Finite Group Databasebeta

← #10697353

Subgroups — #10697353

Order 768.

560 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 109 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×#3152C211yesmaximal, characteristic
2#83320C211yesmaximal, characteristic
3#83223C211yesmaximal, characteristic, complemented
4#71811C211yesmaximal, characteristic
5#71780C211yesmaximal, characteristic, complemented
6#69868C211yesmaximal, characteristic, complemented
7#67668C211yesmaximal, characteristic, complemented
8#10206—31nomaximal, Sylow
9C2×C3×#137—22no—
10C3×#532C2211yescharacteristic
11C2×C3×C4⋊Q8C2211yescharacteristic
12#5470C2211yescharacteristic
13C3×#497—22no—
14C3×#494—21no—
15C3×#492—21no—
16C3×C4×M16C2211yescharacteristic
17#5094—22no—
18#5094—21no—
19#5093—21no—
20#5080—21no—
21#5075—21no—
22#5041—22no—
23#5019—21no—
24#5008—21no—
25#4971—22no—
26C4×C4⋊Dic3C2211yescharacteristic
27#4925C2211yescharacteristic
28C2×C4⋊(C3⋊C8)C2211yescharacteristic
29#4555—22no—
30#4537—22no—
31#4529—22no—
32#3246—31no—
33#3229—31no—
34#3152D311yescharacteristic
35#1988—31no—
36#1986—31no—
37#1502—31no—
38#1283—31no—
39C2×C3×#41—21no—
40C22×C3×Q8—21no—
41C3×#137—44no—
42C3×#137—44no—
43C2×C3×Q16—24no—
44C2×C3×SD16—24no—
45C2×C3×M16D412yes—
46C3×C4⋊Q8—21no—
47C3×C4⋊Q8—21no—
48C3×C4⋊Q8—21no—
49C3×C4⋊Q8D412yes—
50C3×C4⋊Q8D411yescharacteristic
51C3×#127D411yescharacteristic
52C3×#124—21no—
53C3×#121—21no—
54C3×#121—21no—
55C3×C4×D4—21no—
56C3×C4×D4—21no—
57C2×C3×C4⋊C4—21no—
58C2×C3×C4⋊C4—21no—
59C2×C3×C42C2311yescharacteristic
60C2×#214—22no—
61#926—21no—
62C2×C4⋊Dic3—21no—
63#924—21no—
64#924—21no—
65C2×C4×Dic3—21no—
66C22×C3⋊C8D412yes—
67#870—21no—
68#869D411yescharacteristic
69#869D411yescharacteristic
70C4×Dic6—21no—
71C4×Dic6—21no—
72C3×#97—21no—
73C3×C4×C8—21no—
74#832—22no—
75C3⋊#95—21no—
76#830D412yes—
77#810—24no—
78#810—22no—
79#808—24no—
80#808—22no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)—21no—
83C4⋊(C3⋊C8)—21no—
84C4×C3⋊C8—21no—
85C2×#137—22no—
86#533—31no—
87#532C2×D311yescharacteristic
88C2×C4⋊Q8C2×D311yescharacteristic
89#498—61no—
90#497—22no—
91#494—21no—
92#492—21no—
93#487—61no—
94#483—62no—
95#476—61no—
96#475—62no—
97#475—61no—
98#465—62no—
99#461—61no—
100#460—61no—
101#431—31no—
102C2×C4⋊C8—31no—
103C4×M16C2×D311yescharacteristic
104C4×C4⋊C4—31no—
105#340—62no—
106#339—62no—
107#327—62no—
108C3×#41—42no—
109C3×#41—42no—
110C2×C3×Q8—21no—
111C2×C3×Q8—22no—
112C2×C3×Q8—41no—
113C2×C3×Q8—42no—
114C2×C3×Q8—42no—
115C2×C3×D4—21no—
116C2×C3×D4—21no—
117C2×C3×D4—21no—
118C2×C3×D4—41no—
119C22×C3×C4—41no—
120C22×C3×C4—41no—
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×Dic3—41no—
125C2×Dic6—21no—
126C2×Dic6—21no—
127C3×Q16—48no—
128C3×SD16—48no—
129C3×M16—22no—
130C3×M16SD1614yescomplemented
131C2×C3×C8—22no—
132C3×C4⋊C4—22no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—22no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—41no—
137C3×C4⋊C4—41no—
138C3×C22⋊C4—41no—
139C3×C22⋊C4—41no—
140C3×C42C2×D411yescharacteristic
141C3×C42C2×D411yescharacteristic
142C3×C42C2×D411yescharacteristic
143C3×C42#4111yescharacteristic
144#217—41no—
145#214—82no—
146C4⋊Dic3—22no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151C4⋊Dic3—41no—
152Dic3⋊C4—41no—
153Dic3⋊C4—41no—
154C4×Dic3—41no—
155C4×Dic3—42no—
156C2×C3⋊C8—22no—
157C2×C3⋊C8—22no—
158C2×C3⋊C8—42no—
159C2×#41—21no—
160C22×Q8—21no—
161#137—44no—
162#137—44no—
163C2×Q16—24no—
164C2×SD16—24no—
165C2×SD16—62no—
166C2×M16#6712yes—
167C22×C8—32no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—31no—
172C4⋊Q8—31no—
173C4⋊Q8#6712yes—
174C4⋊Q8#6711yescharacteristic
175#127#6711yescharacteristic
176#125—61no—
177#124—21no—
178C22⋊Q8—61no—
179C22⋊Q8—61no—
180#121—21no—
181#121—21no—
182C4×Q8—61no—
183C4×Q8—61no—
184C4×D4—21no—
185C4×D4—21no—
186#117—61no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—61no—
190C2×C42—61no—
191C2×C42C22×D311yescharacteristic
192#106—62no—
193#106—64no—
194C4⋊C8—61no—
195C4⋊C8—61no—
196C4⋊C8—61no—
197#103—62no—
198#103—64no—
199#102—62no—
200C22⋊C8—32no—
201#97—21no—
202C4×C8—21no—
203C4×C8—61no—
204#95—61no—
205C23×C3—41no—
206C23×C3#12711yescharacteristic
207C3×Q8—44no—
208C3×Q8—42no—
209C3×Q8—82no—
210C3×D4—42no—
211C3×D4—41no—
212C3×D4—41no—
213C3×D4—81no—
214C3×D4—81no—
215C3×D4—81no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4—42no—
221C2×C3×C4—81no—
222C2×C3×C4—81no—
223C2×C3×C4—82no—
224C2×C3×C4#12011yescharacteristic
225C2×C3×C4#12011yescharacteristic
226C2×C3×C4#12011yescharacteristic
227C2×C3×C4#12111yescharacteristic
228C2×C3×C4#12111yescharacteristic
229C2×C3×C4#12111yescharacteristic
230C2×C3×C4C2×SD1612yes—
231C2×C3×C4AGL(1,ℤ/8)12yes—
232C2×Dic3—42no—
233C2×Dic3—41no—
234C2×Dic3—41no—
235C2×Dic3—81no—
236C2×Dic3—81no—
237Dic6—42no—
238Dic6—41no—
239Dic6—41no—
240C3×C8—24no—
241C3⋊C8—42no—
242C3⋊C8—42no—
243#41—42no—
244#41—42no—
245C2×Q8—21no—
246C2×Q8—22no—
247C2×Q8—41no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—61no—
251C2×Q8—61no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—121no—
259C22×C4D3×D411yescharacteristic
260C22×C4D3×D411yescharacteristic
261C22×C4C2×#6711yescharacteristic
262Q16—48no—
263SD16—48no—
264SD16—242no—
265M16—22no—
266M16#21414yescomplemented
267C2×C8—22no—
268C2×C8—62no—
269C2×C8—62no—
270C2×C8—122no—
271C4⋊C4—22no—
272C4⋊C4—21no—
273C4⋊C4—22no—
274C4⋊C4—21no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—62no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C4⋊C4—61no—
281C4⋊C4—61no—
282C4⋊C4—121no—
283C4⋊C4—121no—
284C4⋊C4—121no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—121no—
288C42—121no—
289C42—122no—
290C42D3×D411yescharacteristic
291C42#23711yescharacteristic
292C42C2×#6711yescharacteristic
293C42C2×#6711yescharacteristic
294C22×C3—21no—
295C22×C3—21no—
296C22×C3—81no—
297C22×C3—81no—
298C22×C3—81no—
299C22×C3#39211yescharacteristic
300C22×C3#49211yescharacteristic
301C22×C3#49611yescharacteristic
302C3×C4—21no—
303C3×C4—21no—
304C3×C4—21no—
305C3×C4—81no—
306C3×C4—82no—
307C3×C4#45012yes—
308C3×C4#45511yescharacteristic
309C3×C4#45611yescharacteristic
310C3×C4#46012yes—
311Dic3—82no—
312Dic3—81no—
313Dic3—81no—
314C23—41no—
315C23#94011yescharacteristic
316Q8—44no—
317Q8—42no—
318Q8—82no—
319Q8—122no—
320Q8—121no—
321Q8—121no—
322D4—41no—
323D4—41no—
324D4—42no—
325D4—81no—
326D4—81no—
327D4—81no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—82no—
336C2×C4—122no—
337C2×C4—121no—
338C2×C4—121no—
339C2×C4—241no—
340C2×C4—241no—
341C2×C4#93212yes—
342C2×C4C2×#21412yes—
343C2×C4#93711yescharacteristic
344C2×C4#93711yescharacteristic
345C2×C4#93811yescharacteristic
346C2×C4#93911yescharacteristic
347C2×C4#93911yescharacteristic
348C2×C4#95311yescharacteristic
349C8—24no—
350C8—122no—
351C8—122no—
352C2×C3—21no—
353C2×C3—21no—
354C2×C3—161no—
355C2×C3#200711yescharacteristic
356C2×C3#200911yescharacteristic
357C2×C3#201411yescharacteristic
358C22—21no—
359C22—21no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22#507411yescharacteristic
364C22#507611yescharacteristic
365C22#522311yescharacteristic
366C4—21no—
367C4—21no—
368C4—21no—
369C4—81no—
370C4—82no—
371C4—242no—
372C4—241no—
373C4—241no—
374C4#507512yes—
375C4#520211yescharacteristic
376C4#520511yescharacteristic
377C4#521812yes—
378C3#1020611yescharacteristic, complemented, Sylow
379C2—21no—
380C2—21no—
381C2—161no—
382C2#7176811yescharacteristic
383C2#7211011yescharacteristic
384C2#7211111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3152C2
socleC22×C3#392
prime radicalC3×#532C22
hypercenter#3152D3
hypocenterC3#10206
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3152

Complements

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

NH ≅ G/Ntype
3. #83223C2semidirect
5. #71780C2semidirect
6. #69868C2semidirect
7. #67668C2semidirect
130. C3×M16SD16semidirect
266. M16#214semidirect
378. C3#10206semidirect

Maximal subgroups

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

subgroupindex
C3×#31522
#833202
#832232
#718112
#717802
#698682
#676682
#102063