Finite Group Databasebeta

← #10697101

Subgroups — #10697101

Order 768.

574 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 113 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#83317C211yesmaximal, characteristic
3C2×#5093C211yesmaximal, characteristic, complemented
4#71802C211yesmaximal, characteristic
5#71773C211yesmaximal, characteristic, complemented
6#69879C211yesmaximal, characteristic, complemented
7#67667C211yesmaximal, characteristic, complemented
8#10170—31nomaximal, Sylow
9C2×C3×#137—22no—
10C3×#532C2211yescharacteristic
11C2×C3×C4⋊Q8C2211yescharacteristic
12C2×#869C2211yescharacteristic
13C3×#497—22no—
14C3×#494—21no—
15C3×#492—21no—
16C3×C4×M16C2211yescharacteristic
17#5093—21no—
18#5093—22no—
19#5093—22no—
20#5076—21no—
21#5040—22no—
22#5019—22no—
23C2×#810—22no—
24#4925C2211yescharacteristic
25#4924C2211yescharacteristic
26C2×C4×C3⋊C8C2211yescharacteristic
27#4554—22no—
28#4547—22no—
29D4⋊(C3⋊C8)—22no—
30#3206—31no—
31#3152D311yescharacteristic
32C2×#487—31no—
33#1997—31no—
34#1546—31no—
35#1492—31no—
36#1278—31no—
37C2×C3×#41—21no—
38C22×C3×Q8—21no—
39C22×Dic6D412yes—
40C3×#137—44no—
41C3×#137—44no—
42C2×C3×Q16—24no—
43C2×C3×SD16—24no—
44C2×C3×M16D412yes—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—21no—
47C3×C4⋊Q8—21no—
48C3×C4⋊Q8D412yes—
49C3×C4⋊Q8D411yescharacteristic
50C3×#127D411yescharacteristic
51C3×#124—21no—
52C3×#121—21no—
53C3×#121—21no—
54C3×C4×D4—21no—
55C3×C4×D4—21no—
56C2×C3×C4⋊C4—21no—
57C2×C3×C4⋊C4—21no—
58C2×C3×C42C2311yescharacteristic
59C2×#214—24no—
60C2×C4⋊Dic3—21no—
61C2×C4⋊Dic3—21no—
62C2×#208—21no—
63C22×C3⋊C8—21no—
64#869—21no—
65#869—21no—
66#869—21no—
67#869—21no—
68C3×#97—21no—
69C3×C4×C8—21no—
70#830—21no—
71#810—28no—
72#810—24no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)D412yes—
76C4×C3⋊C8—21no—
77C4×C3⋊C8—21no—
78C4×C3⋊C8D412yes—
79C2×#137—22no—
80#532C2×D311yescharacteristic
81C2×C4⋊Q8—31no—
82C2×C4⋊Q8C2×D311yescharacteristic
83#497—22no—
84#494—21no—
85#492—21no—
86#492—61no—
87#487—62no—
88#487—62no—
89#487—61no—
90#461—62no—
91#448—62no—
92#431—31no—
93#423—31no—
94C2×#103—62no—
95C4×M16C2×D311yescharacteristic
96C2×C4×C8—31no—
97#343—62no—
98#332—62no—
99D4⋊C8—62no—
100C3×#41—42no—
101C3×#41—42no—
102C2×C3×Q8—21no—
103C2×C3×Q8—22no—
104C2×C3×Q8—41no—
105C2×C3×Q8—42no—
106C2×C3×Q8—42no—
107C2×C3×D4—21no—
108C2×C3×D4—21no—
109C2×C3×D4—21no—
110C2×C3×D4—41no—
111C22×C3×C4—41no—
112C22×C3×C4—41no—
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4#4111yescharacteristic
116C22×Dic3—22no—
117C2×Dic6—22no—
118C2×Dic6—22no—
119C2×Dic6—44no—
120C3×Q16—48no—
121C3×SD16—48no—
122C3×M16—22no—
123C3×M16SD1614yescomplemented
124C2×C3×C8—22no—
125C3×C4⋊C4—22no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—22no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—41no—
130C3×C4⋊C4—41no—
131C3×C22⋊C4—41no—
132C3×C22⋊C4—41no—
133C3×C42C2×D411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C3×C42C2×D411yescharacteristic
136C3×C42C2×D411yescharacteristic
137#214—84no—
138C4⋊Dic3—41no—
139C4⋊Dic3—41no—
140C4⋊Dic3—41no—
141C4⋊Dic3—41no—
142#208—42no—
143C2×C3⋊C8—22no—
144C2×C3⋊C8—22no—
145C2×C3⋊C8—42no—
146C2×C3⋊C8—41no—
147C2×#41—21no—
148C22×Q8—21no—
149C22×Q8—32no—
150#137—44no—
151#137—44no—
152C2×Q16—24no—
153C2×SD16—24no—
154C2×SD16—64no—
155C2×M16—61no—
156C2×M16#6712yes—
157C22×C8—61no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—61no—
162C4⋊Q8—61no—
163C4⋊Q8—61no—
164C4⋊Q8—61no—
165C4⋊Q8#6712yes—
166C4⋊Q8#6711yescharacteristic
167#127#6711yescharacteristic
168#124—21no—
169#121—21no—
170#121—21no—
171C4×D4—21no—
172C4×D4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—61no—
176C2×C4⋊C4—61no—
177C2×C42C22×D311yescharacteristic
178C4⋊C8—32no—
179C4⋊C8—61no—
180C4⋊C8—61no—
181#103—68no—
182#103—64no—
183C22⋊C8—61no—
184#97—21no—
185C4×C8—21no—
186C4×C8—32no—
187C4×C8—61no—
188C4×C8—61no—
189C23×C3—41no—
190C23×C3#12111yescharacteristic
191C3×Q8—44no—
192C3×Q8—42no—
193C3×Q8—81no—
194C3×Q8—81no—
195C3×D4—42no—
196C3×D4—42no—
197C3×D4—42no—
198C3×D4—82no—
199C2×C3×C4—22no—
200C2×C3×C4—22no—
201C2×C3×C4—41no—
202C2×C3×C4—42no—
203C2×C3×C4—82no—
204C2×C3×C4—81no—
205C2×C3×C4—81no—
206C2×C3×C4#12011yescharacteristic
207C2×C3×C4#12011yescharacteristic
208C2×C3×C4#12011yescharacteristic
209C2×C3×C4#12111yescharacteristic
210C2×C3×C4#12111yescharacteristic
211C2×C3×C4#12711yescharacteristic
212C2×C3×C4C2×SD1612yes—
213C2×C3×C4AGL(1,ℤ/8)12yes—
214C2×Dic3—42no—
215C2×Dic3—82no—
216Dic6—44no—
217Dic6—44no—
218C3×C8—24no—
219C3⋊C8—42no—
220C3⋊C8—42no—
221C3⋊C8—81no—
222#41—42no—
223#41—42no—
224C2×Q8—21no—
225C2×Q8—22no—
226C2×Q8—41no—
227C2×Q8—42no—
228C2×Q8—42no—
229C2×Q8—62no—
230C2×Q8—62no—
231C2×Q8—124no—
232C2×D4—21no—
233C2×D4—21no—
234C2×D4—21no—
235C2×D4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—62no—
239C22×C4D3×D411yescharacteristic
240C22×C4#23711yescharacteristic
241C22×C4C2×#6711yescharacteristic
242Q16—48no—
243SD16—48no—
244SD16—244no—
245M16—22no—
246M16—122no—
247M16#21414yescomplemented
248C2×C8—22no—
249C2×C8—62no—
250C2×C8—62no—
251C2×C8—122no—
252C2×C8—121no—
253C4⋊C4—22no—
254C4⋊C4—21no—
255C4⋊C4—22no—
256C4⋊C4—21no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—121no—
260C4⋊C4—121no—
261C4⋊C4—121no—
262C4⋊C4—121no—
263C22⋊C4—41no—
264C22⋊C4—41no—
265C42D3×D411yescharacteristic
266C42D3×D411yescharacteristic
267C42C2×#6711yescharacteristic
268C42C2×#6711yescharacteristic
269C22×C3—21no—
270C22×C3—21no—
271C22×C3—81no—
272C22×C3—81no—
273C22×C3—81no—
274C22×C3#39211yescharacteristic
275C22×C3#46012yes—
276C3×C4—21no—
277C3×C4—21no—
278C3×C4—81no—
279C3×C4—82no—
280C3×C4#45012yes—
281C3×C4#45512yes—
282C3×C4#47312yes—
283C3×C4#49211yescharacteristic
284C3×C4#49611yescharacteristic
285Dic3—82no—
286C23—41no—
287C23#93811yescharacteristic
288Q8—44no—
289Q8—42no—
290Q8—81no—
291Q8—81no—
292Q8—124no—
293Q8—124no—
294D4—42no—
295D4—42no—
296D4—42no—
297D4—82no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—41no—
301C2×C4—42no—
302C2×C4—82no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—122no—
306C2×C4—242no—
307C2×C4#93212yes—
308C2×C4C2×#21412yes—
309C2×C4#93711yescharacteristic
310C2×C4#93711yescharacteristic
311C2×C4#93911yescharacteristic
312C2×C4#93911yescharacteristic
313C2×C4#94011yescharacteristic
314C2×C4#95311yescharacteristic
315C8—24no—
316C8—122no—
317C8—122no—
318C8—241no—
319C2×C3—22no—
320C2×C3—161no—
321C2×C3#200411yescharacteristic
322C2×C3#201411yescharacteristic
323C2×C3#202011yescharacteristic
324C22—21no—
325C22—21no—
326C22—81no—
327C22—81no—
328C22—81no—
329C22#507512yes—
330C22#522311yescharacteristic
331C4—21no—
332C4—21no—
333C4—81no—
334C4—82no—
335C4—242no—
336C4#507411yescharacteristic
337C4#507611yescharacteristic
338C4#520212yes—
339C4#520412yes—
340C4#521812yes—
341C3#1017011yescharacteristic, complemented, Sylow
342C2—22no—
343C2—161no—
344C2#7176811yescharacteristic
345C2#7210611yescharacteristic
346C2#7210911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3152C2
socleC22×C3#392
prime radicalC3×#532C22
hypercenter#3152D3
hypocenterC3#10170
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. C2×#5093C2semidirect
5. #71773C2semidirect
6. #69879C2semidirect
7. #67667C2semidirect
123. C3×M16SD16semidirect
247. M16#214semidirect
341. C3#10170semidirect

Maximal subgroups

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

subgroupindex
C3×#31522
#833172
C2×#50932
#718022
#717732
#698792
#676672
#101703