Finite Group Databasebeta

← #10690730

Subgroups — #10690730

Order 768.

648 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 115 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#82395C211yesmaximal, characteristic
2#82302C211yesmaximal, characteristic, complemented
3C3×#1459C211yesmaximal, characteristic, complemented
4C2×#4468C212yesmaximal, complemented
5#67677C212yesmaximal, complemented
6#9109—31nomaximal, Sylow
7C2×C3×C4×D4C2211yescharacteristic
8C2×#937C411yescharacteristic
9C22×(C2×D3)⋊C4C2211yescharacteristic
10C2×C3×C22⋊C8C2212yes—
11#5220C411yescharacteristic
12#5213—21no—
13C4×#217C2211yescharacteristic
14#5207—21no—
15C2×#830C2212yes—
16#5011—24no—
17#4990—21no—
18#4957—21no—
19#4955C411yescharacteristic
20#4946C411yescharacteristic
21#4940—21no—
22#4926—21no—
23C3×#324—21no—
24C3×#323—21no—
25#4468—24no—
26#2973—31no—
27#2321—31no—
28#1459D311yescharacteristic, complemented
29C2×#323—32no—
30#1282—32no—
31C24×D3C2×C411yescharacteristic
32C22×C3×D4C2×C411yescharacteristic
33C23×C3×C4D411yescharacteristic
34C23×C3×C4C2311yescharacteristic
35C22×#67—21no—
36C23×Dic3C2×C411yescharacteristic
37C22×C3×C8—22no—
38C3×C4×D4—22no—
39C3×C4×D4—22no—
40C2×C3×C4⋊C4D411yescharacteristic
41C2×C3×C22⋊C4C2×C411yescharacteristic
42C2×C3×C22⋊C4D411yescharacteristic
43C2×C3×C42D411yescharacteristic
44C2×#217—21no—
45C2×#217C2×C411yescharacteristic
46C2×#217C2×C411yescharacteristic
47#937—22no—
48#937—22no—
49C2×(C2×D3)⋊C4—21no—
50C2×(C2×D3)⋊C4—21no—
51C2×(C2×D3)⋊C4—21no—
52C2×(C2×D3)⋊C4—22no—
53C2×(C2×D3)⋊C4—22no—
54C2×(C2×D3)⋊C4—21no—
55C2×(C2×D3)⋊C4—21no—
56C2×Dic3⋊C4—21no—
57C2×C4×Dic3—21no—
58C2×#208—22no—
59C22×C3⋊C8—22no—
60C3×C22⋊C8—22no—
61C3×C22⋊C8—22no—
62C3⋊#95—21no—
63C3⋊#95—21no—
64C3⋊#95—21no—
65#830—22no—
66#830—22no—
67#830D414yescomplemented
68C4⋊(C3⋊C8)—22no—
69#803—22no—
70C2×#120—31no—
71C2×C4×D4C2×D311yescharacteristic
72C22×C22⋊C4—31no—
73#435—64no—
74#432—61no—
75#407—61no—
76C2×C22⋊C8—32no—
77C2×C22⋊C8C2×D312yes—
78#390—61no—
79#388—61no—
80#388—61no—
81#386—31no—
82#386—61no—
83#385—31no—
84#379—31no—
85C4×C22⋊C4—31no—
86#324—21no—
87#323—21no—
88#323—64no—
89C24×C3C22⋊C411yescharacteristic
90C24×C3C22×C411yescharacteristic
91C23×D3—21no—
92C23×D3—22no—
93C23×D3—21no—
94C23×D3—22no—
95C23×D3—81no—
96C23×D3—81no—
97C23×D3M1612yescomplemented
98C2×C3×D4—22no—
99C2×C3×D4—21no—
100C2×C3×D4—21no—
101C2×C3×D4—22no—
102C22×C3×C4—21no—
103C22×C3×C4—21no—
104C22×C3×C4—81no—
105C22×C3×C4—81no—
106C22×C3×C4C22⋊C411yescharacteristic
107C22×C3×C4C22⋊C411yescharacteristic
108C22×C3×C4C22⋊C411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D412yes—
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4#4112yes—
113C2×#67—24no—
114C2×#67—81no—
115C2×#67—81no—
116C22×Dic3—21no—
117C22×Dic3—21no—
118C22×Dic3—21no—
119C22×Dic3—22no—
120C22×Dic3—81no—
121C2×C3×C8—42no—
122C2×C3×C8—82no—
123C3×C4⋊C4—22no—
124C3×C22⋊C4—22no—
125C3×C22⋊C4—22no—
126C3×C42—21no—
127C3×C42—21no—
128#217—21no—
129#217—22no—
130#217—81no—
131#217M1612yescomplemented
132(C2×D3)⋊C4—24no—
133(C2×D3)⋊C4—41no—
134(C2×D3)⋊C4—42no—
135(C2×D3)⋊C4—41no—
136(C2×D3)⋊C4—81no—
137(C2×D3)⋊C4—81no—
138Dic3⋊C4—81no—
139C4×Dic3—81no—
140#208—82no—
141C2×C3⋊C8—24no—
142C2×C3⋊C8—82no—
143C25—31no—
144C22×D4—61no—
145C22×D4C4×D311yescharacteristic
146C23×C4—31no—
147C23×C4D1211yescharacteristic
148C23×C4C22×D311yescharacteristic
149C2×M16—62no—
150C22×C8—22no—
151C22×C8—62no—
152#120—62no—
153#120—62no—
154C4×D4—22no—
155C4×D4—22no—
156C2×C4⋊C4—61no—
157C2×C4⋊C4#6711yescharacteristic
158C2×C22⋊C4—31no—
159C2×C22⋊C4—31no—
160C2×C22⋊C4—61no—
161C2×C22⋊C4—61no—
162C2×C22⋊C4—62no—
163C2×C22⋊C4—62no—
164C2×C22⋊C4—61no—
165C2×C22⋊C4—61no—
166C2×C22⋊C4—61no—
167C2×C22⋊C4—61no—
168C2×C22⋊C4C4×D311yescharacteristic
169C2×C22⋊C4#6711yescharacteristic
170C2×C42—61no—
171C2×C42D1211yescharacteristic
172C4⋊C8—62no—
173C22⋊C8—22no—
174C22⋊C8—22no—
175C22⋊C8—34no—
176C22⋊C8—62no—
177C22⋊C8—62no—
178#97—62no—
179#95—61no—
180#95—61no—
181#95—61no—
182C23×C3—21no—
183C23×C3—21no—
184C23×C3—21no—
185C23×C3—41no—
186C23×C3—41no—
187C23×C3—41no—
188C23×C3—41no—
189C23×C3C2×C22⋊C411yescharacteristic
190C23×C3C4×D412yes—
191C23×C3C2×M1611yescharacteristic
192C23×C3C2×M1611yescharacteristic
193C23×C3#13112yes—
194C22×D3—22no—
195C22×D3—24no—
196C22×D3—41no—
197C22×D3—41no—
198C22×D3—41no—
199C22×D3—41no—
200C22×D3—41no—
201C22×D3—41no—
202C22×D3—42no—
203C22×D3—82no—
204C22×D3—82no—
205C22×D3—82no—
206C22×D3—82no—
207C22×D3—84no—
208C3×D4—42no—
209C3×D4—42no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—21no—
213C2×C3×C4—21no—
214C2×C3×C4—21no—
215C2×C3×C4—42no—
216C2×C3×C4—41no—
217C2×C3×C4—42no—
218C2×C3×C4—41no—
219C2×C3×C4—42no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—81no—
223C2×C3×C4#12011yescharacteristic
224C2×C3×C4#12112yes—
225C2×C3×C4#12311yescharacteristic
226#67—84no—
227C2×Dic3—22no—
228C2×Dic3—41no—
229C2×Dic3—41no—
230C2×Dic3—41no—
231C2×Dic3—41no—
232C2×Dic3—41no—
233C2×Dic3—41no—
234C2×Dic3—41no—
235C2×Dic3—41no—
236C2×Dic3—82no—
237C3×C8—82no—
238C3⋊C8—44no—
239C24—32no—
240C24—61no—
241C24—62no—
242C24—62no—
243C24—61no—
244C24—241no—
245C24—241no—
246C24(C2×D3)⋊C411yescharacteristic
247C24C2×C4×D311yescharacteristic
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—64no—
253C2×D4—241no—
254C2×D4—241no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—61no—
258C22×C4—61no—
259C22×C4—61no—
260C22×C4—62no—
261C22×C4—81no—
262C22×C4—81no—
263C22×C4—241no—
264C22×C4(C2×D3)⋊C411yescharacteristic
265C22×C4(C2×D3)⋊C411yescharacteristic
266C22×C4(C2×D3)⋊C411yescharacteristic
267C22×C4C2×D1211yescharacteristic
268C22×C4D3×D412yes—
269C22×C4#23712yes—
270C22×C4C2×#6711yescharacteristic
271M16—242no—
272C2×C8—42no—
273C2×C8—64no—
274C2×C8—82no—
275C2×C8—242no—
276C4⋊C4—22no—
277C4⋊C4—241no—
278C22⋊C4—22no—
279C22⋊C4—22no—
280C22⋊C4—32no—
281C22⋊C4—64no—
282C22⋊C4—61no—
283C22⋊C4—62no—
284C22⋊C4—121no—
285C22⋊C4—121no—
286C22⋊C4—122no—
287C22⋊C4—241no—
288C22⋊C4—241no—
289C22⋊C4—241no—
290C42—21no—
291C42—21no—
292C42—241no—
293C22×C3—21no—
294C22×C3—22no—
295C22×C3—21no—
296C22×C3—22no—
297C22×C3—41no—
298C22×C3—41no—
299C22×C3—42no—
300C22×C3—42no—
301C22×C3—41no—
302C22×C3—41no—
303C22×C3—41no—
304C22×C3—81no—
305C22×C3#38611yescharacteristic
306C22×C3#40711yescharacteristic
307C22×C3#40811yescharacteristic
308C22×C3#43512yes—
309C22×C3#43612yes—
310C2×D3—42no—
311C2×D3—42no—
312C2×D3—42no—
313C2×D3—81no—
314C2×D3—81no—
315C2×D3—84no—
316C2×D3—81no—
317C2×D3—81no—
318C2×D3—84no—
319C3×C4—21no—
320C3×C4—22no—
321C3×C4—21no—
322C3×C4—41no—
323C3×C4—41no—
324C3×C4—41no—
325Dic3—42no—
326Dic3—81no—
327Dic3—81no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—62no—
336C23—64no—
337C23—121no—
338C23—121no—
339C23—121no—
340C23—121no—
341C23—121no—
342C23—122no—
343C23—121no—
344C23—242no—
345C23—242no—
346C23—242no—
347C23—242no—
348C23—244no—
349C23#67⋊C412yes—
350C23#90112yes—
351C23D3×M1611yescharacteristic
352C23D3×M1611yescharacteristic
353C23C2×(C2×D3)⋊C411yescharacteristic
354D4—42no—
355D4—42no—
356D4—244no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—62no—
370C2×C4—81no—
371C2×C4—121no—
372C2×C4—121no—
373C2×C4—121no—
374C2×C4—121no—
375C2×C4—121no—
376C2×C4—121no—
377C2×C4—121no—
378C2×C4—121no—
379C2×C4—242no—
380C2×C4#88211yescharacteristic
381C2×C4#88611yescharacteristic
382C2×C4#93912yes—
383C8—82no—
384C8—124no—
385C2×C3—21no—
386C2×C3—21no—
387C2×C3—22no—
388C2×C3—22no—
389C2×C3—81no—
390C2×C3#187011yescharacteristic
391C2×C3#191311yescharacteristic
392C2×C3#191411yescharacteristic
393D3—82no—
394C22—21no—
395C22—22no—
396C22—21no—
397C22—22no—
398C22—42no—
399C22—41no—
400C22—41no—
401C22—42no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—81no—
406C22—122no—
407C22—122no—
408C22—122no—
409C22—244no—
410C22—241no—
411C22—241no—
412C22—244no—
413C22—241no—
414C22—241no—
415C22#472812yes—
416C22#472912yes—
417C22#495811yescharacteristic
418C22#511011yescharacteristic
419C22#512611yescharacteristic
420C4—22no—
421C4—21no—
422C4—21no—
423C4—41no—
424C4—41no—
425C4—41no—
426C4—122no—
427C4—241no—
428C4—241no—
429C3#910911yescharacteristic, complemented, Sylow
430C2—21no—
431C2—21no—
432C2—22no—
433C2—22no—
434C2—81no—
435C2—242no—
436C2#6995311yescharacteristic
437C2#7002411yescharacteristic
438C2#7013711yescharacteristic
subgroupG/H
centerC22#4958
commutatorC24×C3C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1459C2
socleC22×C3#386
prime radicalC2×#937C4
cosocleC23×C3×C4C23
hypercenter#1459D3
hypocenterC3#9109
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1459

Complements

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

NH ≅ G/Ntype
2. #82302C2semidirect
3. C3×#1459C2semidirect
4. C2×#4468C2semidirect
5. #67677C2semidirect
28. #1459D3semidirect
67. #830D4semidirect
97. C23×D3M16semidirect
131. #217M16semidirect
429. C3#9109semidirect

Maximal subgroups

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

subgroupindex
#823952
#823022
C3×#14592
C2×#44682
#676772
#91093