Finite Group Databasebeta

← #10692385

Subgroups — #10692385

Order 768.

542 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 107 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×#3094C211yesmaximal, characteristic
2#82240C211yesmaximal, characteristic, complemented
3#71686C211yesmaximal, characteristic
4#71673C211yesmaximal, characteristic
5#71590C211yesmaximal, characteristic, complemented
6#67684C211yesmaximal, characteristic, complemented
7#67683C211yesmaximal, characteristic, complemented
8#9366—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C2×C3×#117C2211yescharacteristic
11C3×#482—22no—
12C3×#481—21no—
13C3×#480—21no—
14C3×#428C412yescomplemented
15C3×#418C412yescomplemented
16C3×#407C2211yescharacteristic
17C2×#834—22no—
18#5212C2211yescharacteristic
19C4×#217C2211yescharacteristic
20C4×#217C2211yescharacteristic
21#5207C2211yescharacteristic
22#4984—21no—
23Dic3×C4⋊C4—21no—
24#4951—22no—
25#4937—21no—
26C4×C4⋊Dic3—21no—
27#4537—21no—
28#4537—21no—
29#4532—22no—
30#3094D311yescharacteristic
31#2347—31no—
32#1964—31no—
33#1826—31no—
34#1507—31no—
35#1305—31no—
36#1297—31no—
37C22×C3×D4—21no—
38C22×C3×D4C2×C411yescharacteristic
39C23×C3×C4C2311yescharacteristic
40C2×C3×M16C2×C412yes—
41C3×#121—21no—
42C3×#121—21no—
43C3×#121—21no—
44C3×#121—21no—
45C3×#117—21no—
46C3×#117—21no—
47C3×#117C2×C412yes—
48C3×#117D412yes—
49C2×C3×C4⋊C4C2×C411yescharacteristic
50C2×C3×C4⋊C4D411yescharacteristic
51C2×C3×C4⋊C4D411yescharacteristic
52C2×C3×C22⋊C4—21no—
53C2×C3×C22⋊C4—21no—
54C2×C3×C42—21no—
55C2×#217—21no—
56C2×#217—21no—
57C2×#217D412yes—
58C2×C4⋊Dic3—22no—
59C2×C4⋊Dic3—21no—
60C2×C4×Dic3—21no—
61C2×C4×Dic3—21no—
62C2×C4×Dic3—22no—
63C2×#208—21no—
64C3×#107—22no—
65C3×#106—22no—
66C3×#102—22no—
67C3×#102—22no—
68C3×C22⋊C8—21no—
69C3×C22⋊C8—21no—
70#834—44no—
71C3⋊#95—22no—
72C3⋊#95—22no—
73C3⋊#95—21no—
74C3⋊#95—21no—
75C3⋊#95D411yescharacteristic
76C3⋊#95D411yescharacteristic
77#830—21no—
78#830—21no—
79C2×#121C2×D311yescharacteristic
80C2×#117C2×D311yescharacteristic
81#482—22no—
82#481—21no—
83#480—21no—
84#428Dic312yescomplemented
85#418Dic312yescomplemented
86C2×#99—62no—
87#407—31no—
88#407C2×D311yescharacteristic
89#389—61no—
90#388—62no—
91#383—61no—
92#380—31no—
93C4×C4⋊C4—61no—
94C4×C4⋊C4—61no—
95C4×C22⋊C4—31no—
96C4×C22⋊C4—31no—
97#339—62no—
98#327—61no—
99#327—61no—
100C24×C3—21no—
101C24×C3C2×D411yescharacteristic
102C2×C3×D4—21no—
103C2×C3×D4—21no—
104C2×C3×D4—42no—
105C2×C3×D4—42no—
106C2×C3×D4—42no—
107C2×C3×D4—41no—
108C2×C3×D4—41no—
109C22×C3×C4—21no—
110C22×C3×C4—21no—
111C22×C3×C4—21no—
112C22×C3×C4—42no—
113C22×C3×C4C22×C411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4#4111yescharacteristic
118C22×C3×C4#4111yescharacteristic
119C22×Dic3—21no—
120C22×Dic3—21no—
121C22×Dic3—22no—
122C22×Dic3—41no—
123C3×M16—42no—
124C2×C3×C8—22no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4C22⋊C412yes—
130C3×C4⋊C4C22⋊C412yes—
131C3×C22⋊C4—22no—
132C3×C22⋊C4—22no—
133C3×C22⋊C4—41no—
134C3×C22⋊C4—41no—
135C3×C42—22no—
136C3×C42—22no—
137#217—22no—
138#217—42no—
139#217—41no—
140#217—42no—
141C4⋊Dic3—44no—
142C4⋊Dic3—42no—
143C4×Dic3—42no—
144C4×Dic3—42no—
145C4×Dic3—82no—
146#208—42no—
147C2×C3⋊C8—41no—
148C22×D4—21no—
149C22×D4C2×Dic311yescharacteristic
150C23×C4C22×D311yescharacteristic
151C2×M16—61no—
152C2×M16C2×Dic312yes—
153#121—21no—
154#121—21no—
155#121—21no—
156#121—21no—
157#117—21no—
158#117—21no—
159#117C2×Dic312yes—
160#117#6712yes—
161C2×C4⋊C4—62no—
162C2×C4⋊C4—61no—
163C2×C4⋊C4C2×Dic311yescharacteristic
164C2×C4⋊C4#6711yescharacteristic
165C2×C4⋊C4#6711yescharacteristic
166C2×C22⋊C4—21no—
167C2×C22⋊C4—21no—
168C2×C22⋊C4—32no—
169C2×C22⋊C4—61no—
170C2×C22⋊C4—61no—
171C2×C42—21no—
172C2×C42—61no—
173C2×C42—61no—
174C2×C42—62no—
175#107—22no—
176#106—22no—
177#102—22no—
178#102—22no—
179#99—124no—
180C22⋊C8—21no—
181C22⋊C8—21no—
182C22⋊C8—61no—
183C22⋊C8—61no—
184#95—31no—
185#95—31no—
186#95—62no—
187#95—62no—
188#95—61no—
189#95—61no—
190C23×C3—21no—
191C23×C3—21no—
192C23×C3—41no—
193C23×C3—41no—
194C23×C3—42no—
195C23×C3—41no—
196C23×C3—81no—
197C23×C3—81no—
198C23×C3C4×D411yescharacteristic
199C23×C3#12011yescharacteristic
200C23×C3#12111yescharacteristic
201C3×D4—42no—
202C3×D4—81no—
203C3×D4—82no—
204C3×D4—81no—
205C3×D4—81no—
206C2×C3×C4—22no—
207C2×C3×C4—22no—
208C2×C3×C4—22no—
209C2×C3×C4—22no—
210C2×C3×C4—22no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—42no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—42no—
218C2×C3×C4—41no—
219C2×C3×C4C2×C22⋊C411yescharacteristic
220C2×C3×C4C4×D411yescharacteristic
221C2×C3×C4#12111yescharacteristic
222C2×C3×C4#12311yescharacteristic
223C2×Dic3—22no—
224C2×Dic3—41no—
225C2×Dic3—41no—
226C2×Dic3—41no—
227C2×Dic3—42no—
228C2×Dic3—42no—
229C2×Dic3—42no—
230C2×Dic3—41no—
231C2×Dic3—42no—
232C2×Dic3—41no—
233C2×Dic3—81no—
234C2×Dic3—81no—
235C3×C8—42no—
236C3⋊C8—81no—
237C24—21no—
238C24D3×D411yescharacteristic
239C2×D4—21no—
240C2×D4—21no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4—42no—
244C2×D4—41no—
245C2×D4—41no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—42no—
250C22×C4—61no—
251C22×C4—61no—
252C22×C4—62no—
253C22×C4—121no—
254C22×C4D3×D411yescharacteristic
255C22×C4#23711yescharacteristic
256C22×C4#23711yescharacteristic
257C22×C4C22×Dic311yescharacteristic
258C22×C4C2×#6711yescharacteristic
259C22×C4C2×#6711yescharacteristic
260M16—42no—
261M16—122no—
262C2×C8—22no—
263C2×C8—121no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—122no—
269C4⋊C4—124no—
270C4⋊C4#21712yes—
271C4⋊C4#21712yes—
272C22⋊C4—22no—
273C22⋊C4—22no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—62no—
277C22⋊C4—122no—
278C22⋊C4—121no—
279C22⋊C4—122no—
280C42—22no—
281C42—22no—
282C42—122no—
283C42—122no—
284C42—242no—
285C22×C3—21no—
286C22×C3—21no—
287C22×C3—21no—
288C22×C3—21no—
289C22×C3—21no—
290C22×C3—21no—
291C22×C3—21no—
292C22×C3—21no—
293C22×C3—81no—
294C22×C3—81no—
295C22×C3—81no—
296C22×C3—81no—
297C22×C3—81no—
298C22×C3—162no—
299C22×C3#38611yescharacteristic
300C22×C3#45311yescharacteristic
301C22×C3#45411yescharacteristic
302C3×C4—21no—
303C3×C4—21no—
304C3×C4—21no—
305C3×C4—42no—
306C3×C4—42no—
307C3×C4—81no—
308C3×C4#41012yes—
309Dic3—42no—
310Dic3—82no—
311Dic3—81no—
312Dic3—82no—
313C23—21no—
314C23—21no—
315C23—41no—
316C23—41no—
317C23—42no—
318C23—41no—
319C23—81no—
320C23—81no—
321C23D4×Dic311yescharacteristic
322C23#93711yescharacteristic
323C23#93911yescharacteristic
324D4—42no—
325D4—81no—
326D4—82no—
327D4—81no—
328D4—81no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—62no—
343C2×C4—121no—
344C2×C4—121no—
345C2×C4—121no—
346C2×C4—122no—
347C2×C4—121no—
348C2×C4—122no—
349C2×C4—122no—
350C2×C4—122no—
351C2×C4—121no—
352C2×C4—241no—
353C2×C4—241no—
354C2×C4D4×Dic311yescharacteristic
355C2×C4#93511yescharacteristic
356C2×C4#93911yescharacteristic
357C2×C4C2×#21711yescharacteristic
358C8—42no—
359C8—241no—
360C2×C3—21no—
361C2×C3—21no—
362C2×C3—41no—
363C2×C3—41no—
364C2×C3—161no—
365C2×C3#188311yescharacteristic
366C2×C3#189911yescharacteristic
367C2×C3#190611yescharacteristic
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22—81no—
381C22—162no—
382C22#519811yescharacteristic
383C22#520111yescharacteristic
384C22#522011yescharacteristic
385C4—21no—
386C4—21no—
387C4—21no—
388C4—42no—
389C4—42no—
390C4—81no—
391C4—122no—
392C4—242no—
393C4—241no—
394C4—242no—
395C4#524012yes—
396C3#936611yescharacteristic, complemented, Sylow
397C2—21no—
398C2—21no—
399C2—41no—
400C2—41no—
401C2—161no—
402C2#7170511yescharacteristic
403C2#7208811yescharacteristic
404C2#7211411yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#3094C2
socleC22×C3#386
prime radicalC2×C3×#121C22
cosocleC23×C3×C4C23
hypercenter#3094D3
hypocenterC3#9366
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3094

Complements

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

NH ≅ G/Ntype
2. #82240C2semidirect
5. #71590C2semidirect
6. #67684C2semidirect
7. #67683C2semidirect
14. C3×#428C4semidirect
15. C3×#418C4semidirect
84. #428Dic3semidirect
85. #418Dic3semidirect
396. C3#9366semidirect

Maximal subgroups

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

subgroupindex
C3×#30942
#822402
#716862
#716732
#715902
#676842
#676832
#93663