Finite Group Databasebeta

← #10692827

Subgroups — #10692827

Order 768.

712 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 111 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×#2432C211yesmaximal, characteristic
2#84074C211yesmaximal, characteristic
3#72139C211yesmaximal, characteristic
4#71689C212yesmaximal, complemented
5#67616C212yesmaximal, complemented
6#9443—31nomaximal, Sylow
7C22×C3×#41C2211yescharacteristic
8C2×C3×C4⋊Q8C411yescharacteristic, complemented
9C2×C3×#124C411yescharacteristic, complemented
10C2×C3×C22⋊Q8C2211yescharacteristic
11C22×#208—21no—
12C3×#394—21no—
13C3×#394—21no—
14C3×#391C411yescharacteristic, complemented
15C3×#390C411yescharacteristic, complemented
16C3×#385—21no—
17C3×#385—21no—
18C3×C4×C22⋊C4C2211yescharacteristic
19#5234—24no—
20C2×#834—21no—
21C2×#834—21no—
22#5211C2212yes—
23#5207—21no—
24#5207C2212yes—
25#4538—24no—
26#2879—31no—
27#2432D311yescharacteristic
28#1782—31no—
29#1511—32no—
30#1284—32no—
31C2×C3×#41—22no—
32C2×C3×#41—22no—
33C2×C3×#41—42no—
34C2×C3×#41—42no—
35C22×C3×Q8C2×C411yescharacteristic
36C22×C3×Q8C2×C411yescharacteristic
37C22×C3×D4C2×C411yescharacteristic
38C22×C3×D4D411yescharacteristic
39C22×C3×D4D411yescharacteristic
40C23×C3×C4D411yescharacteristic
41C23×C3×C4D411yescharacteristic
42C23×C3×C4C2311yescharacteristic
43C3×C4⋊Q8—42no—
44C3×#124—42no—
45C3×C22⋊Q8—22no—
46C3×C22⋊Q8—22no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4C2×C411yescharacteristic
51C2×C3×C22⋊C4—21no—
52C2×C3×C22⋊C4—21no—
53C2×C3×C22⋊C4—21no—
54C2×C3×C22⋊C4—21no—
55C2×C3×C22⋊C4—21no—
56C2×C3×C22⋊C4—21no—
57C2×C3×C42C2×C411yescharacteristic
58C2×C3×C42C2×C411yescharacteristic
59C2×#217—22no—
60C2×#208—22no—
61C2×#208—22no—
62C2×#208—42no—
63C2×#208—42no—
64C22×C3⋊C8—22no—
65C3×#95—21no—
66C3×#95—21no—
67C3×#95—21no—
68#834—42no—
69#834—42no—
70C3⋊#95—22no—
71C3⋊#95—22no—
72#830—24no—
73#830—22no—
74#830D414yes—
75C22×#41C2×D311yescharacteristic
76C22×M16—61no—
77C2×C4⋊Q8Dic311yescharacteristic, complemented
78C2×#124Dic311yescharacteristic, complemented
79C2×C22⋊Q8C2×D311yescharacteristic
80C2×#99—61no—
81C2×#99—61no—
82#408—64no—
83#407—32no—
84#407—61no—
85#394—21no—
86#394—21no—
87#391Dic311yescharacteristic, complemented
88#390Dic311yescharacteristic, complemented
89#385—21no—
90#385—21no—
91#381—32no—
92C4×C22⋊C4C2×D311yescharacteristic
93#328—64no—
94C24×C3C22⋊C411yescharacteristic
95C24×C3C22⋊C411yescharacteristic
96C24×C3C2×D411yescharacteristic
97C3×#41—44no—
98C3×#41—44no—
99C3×#41—48no—
100C2×C3×Q8—22no—
101C2×C3×Q8—22no—
102C2×C3×Q8—22no—
103C2×C3×Q8—41no—
104C2×C3×Q8—41no—
105C2×C3×Q8—41no—
106C2×C3×D4—22no—
107C2×C3×D4—22no—
108C2×C3×D4—22no—
109C2×C3×D4—22no—
110C2×C3×D4—22no—
111C2×C3×D4—22no—
112C2×C3×D4—41no—
113C2×C3×D4—41no—
114C2×C3×D4—41no—
115C22×C3×C4—22no—
116C22×C3×C4—22no—
117C22×C3×C4—21no—
118C22×C3×C4—21no—
119C22×C3×C4—21no—
120C22×C3×C4—21no—
121C22×C3×C4—21no—
122C22×C3×C4—21no—
123C22×C3×C4—21no—
124C22×C3×C4—21no—
125C22×C3×C4—41no—
126C22×C3×C4—41no—
127C22×C3×C4—41no—
128C22×C3×C4—41no—
129C22×C3×C4C22⋊C411yescharacteristic
130C22×C3×C4C22⋊C411yescharacteristic
131C22×C3×C4C22×C411yescharacteristic
132C22×C3×C4C2×D412yes—
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4#4112yes—
135C22×Dic3—42no—
136C3×C4⋊C4—22no—
137C3×C4⋊C4—42no—
138C3×C4⋊C4—42no—
139C3×C4⋊C4—41no—
140C3×C4⋊C4—41no—
141C3×C22⋊C4—42no—
142C3×C22⋊C4—42no—
143C3×C22⋊C4—42no—
144C3×C22⋊C4—42no—
145C3×C22⋊C4—42no—
146C3×C22⋊C4—42no—
147C3×C42—41no—
148C3×C42—41no—
149#217—42no—
150#217—42no—
151#208—44no—
152#208—42no—
153#208—42no—
154C2×C3⋊C8—24no—
155C2×C3⋊C8—44no—
156C2×#41—22no—
157C2×#41—22no—
158C2×#41—42no—
159C2×#41—42no—
160C22×Q8C2×Dic311yescharacteristic
161C22×Q8C2×Dic311yescharacteristic
162C22×D4C2×Dic311yescharacteristic
163C22×D4#6711yescharacteristic
164C22×D4#6711yescharacteristic
165C23×C4#6711yescharacteristic
166C23×C4#6711yescharacteristic
167C23×C4C22×D311yescharacteristic
168C2×M16—62no—
169C2×M16—62no—
170C2×M16—122no—
171C2×M16—122no—
172C22×C8—62no—
173C4⋊Q8—42no—
174#124—42no—
175C22⋊Q8—22no—
176C22⋊Q8—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4C2×Dic311yescharacteristic
181C2×C22⋊C4—21no—
182C2×C22⋊C4—21no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—62no—
188C2×C42C2×Dic311yescharacteristic
189C2×C42C2×Dic311yescharacteristic
190#99—122no—
191#99—122no—
192C22⋊C8—34no—
193C22⋊C8—64no—
194C22⋊C8—62no—
195#95—21no—
196#95—21no—
197#95—21no—
198#95—62no—
199#95—62no—
200C23×C3—21no—
201C23×C3—21no—
202C23×C3—21no—
203C23×C3—21no—
204C23×C3—21no—
205C23×C3—21no—
206C23×C3—21no—
207C23×C3—41no—
208C23×C3—41no—
209C23×C3—41no—
210C23×C3—41no—
211C23×C3—41no—
212C23×C3—41no—
213C23×C3#9912yes—
214C23×C3C2×C22⋊C411yescharacteristic
215C23×C3#12112yes—
216C3×Q8—44no—
217C3×Q8—82no—
218C3×D4—44no—
219C3×D4—44no—
220C3×D4—44no—
221C2×C3×C4—22no—
222C2×C3×C4—22no—
223C2×C3×C4—24no—
224C2×C3×C4—21no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—21no—
230C2×C3×C4—22no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—42no—
235C2×C3×C4—42no—
236C2×C3×C4—42no—
237C2×C3×C4—42no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4#9912yes—
250C2×C3×C4C4×D412yes—
251C2×C3×C4#12011yescharacteristic
252C2×C3×C4#12311yescharacteristic
253C2×Dic3—82no—
254C2×Dic3—82no—
255C2×Dic3—82no—
256C3⋊C8—44no—
257C24#21711yescharacteristic
258C24#21711yescharacteristic
259C24C2×#6711yescharacteristic
260#41—44no—
261#41—44no—
262#41—48no—
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—22no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×D4—22no—
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—22no—
274C2×D4—22no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—122no—
293C22×C4#21711yescharacteristic
294C22×C4#21711yescharacteristic
295C22×C4D3×D412yes—
296C22×C4#23712yes—
297C22×C4C22×Dic311yescharacteristic
298C22×C4C2×#6711yescharacteristic
299M16—124no—
300M16—122no—
301M16—122no—
302C2×C8—64no—
303C2×C8—124no—
304C4⋊C4—22no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C22⋊C4—42no—
312C22⋊C4—42no—
313C22⋊C4—42no—
314C22⋊C4—42no—
315C22⋊C4—122no—
316C22⋊C4—122no—
317C42—41no—
318C42—41no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—22no—
322C22×C3—22no—
323C22×C3—22no—
324C22×C3—22no—
325C22×C3—42no—
326C22×C3—42no—
327C22×C3—41no—
328C22×C3—41no—
329C22×C3—41no—
330C22×C3—41no—
331C22×C3—41no—
332C22×C3—81no—
333C22×C3—81no—
334C22×C3—81no—
335C22×C3—81no—
336C22×C3#38611yescharacteristic
337C22×C3C2×#9911yescharacteristic
338C22×C3C2×#9911yescharacteristic
339C3×C4—21no—
340C3×C4—21no—
341C3×C4—22no—
342C3×C4—22no—
343C3×C4—42no—
344C3×C4—41no—
345C3×C4—81no—
346C3×C4—81no—
347C3×C4—81no—
348Dic3—162no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23#83412yes—
363C23#93912yes—
364C23C2×#21711yescharacteristic
365Q8—44no—
366Q8—82no—
367D4—44no—
368D4—44no—
369D4—44no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—24no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—22no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—242no—
399C2×C4—242no—
400C2×C4—242no—
401C2×C4#83412yes—
402C2×C4D4×Dic312yes—
403C2×C4#93511yescharacteristic
404C2×C4#93711yescharacteristic
405C8—124no—
406C2×C3—21no—
407C2×C3—21no—
408C2×C3—42no—
409C2×C3—42no—
410C2×C3—81no—
411C2×C3#188411yescharacteristic
412C2×C3#189811yescharacteristic
413C2×C3#190911yescharacteristic
414C22—21no—
415C22—21no—
416C22—22no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22—42no—
421C22—42no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—41no—
427C22—81no—
428C22—81no—
429C22—81no—
430C22—81no—
431C22C2×#83411yescharacteristic
432C22C2×#83411yescharacteristic
433C22#522011yescharacteristic
434C4—21no—
435C4—21no—
436C4—22no—
437C4—22no—
438C4—42no—
439C4—41no—
440C4—81no—
441C4—81no—
442C4—81no—
443C4—482no—
444C3#944311yescharacteristic, complemented, Sylow
445C2—21no—
446C2—21no—
447C2—42no—
448C2—42no—
449C2—81no—
450C2#7167011yescharacteristic
451C2#7181711yescharacteristic
452C2#7186511yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2432C2
socleC22×C3#386
prime radicalC22×C3×#41C22
cosocleC23×C3×C4C23
hypercenter#2432D3
hypocenterC3#9443
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2432

Complements

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

NH ≅ G/Ntype
4. #71689C2semidirect
5. #67616C2semidirect
8. C2×C3×C4⋊Q8C4semidirect
9. C2×C3×#124C4semidirect
14. C3×#391C4semidirect
15. C3×#390C4semidirect
77. C2×C4⋊Q8Dic3semidirect
78. C2×#124Dic3semidirect
87. #391Dic3semidirect
88. #390Dic3semidirect
444. C3#9443semidirect

Maximal subgroups

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

subgroupindex
C3×#24322
#840742
#721392
#716892
#676162
#94433