Finite Group Databasebeta

← #10692993

Subgroups — #10692993

Order 768.

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

normal subgroups only

This group has 734 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#82282C211yesmaximal, characteristic
2#73560C211yesmaximal, characteristic
3C3×#1826C211yesmaximal, characteristic, complemented
4#71590C211yesmaximal, characteristic, complemented
5#69967C211yesmaximal, characteristic
6#67682C211yesmaximal, characteristic, complemented
7#67616C211yesmaximal, characteristic, complemented
8#9477—31nomaximal, Sylow
9C2×C3×#117C2211yescharacteristic
10C22×(C2×D3)⋊C4C2211yescharacteristic
11C2×D3×C22⋊C4—21no—
12C3×#407C2211yescharacteristic
13C3×C4×C22⋊C4C2211yescharacteristic
14#5211C2211yescharacteristic
15C4×#217C2211yescharacteristic
16#5207C2211yescharacteristic
17#4993C412yescomplemented
18#4992C412yescomplemented
19#4955—21no—
20#4946—21no—
21Dic3×C22⋊C4—21no—
22#4940—22no—
23C4×(C2×D3)⋊C4—22no—
24#4666—22no—
25C3×#339—22no—
26#4532—22no—
27#4468—22no—
28#2482—31no—
29#2273—31no—
30#1826—31no—
31#1826D311yescharacteristic, complemented
32#1462—31no—
33#1284—31no—
34#1284—31no—
35C24×D3D411yescharacteristic
36C23×C3×C4C2311yescharacteristic
37C23×Dic3D411yescharacteristic
38C22×C4×D3—22no—
39C2×C3×M16—21no—
40C3×#117—21no—
41C3×#117—21no—
42C3×#117—21no—
43C3×#117—21no—
44C2×C3×C4⋊C4C2×C411yescharacteristic, complemented
45C2×C3×C4⋊C4C2×C411yescharacteristic, complemented
46C2×C3×C22⋊C4—21no—
47C2×C3×C22⋊C4—21no—
48C2×C3×C42—21no—
49C2×C3×C42D412yes—
50C2×#217—21no—
51C2×#217—21no—
52C2×(C2×D3)⋊C4—22no—
53C2×(C2×D3)⋊C4—21no—
54C2×(C2×D3)⋊C4—21no—
55C2×(C2×D3)⋊C4—21no—
56C2×(C2×D3)⋊C4—21no—
57C2×(C2×D3)⋊C4—22no—
58C2×(C2×D3)⋊C4—22no—
59C2×(C2×D3)⋊C4C2×C412yes—
60C2×(C2×D3)⋊C4D412yes—
61C2×Dic3⋊C4—22no—
62C2×C4×Dic3—22no—
63C2×C4×Dic3D412yes—
64C2×#208—21no—
65D3×C22⋊C4—28no—
66C3×C22⋊C8—21no—
67C3×C22⋊C8—21no—
68C3×#95—21no—
69C3⋊#95—22no—
70C3⋊#95—21no—
71C3⋊#95—21no—
72C3⋊#95—22no—
73C3⋊#95C2×C412yescomplemented
74#830—21no—
75#830—21no—
76C2×#117C2×D311yescharacteristic
77C22×C22⋊C4—31no—
78C22×C22⋊C4—61no—
79#407—31no—
80#407C2×D311yescharacteristic
81#399—32no—
82#397—32no—
83#388—62no—
84#385—61no—
85#381—31no—
86#379—61no—
87C4×C22⋊C4—31no—
88C4×C22⋊C4—62no—
89C4×C22⋊C4—61no—
90C4×C22⋊C4—62no—
91C4×C22⋊C4C2×D311yescharacteristic
92#339—22no—
93#339—62no—
94#323—62no—
95C24×C3C2×D411yescharacteristic
96C23×D3—21no—
97C23×D3—21no—
98C23×D3—24no—
99C23×D3—42no—
100C23×D3—81no—
101C23×D3C22⋊C412yes—
102C22×C3×C4—22no—
103C22×C3×C4—21no—
104C22×C3×C4—21no—
105C22×C3×C4—42no—
106C22×C3×C4C22×C411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4#4111yescharacteristic
111C22×C3×C4#4111yescharacteristic
112C22×Dic3—21no—
113C22×Dic3—22no—
114C22×Dic3—21no—
115C22×Dic3—22no—
116C22×Dic3—81no—
117C22×Dic3C22⋊C412yes—
118C2×C4×D3—28no—
119C2×C4×D3—82no—
120C2×C4×D3—82no—
121C3×M16—42no—
122C2×C3×C8—41no—
123C3×C4⋊C4—21no—
124C3×C4⋊C4—21no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—21no—
127C3×C22⋊C4—24no—
128C3×C22⋊C4—41no—
129C3×C22⋊C4—41no—
130C3×C42—42no—
131C3×C42—41no—
132C3×C42—41no—
133#217—24no—
134#217—42no—
135(C2×D3)⋊C4—28no—
136(C2×D3)⋊C4—22no—
137(C2×D3)⋊C4—22no—
138(C2×D3)⋊C4—22no—
139(C2×D3)⋊C4—22no—
140(C2×D3)⋊C4—44no—
141(C2×D3)⋊C4—82no—
142Dic3⋊C4—82no—
143C4×Dic3—42no—
144C4×Dic3—82no—
145#208—42no—
146C2×C3⋊C8—41no—
147C25—31no—
148C23×C4—31no—
149C23×C4—62no—
150C23×C4C22×D311yescharacteristic
151C2×M16—21no—
152C2×M16—61no—
153#117—21no—
154#117—21no—
155#117—21no—
156#117—21no—
157C2×C4⋊C4—62no—
158C2×C4⋊C4C4×D311yescharacteristic, complemented
159C2×C4⋊C4C4×D311yescharacteristic, complemented
160C2×C22⋊C4—21no—
161C2×C22⋊C4—21no—
162C2×C22⋊C4—32no—
163C2×C22⋊C4—32no—
164C2×C22⋊C4—68no—
165C2×C22⋊C4—62no—
166C2×C22⋊C4—61no—
167C2×C22⋊C4—61no—
168C2×C22⋊C4—61no—
169C2×C22⋊C4—61no—
170C2×C22⋊C4—61no—
171C2×C22⋊C4—62no—
172C2×C22⋊C4—61no—
173C2×C22⋊C4—62no—
174C2×C42—21no—
175C2×C42—32no—
176C2×C42—62no—
177C2×C42#6712yes—
178C22⋊C8—21no—
179C22⋊C8—21no—
180C22⋊C8—61no—
181C22⋊C8—61no—
182#95—21no—
183#95—32no—
184#95—62no—
185#95—62no—
186#95—61no—
187#95—61no—
188C23×C3—22no—
189C23×C3—22no—
190C23×C3—41no—
191C23×C3C2×C22⋊C411yescharacteristic
192C23×C3#12011yescharacteristic
193C23×C3#12311yescharacteristic
194C22×D3—22no—
195C22×D3—22no—
196C22×D3—22no—
197C22×D3—24no—
198C22×D3—22no—
199C22×D3—42no—
200C22×D3—42no—
201C22×D3—42no—
202C22×D3—42no—
203C22×D3—88no—
204C22×D3—81no—
205C22×D3—81no—
206C22×D3#10414yes—
207C2×C3×C4—22no—
208C2×C3×C4—22no—
209C2×C3×C4—24no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—42no—
215C2×C3×C4—42no—
216C2×C3×C4—42no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—42no—
220C2×C3×C4C4×D411yescharacteristic
221C2×C3×C4C4×D411yescharacteristic
222C2×C3×C4#12111yescharacteristic
223C2×C3×C4#12111yescharacteristic
224C2×Dic3—24no—
225C2×Dic3—22no—
226C2×Dic3—42no—
227C2×Dic3—42no—
228C2×Dic3—42no—
229C2×Dic3—42no—
230C2×Dic3—42no—
231C2×Dic3—42no—
232C2×Dic3—42no—
233C2×Dic3—81no—
234C4×D3—88no—
235C3×C8—81no—
236C3⋊C8—81no—
237C24—32no—
238C24—64no—
239C24—61no—
240C24—61no—
241C24—122no—
242C24—241no—
243C24D3×D411yescharacteristic
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—32no—
248C22×C4—42no—
249C22×C4—68no—
250C22×C4—62no—
251C22×C4—61no—
252C22×C4—61no—
253C22×C4—62no—
254C22×C4—242no—
255C22×C4—242no—
256C22×C4—241no—
257C22×C4C2×C4×D311yescharacteristic
258C22×C4#23511yescharacteristic
259C22×C4D3×D411yescharacteristic
260C22×C4D3×D411yescharacteristic
261C22×C4#23711yescharacteristic
262C22×C4C2×#6711yescharacteristic
263M16—42no—
264M16—122no—
265C2×C8—41no—
266C2×C8—121no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—242no—
272C22⋊C4—24no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—68no—
276C22⋊C4—64no—
277C22⋊C4—62no—
278C22⋊C4—62no—
279C22⋊C4—62no—
280C22⋊C4—62no—
281C22⋊C4—124no—
282C22⋊C4—122no—
283C22⋊C4—242no—
284C42—42no—
285C42—41no—
286C42—41no—
287C42—122no—
288C42—242no—
289C22×C3—21no—
290C22×C3—21no—
291C22×C3—21no—
292C22×C3—21no—
293C22×C3—21no—
294C22×C3—21no—
295C22×C3—21no—
296C22×C3—44no—
297C22×C3#38611yescharacteristic
298C22×C3C2×#10412yes—
299C22×C3#45812yes—
300C2×D3—24no—
301C2×D3—44no—
302C2×D3—44no—
303C2×D3—42no—
304C2×D3—42no—
305C2×D3—44no—
306C2×D3—82no—
307C2×D3—84no—
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—44no—
313C3×C4—81no—
314C3×C4—81no—
315Dic3—44no—
316Dic3—42no—
317Dic3—82no—
318C23—22no—
319C23—22no—
320C23—34no—
321C23—41no—
322C23—62no—
323C23—62no—
324C23—64no—
325C23—62no—
326C23—62no—
327C23—122no—
328C23—122no—
329C23—122no—
330C23—122no—
331C23—241no—
332C23—241no—
333C23—248no—
334C23D3×C22⋊C411yescharacteristic
335C23#88311yescharacteristic
336C23#93711yescharacteristic
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—24no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—64no—
351C2×C4—62no—
352C2×C4—122no—
353C2×C4—122no—
354C2×C4—122no—
355C2×C4—122no—
356C2×C4—122no—
357C2×C4—122no—
358C2×C4—122no—
359C2×C4—248no—
360C2×C4—241no—
361C2×C4#67⋊C411yescharacteristic
362C2×C4#88411yescharacteristic
363C2×C4C4×#6711yescharacteristic
364C2×C4#93811yescharacteristic
365C8—81no—
366C8—241no—
367C2×C3—22no—
368C2×C3—22no—
369C2×C3—41no—
370C2×C3#189011yescharacteristic
371C2×C3#189611yescharacteristic
372C2×C3#191111yescharacteristic
373D3—44no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—44no—
382C22—64no—
383C22—124no—
384C22—124no—
385C22—124no—
386C22—122no—
387C22—122no—
388C22—242no—
389C22—244no—
390C22D3×#10412yes—
391C22#495611yescharacteristic
392C22#505312yes—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—44no—
398C4—81no—
399C4—81no—
400C4—124no—
401C4—122no—
402C4—242no—
403C3#947711yescharacteristic, complemented, Sylow
404C2—22no—
405C2—22no—
406C2—41no—
407C2—124no—
408C2#6953811yescharacteristic
409C2#7038211yescharacteristic
410C2#7040811yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1826C2
socleC22×C3#386
prime radicalC24×D3D4
cosocleC23×C3×C4C23
hypercenter#1826D3
hypocenterC3#9477
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1826

Complements

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

NH ≅ G/Ntype
3. C3×#1826C2semidirect
4. #71590C2semidirect
6. #67682C2semidirect
7. #67616C2semidirect
17. #4993C4semidirect
18. #4992C4semidirect
31. #1826D3semidirect
44. C2×C3×C4⋊C4C2×C4semidirect
45. C2×C3×C4⋊C4C2×C4semidirect
73. C3⋊#95C2×C4semidirect
158. C2×C4⋊C4C4×D3semidirect
159. C2×C4⋊C4C4×D3semidirect
403. C3#9477semidirect

Maximal subgroups

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

subgroupindex
#822822
#735602
C3×#18262
#715902
#699672
#676822
#676162
#94773