
73C3×C22⋊C8D412yescomplemented
74C3×#95—21no—
75C3⋊#95—21no—
76C3⋊#95—21no—
77C3⋊#95—21no—
78C3⋊#95—21no—
79C3⋊#95C2×C411yescharacteristic
80C3⋊#95C2×C411yescharacteristic
81#830—21no—
82#830D412yescomplemented
83(C2×D3)⋊C8—22no—
84#815—22no—
85Dic3⋊C8—22no—
86#806—24no—
87C2×#121C2×D311yescharacteristic
88C2×C4×D4—31no—
89#435—62no—
90#435—62no—
91#432—61no—
92C2×#99—61no—
93#407—31no—
94#407C2×D311yescharacteristic
95C2×C22⋊C8—61no—
96#400—31no—
97#399—31no—
98#394—31no—
99#394—61no—
100#392—31no—
101#392—61no—
102#388—61no—
103#386—61no—
104#381—31no—
105C4×C22⋊C4—31no—
106C4×C22⋊C4C2×D311yescharacteristic
107#342—61no—
108#327—22no—
109#327—62no—
110C24×C3—21no—
111C24×C3C2×D411yescharacteristic
112C23×D3C22⋊C411yescharacteristic
113C2×C3×D4—21no—
114C2×C3×D4—21no—
115C2×C3×D4—41no—
116C2×C3×D4—41no—
117C2×C3×D4—81no—
118C2×C3×D4—81no—
119C2×C3×D4—81no—
120C22×C3×C4—21no—
121C22×C3×C4—41no—
122C22×C3×C4—81no—
123C22×C3×C4C22×C411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4#4111yescharacteristic
128C22×C3×C4#4111yescharacteristic
129C2×#67—21no—
130C2×#67—21no—
131C2×#67—22no—
132C2×#67—41no—
133C2×#67—42no—
134C2×#67—81no—
135C2×#67—81no—
136C22×Dic3—21no—
137C22×Dic3—21no—
138C22×Dic3—41no—
139C22×Dic3C22⋊C411yescharacteristic
140C22×Dic3C22⋊C411yescharacteristic
141C22×Dic3C22⋊C411yescharacteristic
142C2×C4×D3—21no—
143C2×C4×D3—21no—
144C2×C4×D3—81no—
145C3×M16—81no—
146C2×C3×C8—22no—
147C3×C4⋊C4—21no—
148C3×C4⋊C4—21no—
149C3×C22⋊C4—22no—
150C3×C22⋊C4—41no—
151C3×C22⋊C4—41no—
152C3×C22⋊C4—41no—
153C3×C42—81no—
154#217—21no—
155#217—21no—
156#217—22no—
157#217—42no—
158#217—41no—
159#217—81no—
160#217—81no—
161#217—81no—
162(C2×D3)⋊C4—21no—
163(C2×D3)⋊C4—21no—
164(C2×D3)⋊C4—81no—
165Dic3⋊C4—21no—
166Dic3⋊C4—21no—
167Dic3⋊C4—81no—
168C4×Dic3—21no—
169C4×Dic3—21no—
170C4×Dic3—81no—
171#208—81no—
172C2×C3⋊C8—22no—
173D3×C8—82no—
174C22×D4—21no—
175C22×D4—31no—
176C22×D4—61no—
177C22×D4C4×D311yescharacteristic
178C23×C4—31no—
179C23×C4C22×D311yescharacteristic
180C2×M16—21no—
181C2×M16—61no—
182C22×C8—62no—
183#121—21no—
184#121—21no—
185#121—21no—
186#121—21no—
187C4×D4—61no—
188C4×D4—61no—
189C4×D4—61no—
190C4×D4—61no—
191C2×C4⋊C4—31no—
192C2×C4⋊C4—61no—
193C2×C4⋊C4C4×D311yescharacteristic
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—31no—
197C2×C22⋊C4—31no—
198C2×C22⋊C4—61no—
199C2×C22⋊C4—61no—
200C2×C22⋊C4—61no—
201C2×C22⋊C4—61no—
202C2×C22⋊C4—61no—
203C2×C22⋊C4—61no—
204C2×C42—21no—
205C2×C42—31no—
206C2×C42—61no—
207C4⋊C8—62no—
208#99—64no—
209C22⋊C8—21no—
210C22⋊C8—32no—
211C22⋊C8—62no—
212C22⋊C8—61no—
213C22⋊C8#6712yescomplemented
214#97—62no—
215#95—21no—
216#95—31no—
217#95—31no—
218#95—61no—
219#95—61no—
220#95—61no—
221#95—61no—
222C23×C3—21no—
223C23×C3—21no—
224C23×C3—41no—
225C23×C3—41no—
226C23×C3—41no—
227C23×C3—41no—
228C23×C3—41no—
229C23×C3—41no—
230C23×C3—41no—
231C23×C3—41no—
232C23×C3—41no—
233C23×C3C2×C22⋊C411yescharacteristic
234C23×C3#12111yescharacteristic
235C23×C3#12111yescharacteristic
236C22×D3—21no—
237C22×D3—21no—
238C22×D3—41no—
239C22×D3—41no—
240C22×D3#10412yes—
241C3×D4—81no—
242C3×D4—81no—
243C3×D4—82no—
244C3×D4—82no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—42no—
254C2×C3×C4—41no—
255C2×C3×C4—81no—
256C2×C3×C4—81no—
257C2×C3×C4C4×D411yescharacteristic
258C2×C3×C4C4×D411yescharacteristic
259C2×C3×C4#12011yescharacteristic
260C2×C3×C4#12311yescharacteristic
261#67—42no—
262#67—42no—
263#67—84no—
264C2×Dic3—21no—
265C2×Dic3—21no—
266C2×Dic3—21no—
267C2×Dic3—21no—
268C2×Dic3—21no—
269C2×Dic3—41no—
270C2×Dic3—41no—
271C2×Dic3—41no—
272C2×Dic3—42no—
273C2×Dic3—41no—
274C2×Dic3—41no—
275C2×Dic3—41no—
276C2×Dic3—41no—
277C2×Dic3—41no—
278C2×Dic3—81no—
279C2×Dic3—81no—
280C4×D3—42no—
281C4×D3—81no—
282C3×C8—42no—
283C3⋊C8—42no—
284C24—21no—
285C24—31no—
286C24D3×D411yescharacteristic
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—62no—
292C2×D4—61no—
293C2×D4—61no—
294C2×D4—81no—
295C2×D4—81no—
296C2×D4—81no—
297C2×D4—121no—
298C2×D4—122no—
299C2×D4—241no—
300C2×D4—241no—
301C22×C4—21no—
302C22×C4—31no—
303C22×C4—31no—
304C22×C4—31no—
305C22×C4—41no—
306C22×C4—61no—
307C22×C4—61no—
308C22×C4—61no—
309C22×C4—61no—
310C22×C4—81no—
311C22×C4—121no—
312C22×C4—241no—
313C22×C4C2×C4×D311yescharacteristic
314C22×C4#23511yescharacteristic
315C22×C4D3×D411yescharacteristic
316C22×C4D3×D411yescharacteristic
317C22×C4#23711yescharacteristic
318C22×C4C2×#6711yescharacteristic
319M16—81no—
320M16—241no—
321C2×C8—22no—
322C2×C8—62no—
323C2×C8—242no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—61no—
327C4⋊C4—61no—
328C4⋊C4—241no—
329C22⋊C4—22no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—61no—
334C22⋊C4—61no—
335C22⋊C4—61no—
336C22⋊C4—61no—
337C22⋊C4—62no—
338C22⋊C4—122no—
339C22⋊C4—121no—
340C22⋊C4—241no—
341C22⋊C4—241no—
342C22⋊C4—241no—
343C22⋊C4—241no—
344C42—61no—
345C42—61no—
346C42—81no—
347C42—241no—
348C22×C3—21no—
349C22×C3—21no—
350C22×C3—21no—
351C22×C3—21no—
352C22×C3—21no—
353C22×C3—21no—
354C22×C3—21no—
355C22×C3—21no—
356C22×C3—81no—
357C22×C3—81no—
358C22×C3—81no—
359C22×C3—81no—
360C22×C3—81no—
361C22×C3—81no—
362C22×C3—81no—
363C22×C3—81no—
364C22×C3—81no—
365C22×C3#38611yescharacteristic
366C22×C3C2×#10411yescharacteristic
367C22×C3#42111yescharacteristic
368C2×D3—22no—
369C2×D3—41no—
370C2×D3—41no—
371C2×D3—42no—
372C2×D3—81no—
373C3×C4—21no—
374C3×C4—21no—
375C3×C4—21no—
376C3×C4—21no—
377C3×C4—82no—
378C3×C4—81no—
379Dic3—41no—
380Dic3—41no—
381Dic3—41no—
382Dic3—81no—
383Dic3—81no—
384Dic3—82no—
385C23—21no—
386C23—21no—
387C23—32no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—61no—
398C23—61no—
399C23—121no—
400C23—121no—
401C23D3×C22⋊C411yescharacteristic
402C23#88411yescharacteristic
403C23#93811yescharacteristic
404D4—81no—
405D4—81no—
406D4—82no—
407D4—82no—
408D4—122no—
409D4—122no—
410D4—244no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—61no—
422C2×C4—61no—
423C2×C4—61no—
424C2×C4—61no—
425C2×C4—61no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—122no—
429C2×C4—121no—
430C2×C4—121no—
431C2×C4—122no—
432C2×C4—121no—
433C2×C4—121no—
434C2×C4—121no—
435C2×C4—121no—
436C2×C4—121no—
437C2×C4—121no—
438C2×C4—241no—
439C2×C4—241no—
440C2×C4—241no—
441C2×C4#67⋊C411yescharacteristic
442C2×C4#88311yescharacteristic
443C2×C4C4×#6711yescharacteristic
444C2×C4#93711yescharacteristic
445C8—42no—
446C8—122no—
447C2×C3—21no—
448C2×C3—21no—
449C2×C3—41no—
450C2×C3—41no—
451C2×C3—161no—
452C2×C3#187111yescharacteristic
453C2×C3#191011yescharacteristic
454C2×C3#191311yescharacteristic
455D3—42no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—62no—
465C22—81no—
466C22—81no—
467C22—81no—
468C22—81no—
469C22—81no—
470C22—81no—
471C22—81no—
472C22—81no—
473C22—81no—
474C22—122no—
475C22—121no—
476C22—121no—
477C22—241no—
478C22D3×#10411yescharacteristic
479C22#482111yescharacteristic
480C22#495611yescharacteristic
481C4—21no—
482C4—21no—
483C4—21no—
484C4—21no—
485C4—82no—
486C4—81no—
487C4—121no—
488C4—121no—
489C4—121no—
490C4—241no—
491C4—241no—
492C4—242no—
493C3#918511yescharacteristic, complemented, Sylow
494C2—21no—
495C2—21no—
496C2—41no—
497C2—41no—
498C2—122no—
499C2—161no—
500C2#7003411yescharacteristic
501C2#7033011yescharacteristic
502C2#7038111yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1507C2
socleC22×C3#386
prime radical#82711C2
cosocleC23×C3×C4C23
hypercenter#1507D3
hypocenterC3#9185
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1507

Complements

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

NH ≅ G/Ntype
2. #78632C2semidirect
3. C3×#1507C2semidirect
4. #71673C2semidirect
5. C4×#806C2semidirect
6. #67682C2semidirect
7. #67616C2semidirect
14. #5223C4semidirect
16. #5222C4semidirect
21. #4993C4semidirect
22. #4985C4semidirect
36. #1507D3semidirect
73. C3×C22⋊C8D4semidirect
82. #830D4semidirect
213. C22⋊C8#67semidirect
493. C3#9185semidirect

Maximal subgroups

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

subgroupindex
#827112
#786322
C3×#15072
#716732
C4×#8062
#676822
#676162
#91853