Finite Group Databasebeta

← #10692983

Subgroups — #10692983

Order 768.

958 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 958 conjugacy classes of subgroups — too many to draw. Only its 97 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#84096C211yesmaximal, characteristic
2#73560C211yesmaximal, characteristic, complemented
3C3×#1789C211yesmaximal, characteristic, complemented
4#72138C211yesmaximal, characteristic, complemented
5#69967C211yesmaximal, characteristic, complemented
6#67682C211yesmaximal, characteristic, complemented
7#67616C211yesmaximal, characteristic, complemented
8#9475—31nomaximal, Sylow
9C22×C3×#41C2211yescharacteristic
10C2×#937—21no—
11C22×(C2×D3)⋊C4C2211yescharacteristic
12C2×D3×C22⋊C4—21no—
13C2×C3×#104—22no—
14C3×#407C2211yescharacteristic
15C3×C4×C22⋊C4C2211yescharacteristic
16C2×#837—22no—
17#5231C412yescomplemented
18#5223C412yescomplemented
19#5220—21no—
20#5213—22no—
21#5211C2211yescharacteristic
22C4×#217C2211yescharacteristic
23#5207C2211yescharacteristic
24Dic3×C22⋊C4—21no—
25C4×(C2×D3)⋊C4—22no—
26#4666—22no—
27#4468—22no—
28#2404—31no—
29#2273—31no—
30#1789—31no—
31#1789D311yescharacteristic, complemented
32#1462—31no—
33#1284—31no—
34#1284—31no—
35C24×D3D411yescharacteristic
36C2×C3×#41—21no—
37C2×C3×#41—21no—
38C2×C3×#41—21no—
39C2×C3×#41—21no—
40C2×C3×#41—44no—
41C22×C3×Q8C2×C411yescharacteristic, complemented
42C22×C3×D4—21no—
43C22×C3×D4C2×C411yescharacteristic, complemented
44C23×C3×C4—21no—
45C23×C3×C4C2311yescharacteristic
46C22×#67—22no—
47C23×Dic3D411yescharacteristic
48C22×C4×D3—22no—
49C2×C3×M16—21no—
50C2×C3×C22⋊C4—21no—
51C2×C3×C42D412yescomplemented
52C2×#217—21no—
53C2×#217—21no—
54C2×#217—22no—
55#937—44no—
56C2×(C2×D3)⋊C4—22no—
57C2×(C2×D3)⋊C4—21no—
58C2×(C2×D3)⋊C4—21no—
59C2×(C2×D3)⋊C4—21no—
60C2×(C2×D3)⋊C4—21no—
61C2×(C2×D3)⋊C4—22no—
62C2×(C2×D3)⋊C4C2×C412yes—
63C2×(C2×D3)⋊C4D412yes—
64C2×C4×Dic3—22no—
65C2×C4×Dic3D412yescomplemented
66C2×#208—21no—
67D3×C22⋊C4—28no—
68C3×#104—44no—
69C3×C22⋊C8—21no—
70C3×C22⋊C8—21no—
71C3×#95—21no—
72#837—44no—
73C3⋊#95—22no—
74C3⋊#95—21no—
75C3⋊#95—21no—
76C3⋊#95—22no—
77C3⋊#95C2×C412yescomplemented
78#830—21no—
79#830—21no—
80C22×#41C2×D311yescharacteristic
81C2×#120—61no—
82C22×C22⋊C4—31no—
83C22×C22⋊C4—61no—
84C2×#104—22no—
85C2×#104—62no—
86#407—31no—
87#407C2×D311yescharacteristic
88#393—32no—
89#392—32no—
90#386—62no—
91#386—61no—
92#381—31no—
93C4×C22⋊C4—31no—
94C4×C22⋊C4—62no—
95C4×C22⋊C4—61no—
96C4×C22⋊C4—62no—
97C4×C22⋊C4C2×D311yescharacteristic
98#323—62no—
99C24×C3—21no—
100C24×C3C2×D411yescharacteristic
101C23×D3—21no—
102C23×D3—21no—
103C23×D3—24no—
104C23×D3—42no—
105C23×D3—81no—
106C23×D3C22⋊C412yes—
107C3×#41—24no—
108C3×#41—24no—
109C3×#41—42no—
110C3×#41—42no—
111C3×#41—42no—
112C3×#41—42no—
113C3×#41—42no—
114C3×#41—42no—
115C2×C3×Q8—21no—
116C2×C3×Q8—21no—
117C2×C3×Q8—42no—
118C2×C3×D4—24no—
119C2×C3×D4—21no—
120C2×C3×D4—21no—
121C2×C3×D4—42no—
122C2×C3×D4—42no—
123C2×C3×D4—41no—
124C2×C3×D4—41no—
125C22×C3×C4—21no—
126C22×C3×C4—22no—
127C22×C3×C4—42no—
128C22×C3×C4—42no—
129C22×C3×C4—42no—
130C22×C3×C4—41no—
131C22×C3×C4—41no—
132C22×C3×C4C22×C411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4#4111yescharacteristic
137C22×C3×C4#4111yescharacteristic
138C2×#67—44no—
139C2×#67—44no—
140C2×#67—82no—
141C22×Dic3—21no—
142C22×Dic3—22no—
143C22×Dic3—21no—
144C22×Dic3—22no—
145C22×Dic3—81no—
146C22×Dic3C22⋊C412yes—
147C2×C4×D3—28no—
148C2×C4×D3—82no—
149C2×C4×D3—82no—
150C3×M16—42no—
151C2×C3×C8—41no—
152C3×C22⋊C4—24no—
153C3×C42—42no—
154#217—24no—
155#217—42no—
156#217—82no—
157(C2×D3)⋊C4—28no—
158(C2×D3)⋊C4—22no—
159(C2×D3)⋊C4—22no—
160(C2×D3)⋊C4—22no—
161(C2×D3)⋊C4—22no—
162(C2×D3)⋊C4—44no—
163C4×Dic3—42no—
164C4×Dic3—82no—
165#208—42no—
166C2×C3⋊C8—41no—
167C25—31no—
168C2×#41—21no—
169C2×#41—21no—
170C2×#41—21no—
171C2×#41—21no—
172C2×#41—44no—
173C22×Q8C4×D311yescharacteristic, complemented
174C22×D4—21no—
175C22×D4—62no—
176C22×D4C4×D311yescharacteristic, complemented
177C23×C4—21no—
178C23×C4—31no—
179C23×C4—62no—
180C23×C4C22×D311yescharacteristic
181C2×M16—21no—
182C2×M16—61no—
183#120—124no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—32no—
186C2×C22⋊C4—32no—
187C2×C22⋊C4—68no—
188C2×C22⋊C4—62no—
189C2×C22⋊C4—61no—
190C2×C22⋊C4—61no—
191C2×C22⋊C4—61no—
192C2×C22⋊C4—61no—
193C2×C22⋊C4—61no—
194C2×C22⋊C4—62no—
195C2×C22⋊C4—61no—
196C2×C22⋊C4—62no—
197C2×C42—32no—
198C2×C42—62no—
199C2×C42#6712yescomplemented
200#104—44no—
201#104—124no—
202C22⋊C8—21no—
203C22⋊C8—21no—
204C22⋊C8—61no—
205C22⋊C8—61no—
206#95—21no—
207#95—32no—
208#95—62no—
209#95—62no—
210#95—61no—
211#95—61no—
212C23×C3—22no—
213C23×C3—22no—
214C23×C3—22no—
215C23×C3—41no—
216C23×C3—42no—
217C23×C3—41no—
218C23×C3—41no—
219C23×C3—42no—
220C23×C3C2×C22⋊C411yescharacteristic
221C23×C3#12011yescharacteristic
222C23×C3#12311yescharacteristic
223C22×D3—22no—
224C22×D3—22no—
225C22×D3—22no—
226C22×D3—24no—
227C22×D3—22no—
228C22×D3—42no—
229C22×D3—42no—
230C22×D3—42no—
231C22×D3—42no—
232C22×D3—88no—
233C22×D3—81no—
234C22×D3—81no—
235C22×D3#10414yescomplemented
236C3×Q8—22no—
237C3×Q8—22no—
238C3×Q8—41no—
239C3×Q8—41no—
240C3×D4—22no—
241C3×D4—22no—
242C3×D4—41no—
243C3×D4—41no—
244C3×D4—44no—
245C3×D4—44no—
246C2×C3×C4—22no—
247C2×C3×C4—22no—
248C2×C3×C4—24no—
249C2×C3×C4—22no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—44no—
255C2×C3×C4—44no—
256C2×C3×C4—42no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4—42no—
260C2×C3×C4C4×D411yescharacteristic
261C2×C3×C4C4×D411yescharacteristic
262C2×C3×C4#12111yescharacteristic
263C2×C3×C4#12111yescharacteristic
264#67—88no—
265C2×Dic3—24no—
266C2×Dic3—22no—
267C2×Dic3—42no—
268C2×Dic3—42no—
269C2×Dic3—42no—
270C2×Dic3—42no—
271C2×Dic3—42no—
272C2×Dic3—42no—
273C2×Dic3—42no—
274C2×Dic3—81no—
275C4×D3—88no—
276C3×C8—81no—
277C3⋊C8—81no—
278C24—21no—
279C24—32no—
280C24—64no—
281C24—61no—
282C24—61no—
283C24—122no—
284C24—241no—
285C24D3×D411yescharacteristic
286#41—24no—
287#41—24no—
288#41—42no—
289#41—42no—
290#41—42no—
291#41—42no—
292#41—42no—
293#41—42no—
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—42no—
297C2×D4—24no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—124no—
305C2×D4—124no—
306C2×D4—242no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—32no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—68no—
316C22×C4—62no—
317C22×C4—61no—
318C22×C4—61no—
319C22×C4—62no—
320C22×C4—242no—
321C22×C4—242no—
322C22×C4—241no—
323C22×C4C2×C4×D311yescharacteristic
324C22×C4#23511yescharacteristic
325C22×C4D3×D411yescharacteristic
326C22×C4D3×D411yescharacteristic
327C22×C4#23711yescharacteristic
328C22×C4C2×#6711yescharacteristic
329M16—42no—
330M16—122no—
331C2×C8—41no—
332C2×C8—121no—
333C22⋊C4—24no—
334C22⋊C4—68no—
335C22⋊C4—64no—
336C22⋊C4—62no—
337C22⋊C4—62no—
338C22⋊C4—62no—
339C22⋊C4—62no—
340C22⋊C4—124no—
341C22⋊C4—122no—
342C22⋊C4—242no—
343C42—42no—
344C42—122no—
345C42—242no—
346C22×C3—21no—
347C22×C3—21no—
348C22×C3—21no—
349C22×C3—21no—
350C22×C3—21no—
351C22×C3—21no—
352C22×C3—21no—
353C22×C3—44no—
354C22×C3—44no—
355C22×C3—42no—
356C22×C3—82no—
357C22×C3—81no—
358C22×C3—81no—
359C22×C3#38611yescharacteristic
360C22×C3C2×#10412yes—
361C22×C3B(4,𝔽2)12yes—
362C2×D3—24no—
363C2×D3—44no—
364C2×D3—44no—
365C2×D3—42no—
366C2×D3—42no—
367C2×D3—44no—
368C2×D3—82no—
369C2×D3—84no—
370C3×C4—21no—
371C3×C4—21no—
372C3×C4—21no—
373C3×C4—21no—
374C3×C4—44no—
375C3×C4—42no—
376Dic3—44no—
377Dic3—42no—
378Dic3—82no—
379C23—22no—
380C23—22no—
381C23—22no—
382C23—34no—
383C23—41no—
384C23—42no—
385C23—41no—
386C23—41no—
387C23—42no—
388C23—62no—
389C23—62no—
390C23—64no—
391C23—62no—
392C23—62no—
393C23—122no—
394C23—122no—
395C23—122no—
396C23—122no—
397C23—241no—
398C23—241no—
399C23—248no—
400C23D3×C22⋊C411yescharacteristic
401C23#88311yescharacteristic
402C23#93711yescharacteristic
403Q8—22no—
404Q8—22no—
405Q8—41no—
406Q8—41no—
407D4—22no—
408D4—22no—
409D4—41no—
410D4—41no—
411D4—44no—
412D4—44no—
413D4—248no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—24no—
417C2×C4—22no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—64no—
429C2×C4—62no—
430C2×C4—122no—
431C2×C4—122no—
432C2×C4—122no—
433C2×C4—122no—
434C2×C4—122no—
435C2×C4—122no—
436C2×C4—122no—
437C2×C4—248no—
438C2×C4—241no—
439C2×C4#67⋊C411yescharacteristic
440C2×C4#88411yescharacteristic
441C2×C4C4×#6711yescharacteristic
442C2×C4#93811yescharacteristic
443C8—81no—
444C8—241no—
445C2×C3—22no—
446C2×C3—22no—
447C2×C3—41no—
448C2×C3—82no—
449C2×C3#188911yescharacteristic
450C2×C3#189611yescharacteristic
451C2×C3#191111yescharacteristic
452D3—44no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—44no—
461C22—44no—
462C22—42no—
463C22—64no—
464C22—82no—
465C22—81no—
466C22—81no—
467C22—124no—
468C22—124no—
469C22—124no—
470C22—122no—
471C22—122no—
472C22—242no—
473C22—244no—
474C22D3×#10412yes—
475C22#495611yescharacteristic
476C22#503212yes—
477C4—21no—
478C4—21no—
479C4—21no—
480C4—21no—
481C4—44no—
482C4—42no—
483C4—124no—
484C4—122no—
485C4—242no—
486C3#947511yescharacteristic, complemented, Sylow
487C2—22no—
488C2—22no—
489C2—41no—
490C2—82no—
491C2—124no—
492C2#6949911yescharacteristic
493C2#7038211yescharacteristic
494C2#7040811yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1789C2
socleC22×C3#386
prime radical#84096C2
cosocleC23×C3×C4C23
hypercenter#1789D3
hypocenterC3#9475
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1789

Complements

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

NH ≅ G/Ntype
2. #73560C2semidirect
3. C3×#1789C2semidirect
4. #72138C2semidirect
5. #69967C2semidirect
6. #67682C2semidirect
7. #67616C2semidirect
17. #5231C4semidirect
18. #5223C4semidirect
31. #1789D3semidirect
41. C22×C3×Q8C2×C4semidirect
43. C22×C3×D4C2×C4semidirect
51. C2×C3×C42D4semidirect
65. C2×C4×Dic3D4semidirect
77. C3⋊#95C2×C4semidirect
173. C22×Q8C4×D3semidirect
176. C22×D4C4×D3semidirect
199. C2×C42#67semidirect
235. C22×D3#104semidirect
486. C3#9475semidirect

Maximal subgroups

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

subgroupindex
#840962
#735602
C3×#17892
#721382
#699672
#676822
#676162
#94753