Finite Group Databasebeta

← #10692918

Subgroups — #10692918

Order 768.

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

normal subgroups only

This group has 700 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#82760C211yesmaximal, characteristic
2#74053C211yesmaximal, characteristic
3C3×#1511C211yesmaximal, characteristic
4#71689C211yesmaximal, characteristic
5#69990C211yesmaximal, characteristic
6#67682C211yesmaximal, characteristic
7#67628C211yesmaximal, characteristic
8#9462—31nomaximal, Sylow
9C23×Dic6C2211yescharacteristic
10C2×C3×C22⋊Q8C2211yescharacteristic
11C2×#944C412yescomplemented
12C2×#877—21no—
13C3×#407C2211yescharacteristic
14C3×#381C2211yescharacteristic
15#5211C2211yescharacteristic
16C4×#217C2211yescharacteristic
17#5207C2211yescharacteristic
18C2×#824—22no—
19#4979—22no—
20#4973C412yescomplemented
21#4946—21no—
22#4946—21no—
23#4943—21no—
24#4646—22no—
25#4646—22no—
26C3×#328—22no—
27#4538—22no—
28#2434—31no—
29#2310—31no—
30#1784—31no—
31#1511—31no—
32#1511D311yescharacteristic
33#1284—31no—
34#1284—31no—
35C22×C3×Q8C2×C411yescharacteristic
36C23×C3×C4C2311yescharacteristic
37C23×Dic3D411yescharacteristic
38C23×Dic3D411yescharacteristic
39C22×Dic6—21no—
40C22×Dic6—21no—
41C22×Dic6—21no—
42C22×Dic6—21no—
43C22×Dic6—44no—
44C22×Dic6C2×C412yes—
45C22×Dic6D412yes—
46C2×C3×M16—21no—
47C3×C22⋊Q8—21no—
48C3×C22⋊Q8—21no—
49C3×C22⋊Q8—21no—
50C3×C22⋊Q8—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4C2×C411yescharacteristic
53C2×C3×C22⋊C4—21no—
54C2×C3×C22⋊C4—21no—
55C2×#217—21no—
56C2×#217—21no—
57#944—44no—
58C2×Dic3⋊C4—22no—
59C2×Dic3⋊C4—22no—
60C2×Dic3⋊C4—22no—
61C2×C4×Dic3—22no—
62C2×C4×Dic3C2×C412yes—
63C2×#208—21no—
64#877—44no—
65C3×C22⋊C8—21no—
66C3×C22⋊C8—21no—
67C3×#95—21no—
68C3×#95D412yes—
69C3⋊#95—24no—
70C3⋊#95—21no—
71C3⋊#95—21no—
72C3⋊#95—22no—
73C3⋊#95—22no—
74C3⋊#95D412yes—
75#830—21no—
76#830—21no—
77#824—44no—
78C23×Q8—31no—
79C2×C4⋊Q8—32no—
80C2×C22⋊Q8C2×D311yescharacteristic
81C2×#117—61no—
82C2×#101—62no—
83#407—31no—
84#407C2×D311yescharacteristic
85#395—62no—
86#391—32no—
87#391—62no—
88#391—62no—
89#385—61no—
90#385—61no—
91#381—31no—
92#381—61no—
93#381C2×D311yescharacteristic
94C4×C22⋊C4—31no—
95#328—22no—
96#328—62no—
97C24×C3C2×D411yescharacteristic
98C2×C3×Q8—21no—
99C2×C3×Q8—21no—
100C2×C3×Q8—42no—
101C22×C3×C4—21no—
102C22×C3×C4—21no—
103C22×C3×C4—22no—
104C22×C3×C4—42no—
105C22×C3×C4C22×C411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4#4111yescharacteristic
110C22×C3×C4#4111yescharacteristic
111C22×Dic3—21no—
112C22×Dic3—21no—
113C22×Dic3—24no—
114C22×Dic3—22no—
115C22×Dic3—21no—
116C22×Dic3—21no—
117C22×Dic3—22no—
118C22×Dic3—42no—
119C22×Dic3C22⋊C412yes—
120C22×Dic3C22⋊C412yes—
121C2×Dic6—22no—
122C2×Dic6—22no—
123C2×Dic6—22no—
124C2×Dic6—22no—
125C2×Dic6—24no—
126C2×Dic6—24no—
127C2×Dic6—42no—
128C2×Dic6—42no—
129C2×Dic6—42no—
130C2×Dic6—42no—
131C2×Dic6—42no—
132C2×Dic6—42no—
133C2×Dic6—44no—
134C2×Dic6—44no—
135C3×M16—42no—
136C2×C3×C8—41no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—41no—
140C3×C4⋊C4—41no—
141C3×C22⋊C4—42no—
142C3×C22⋊C4—41no—
143C3×C22⋊C4—41no—
144#217—24no—
145#217—42no—
146Dic3⋊C4—44no—
147Dic3⋊C4—44no—
148Dic3⋊C4—44no—
149C4×Dic3—44no—
150C4×Dic3—42no—
151#208—42no—
152C2×C3⋊C8—41no—
153C22×Q8—32no—
154C22×Q8—32no—
155C22×Q8—61no—
156C22×Q8—61no—
157C22×Q8—61no—
158C22×Q8—61no—
159C22×Q8—124no—
160C22×Q8C4×D311yescharacteristic
161C23×C4—31no—
162C23×C4—31no—
163C23×C4C22×D311yescharacteristic
164C2×M16—21no—
165C2×M16—61no—
166C4⋊Q8—124no—
167C22⋊Q8—21no—
168C22⋊Q8—21no—
169C22⋊Q8—21no—
170C22⋊Q8—21no—
171#117—124no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—62no—
174C2×C4⋊C4—62no—
175C2×C4⋊C4—62no—
176C2×C4⋊C4C4×D311yescharacteristic
177C2×C22⋊C4—21no—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—61no—
180C2×C22⋊C4—61no—
181C2×C42—32no—
182C2×C42—62no—
183#101—124no—
184C22⋊C8—21no—
185C22⋊C8—21no—
186C22⋊C8—61no—
187C22⋊C8—61no—
188#95—21no—
189#95—32no—
190#95—62no—
191#95—62no—
192#95—64no—
193#95—61no—
194#95—61no—
195#95#6712yes—
196C23×C3—22no—
197C23×C3—22no—
198C23×C3—41no—
199C23×C3C2×C22⋊C411yescharacteristic
200C23×C3#12011yescharacteristic
201C23×C3#12311yescharacteristic
202C3×Q8—81no—
203C3×Q8—81no—
204C2×C3×C4—22no—
205C2×C3×C4—22no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—42no—
211C2×C3×C4—42no—
212C2×C3×C4—42no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4—42no—
216C2×C3×C4—42no—
217C2×C3×C4C4×D411yescharacteristic
218C2×C3×C4C4×D411yescharacteristic
219C2×C3×C4#12111yescharacteristic
220C2×C3×C4#12111yescharacteristic
221C2×Dic3—22no—
222C2×Dic3—22no—
223C2×Dic3—24no—
224C2×Dic3—22no—
225C2×Dic3—22no—
226C2×Dic3—24no—
227C2×Dic3—22no—
228C2×Dic3—22no—
229C2×Dic3—22no—
230C2×Dic3—41no—
231C2×Dic3—41no—
232C2×Dic3—42no—
233C2×Dic3—42no—
234C2×Dic3—42no—
235C2×Dic3—42no—
236C2×Dic3—42no—
237C2×Dic3—42no—
238C2×Dic3#9914yes—
239Dic6—44no—
240Dic6—44no—
241Dic6—44no—
242Dic6—44no—
243C3×C8—81no—
244C3⋊C8—81no—
245C24D3×D411yescharacteristic
246C2×Q8—21no—
247C2×Q8—21no—
248C2×Q8—42no—
249C2×Q8—64no—
250C2×Q8—64no—
251C2×Q8—62no—
252C2×Q8—62no—
253C2×Q8—62no—
254C2×Q8—62no—
255C2×Q8—122no—
256C2×Q8—122no—
257C2×Q8—124no—
258C2×Q8—122no—
259C2×Q8—122no—
260C2×Q8—124no—
261C2×Q8—122no—
262C2×Q8—122no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—32no—
267C22×C4—32no—
268C22×C4—42no—
269C22×C4—64no—
270C22×C4—61no—
271C22×C4—61no—
272C22×C4—62no—
273C22×C4—61no—
274C22×C4—61no—
275C22×C4—62no—
276C22×C4—122no—
277C22×C4C2×C4×D311yescharacteristic
278C22×C4#23511yescharacteristic
279C22×C4D3×D411yescharacteristic
280C22×C4D3×D411yescharacteristic
281C22×C4#23711yescharacteristic
282C22×C4C2×#6711yescharacteristic
283M16—42no—
284M16—122no—
285C2×C8—41no—
286C2×C8—121no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—124no—
292C4⋊C4—124no—
293C4⋊C4—124no—
294C22⋊C4—42no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—64no—
298C22⋊C4—122no—
299C42—122no—
300C42—124no—
301C22×C3—21no—
302C22×C3—21no—
303C22×C3—21no—
304C22×C3—21no—
305C22×C3—21no—
306C22×C3—21no—
307C22×C3—21no—
308C22×C3—44no—
309C22×C3#38611yescharacteristic
310C22×C3C2×#9912yes—
311C22×C3#45612yes—
312C3×C4—21no—
313C3×C4—21no—
314C3×C4—21no—
315C3×C4—21no—
316C3×C4—82no—
317C3×C4—81no—
318C3×C4—81no—
319Dic3—24no—
320Dic3—44no—
321Dic3—42no—
322Dic3—82no—
323C23—22no—
324C23—22no—
325C23—41no—
326C23D3×C22⋊C411yescharacteristic
327C23#88311yescharacteristic
328C23#93711yescharacteristic
329Q8—81no—
330Q8—81no—
331Q8—124no—
332Q8—124no—
333Q8—124no—
334Q8—124no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—34no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—62no—
350C2×C4—62no—
351C2×C4—64no—
352C2×C4—62no—
353C2×C4—62no—
354C2×C4—64no—
355C2×C4—62no—
356C2×C4—62no—
357C2×C4—62no—
358C2×C4—121no—
359C2×C4—121no—
360C2×C4—122no—
361C2×C4—122no—
362C2×C4—122no—
363C2×C4—122no—
364C2×C4—122no—
365C2×C4—122no—
366C2×C4#67⋊C411yescharacteristic
367C2×C4#88411yescharacteristic
368C2×C4C4×#6711yescharacteristic
369C2×C4#93811yescharacteristic
370C8—81no—
371C8—241no—
372C2×C3—22no—
373C2×C3—22no—
374C2×C3—41no—
375C2×C3#188611yescharacteristic
376C2×C3#189811yescharacteristic
377C2×C3#191211yescharacteristic
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—44no—
386C22D3×#9912yes—
387C22#495611yescharacteristic
388C22#520512yes—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—64no—
394C4—82no—
395C4—81no—
396C4—81no—
397C4—124no—
398C4—122no—
399C4—242no—
400C3#946211yescharacteristic, complemented, Sylow
401C2—22no—
402C2—22no—
403C2—41no—
404C2#7003611yescharacteristic
405C2#7034511yescharacteristic
406C2#7041111yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1511C2
socleC22×C3#386
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenter#1511D3
hypocenterC3#9462
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1511

Complements

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

NH ≅ G/Ntype
11. C2×#944C4semidirect
20. #4973C4semidirect
400. C3#9462semidirect

Maximal subgroups

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

subgroupindex
#827602
#740532
C3×#15112
#716892
#699902
#676822
#676282
#94623