Finite Group Databasebeta

← #10692224

Subgroups — #10692224

Order 768.

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

normal subgroups only

This group has 576 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×#3192C211yesmaximal, characteristic
2#82639C211yesmaximal, characteristic, complemented
3#71686C211yesmaximal, characteristic, complemented
4#71674C211yesmaximal, characteristic, complemented
5#71673C211yesmaximal, characteristic, complemented
6#67685C211yesmaximal, characteristic, complemented
7#67682C211yesmaximal, characteristic, complemented
8#9343—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C2×C3×#121C2211yescharacteristic
11C3×#481—21no—
12C3×#481C412yescomplemented
13C3×#469—22no—
14C3×#447—21no—
15C3×#447C412yescomplemented
16C3×#407C2211yescharacteristic
17#5221—21no—
18#5220—21no—
19#5220—21no—
20C2×#834—21no—
21C2×#834—21no—
22#5212C2211yescharacteristic
23#5211C2211yescharacteristic
24C4×#217C2211yescharacteristic
25#5207C2211yescharacteristic
26#4980—21no—
27#4951—21no—
28#4950—21no—
29#4537—22no—
30#4537—21no—
31#4537—21no—
32#3192D311yescharacteristic
33#2377—31no—
34#1964—31no—
35#1510—31no—
36#1507—31no—
37#1305—31no—
38#1284—31no—
39C22×C3×D4—21no—
40C22×C3×D4—21no—
41C22×C3×D4C2×C411yescharacteristic, complemented
42C22×C3×D4D411yescharacteristic
43C23×C3×C4C2311yescharacteristic
44C23×Dic3—21no—
45C2×C3×D8—22no—
46C2×C3×M16—21no—
47C3×#121—21no—
48C3×#121—21no—
49C3×#121—21no—
50C3×#121—21no—
51C3×#121—21no—
52C3×#121—21no—
53C3×#121C2×C412yes—
54C3×#121D412yes—
55C2×C3×C4⋊C4C2×C411yescharacteristic, complemented
56C2×C3×C4⋊C4D411yescharacteristic
57C2×C3×C22⋊C4—21no—
58C2×C3×C22⋊C4—21no—
59C2×#217—21no—
60C2×#217—21no—
61C2×#217—21no—
62C2×#217—21no—
63C2×#217—21no—
64C2×#217D411yescharacteristic
65C2×#217D411yescharacteristic
66C2×C4⋊Dic3—21no—
67C2×C4⋊Dic3—21no—
68C2×C4×Dic3—21no—
69C2×C4×Dic3—21no—
70C2×#208—21no—
71C3×#106—22no—
72C3×#102—22no—
73C3×#102—22no—
74C3×C22⋊C8—21no—
75C3×C22⋊C8C2×C412yescomplemented
76#834—42no—
77#834—42no—
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×#121C2×D311yescharacteristic
88C2×#121C2×D311yescharacteristic
89#481—21no—
90#481Dic312yescomplemented
91#469—22no—
92#447—21no—
93#447Dic312yescomplemented
94C2×#99—61no—
95C2×#99—61no—
96#407—31no—
97#407C2×D311yescharacteristic
98#390—61no—
99#388—61no—
100#387—61no—
101#386—61no—
102#386—61no—
103#385—61no—
104#381—31no—
105#380—31no—
106C4×C22⋊C4—31no—
107#327—61no—
108#327—61no—
109#327—62no—
110C24×C3—21no—
111C24×C3—21no—
112C24×C3C2×D411yescharacteristic
113C2×C3×D4—22no—
114C2×C3×D4—22no—
115C2×C3×D4—21no—
116C2×C3×D4—21no—
117C2×C3×D4—21no—
118C2×C3×D4—41no—
119C2×C3×D4—41no—
120C2×C3×D4—41no—
121C2×C3×D4—41no—
122C2×C3×D4—41no—
123C2×C3×D4—41no—
124C2×C3×D4—81no—
125C2×C3×D4—81no—
126C2×C3×D4—81no—
127C2×C3×D4—81no—
128C2×C3×D4C22⋊C412yes—
129C22×C3×C4—21no—
130C22×C3×C4—21no—
131C22×C3×C4—81no—
132C22×C3×C4C22×C411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4#4111yescharacteristic
137C22×C3×C4#4111yescharacteristic
138C22×Dic3—21no—
139C22×Dic3—21no—
140C22×Dic3—21no—
141C22×Dic3—21no—
142C22×Dic3—41no—
143C22×Dic3—81no—
144C22×Dic3—81no—
145C22×Dic3—81no—
146C3×D8—82no—
147C3×M16—81no—
148C2×C3×C8—22no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4—21no—
152C3×C4⋊C4C22⋊C412yes—
153C3×C22⋊C4—22no—
154C3×C22⋊C4—22no—
155C3×C22⋊C4—41no—
156C3×C22⋊C4—41no—
157#217—22no—
158#217—41no—
159#217—41no—
160#217—41no—
161#217—81no—
162#217—81no—
163#217—81no—
164#217—81no—
165C4⋊Dic3—81no—
166C4⋊Dic3—81no—
167C4×Dic3—81no—
168C4×Dic3—81no—
169#208—81no—
170C2×C3⋊C8—41no—
171C22×D4—21no—
172C22×D4—21no—
173C22×D4C2×Dic311yescharacteristic, complemented
174C22×D4#6711yescharacteristic
175C23×C4—61no—
176C23×C4C22×D311yescharacteristic
177C2×D8—22no—
178C2×M16—21no—
179C2×M16—61no—
180#121—21no—
181#121—21no—
182#121—21no—
183#121—21no—
184#121—21no—
185#121—21no—
186#121C2×Dic312yes—
187#121#6712yes—
188C2×C4⋊C4—61no—
189C2×C4⋊C4—61no—
190C2×C4⋊C4C2×Dic311yescharacteristic, complemented
191C2×C4⋊C4#6711yescharacteristic
192C2×C22⋊C4—21no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—31no—
195C2×C22⋊C4—31no—
196C2×C22⋊C4—61no—
197C2×C22⋊C4—61no—
198C2×C22⋊C4—61no—
199C2×C22⋊C4—61no—
200C2×C22⋊C4—61no—
201C2×C42—61no—
202C2×C42—61no—
203#106—22no—
204#102—22no—
205#102—22no—
206#99—122no—
207#99—122no—
208C22⋊C8—21no—
209C22⋊C8—61no—
210C22⋊C8—61no—
211C22⋊C8C2×Dic312yescomplemented
212#95—31no—
213#95—31no—
214#95—61no—
215#95—61no—
216#95—61no—
217#95—61no—
218#95—61no—
219C23×C3—21no—
220C23×C3—21no—
221C23×C3—22no—
222C23×C3—41no—
223C23×C3—41no—
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×C3—81no—
234C23×C3—81no—
235C23×C3C4×D411yescharacteristic
236C23×C3#12011yescharacteristic
237C23×C3#12111yescharacteristic
238C3×D4—42no—
239C3×D4—42no—
240C3×D4—81no—
241C3×D4—81no—
242C3×D4—82no—
243C3×D4—82no—
244C3×D4—81no—
245C3×D4—81no—
246C3×D4—81no—
247C3×D4—81no—
248C2×C3×C4—22no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—41no—
255C2×C3×C4—41no—
256C2×C3×C4—41no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4—41no—
260C2×C3×C4C2×C22⋊C411yescharacteristic
261C2×C3×C4C4×D411yescharacteristic
262C2×C3×C4#12111yescharacteristic
263C2×C3×C4#12311yescharacteristic
264C2×Dic3—41no—
265C2×Dic3—41no—
266C2×Dic3—41no—
267C2×Dic3—41no—
268C2×Dic3—41no—
269C2×Dic3—42no—
270C2×Dic3—41no—
271C2×Dic3—41no—
272C2×Dic3—41no—
273C2×Dic3—41no—
274C2×Dic3—41no—
275C2×Dic3—41no—
276C2×Dic3—41no—
277C2×Dic3—81no—
278C2×Dic3—81no—
279C2×Dic3—84no—
280C3×C8—42no—
281C3⋊C8—81no—
282C24—21no—
283C24—21no—
284C24D3×D411yescharacteristic
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—81no—
297C2×D4—81no—
298C2×D4—81no—
299C2×D4—81no—
300C2×D4#21712yes—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—61no—
304C22×C4—61no—
305C22×C4—61no—
306C22×C4—61no—
307C22×C4—81no—
308C22×C4—121no—
309C22×C4—241no—
310C22×C4—241no—
311C22×C4—241no—
312C22×C4D3×D411yescharacteristic
313C22×C4#23711yescharacteristic
314C22×C4#23711yescharacteristic
315C22×C4C22×Dic311yescharacteristic
316C22×C4C2×#6711yescharacteristic
317C22×C4C2×#6711yescharacteristic
318D8—82no—
319M16—81no—
320M16—241no—
321C2×C8—22no—
322C2×C8—121no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—241no—
327C4⋊C4—241no—
328C4⋊C4#21712yes—
329C22⋊C4—22no—
330C22⋊C4—22no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—62no—
334C22⋊C4—121no—
335C22⋊C4—121no—
336C22⋊C4—121no—
337C22⋊C4—241no—
338C22⋊C4—241no—
339C22⋊C4—241no—
340C22⋊C4—241no—
341C42—241no—
342C42—241no—
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—21no—
346C22×C3—21no—
347C22×C3—21no—
348C22×C3—21no—
349C22×C3—21no—
350C22×C3—21no—
351C22×C3—42no—
352C22×C3—42no—
353C22×C3—42no—
354C22×C3—81no—
355C22×C3—81no—
356C22×C3—81no—
357C22×C3—81no—
358C22×C3—81no—
359C22×C3—81no—
360C22×C3—81no—
361C22×C3—81no—
362C22×C3—81no—
363C22×C3—161no—
364C22×C3—161no—
365C22×C3#38611yescharacteristic
366C22×C3#45311yescharacteristic
367C22×C3#45411yescharacteristic
368C3×C4—21no—
369C3×C4—21no—
370C3×C4—21no—
371C3×C4—21no—
372C3×C4—42no—
373C3×C4—81no—
374Dic3—81no—
375Dic3—81no—
376Dic3—81no—
377Dic3—81no—
378Dic3—82no—
379C23—21no—
380C23—21no—
381C23—22no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—81no—
394C23—81no—
395C23D4×Dic311yescharacteristic
396C23#93711yescharacteristic
397C23#93911yescharacteristic
398D4—42no—
399D4—42no—
400D4—81no—
401D4—81no—
402D4—82no—
403D4—82no—
404D4—81no—
405D4—81no—
406D4—81no—
407D4—81no—
408C2×C4—22no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—121no—
421C2×C4—121no—
422C2×C4—122no—
423C2×C4—121no—
424C2×C4—121no—
425C2×C4—121no—
426C2×C4—121no—
427C2×C4—121no—
428C2×C4—121no—
429C2×C4—121no—
430C2×C4—121no—
431C2×C4—121no—
432C2×C4—121no—
433C2×C4—241no—
434C2×C4—241no—
435C2×C4—244no—
436C2×C4D4×Dic311yescharacteristic
437C2×C4#93511yescharacteristic
438C2×C4#93911yescharacteristic
439C2×C4C2×#21711yescharacteristic
440C8—42no—
441C8—241no—
442C2×C3—21no—
443C2×C3—21no—
444C2×C3—41no—
445C2×C3—41no—
446C2×C3—82no—
447C2×C3—161no—
448C2×C3#188311yescharacteristic
449C2×C3#190411yescharacteristic
450C2×C3#191311yescharacteristic
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—42no—
460C22—42no—
461C22—42no—
462C22—81no—
463C22—81no—
464C22—81no—
465C22—81no—
466C22—81no—
467C22—81no—
468C22—81no—
469C22—81no—
470C22—81no—
471C22—161no—
472C22—161no—
473C22#519811yescharacteristic
474C22#520111yescharacteristic
475C22#522011yescharacteristic
476C4—21no—
477C4—21no—
478C4—21no—
479C4—21no—
480C4—42no—
481C4—81no—
482C4—241no—
483C4—241no—
484C4—241no—
485C4—241no—
486C4—242no—
487C3#934311yescharacteristic, complemented, Sylow
488C2—21no—
489C2—21no—
490C2—41no—
491C2—41no—
492C2—82no—
493C2—161no—
494C2#7184811yescharacteristic
495C2#7208811yescharacteristic
496C2#7208911yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
Fitting, prime radicalC3×#3192C2
socleC22×C3#386
cosocleC23×C3×C4C23
hypercenter#3192D3
hypocenterC3#9343
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3192

Complements

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

NH ≅ G/Ntype
2. #82639C2semidirect
3. #71686C2semidirect
4. #71674C2semidirect
5. #71673C2semidirect
6. #67685C2semidirect
7. #67682C2semidirect
12. C3×#481C4semidirect
15. C3×#447C4semidirect
41. C22×C3×D4C2×C4semidirect
55. C2×C3×C4⋊C4C2×C4semidirect
75. C3×C22⋊C8C2×C4semidirect
90. #481Dic3semidirect
93. #447Dic3semidirect
173. C22×D4C2×Dic3semidirect
190. C2×C4⋊C4C2×Dic3semidirect
211. C22⋊C8C2×Dic3semidirect
487. C3#9343semidirect

Maximal subgroups

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

subgroupindex
C3×#31922
#826392
#716862
#716742
#716732
#676852
#676822
#93433