Finite Group Databasebeta

← #10708210

Subgroups — #10708210

Order 768.

342 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 57 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#77927C211yesmaximal, characteristic
2#69713C211yesmaximal, characteristic
3#68979C211yesmaximal, characteristic
4#68140C211yesmaximal, characteristic
5C3×#1346C211yesmaximal, characteristic
6#67495C211yesmaximal, characteristic
7#67452C211yesmaximal, characteristic
8#12251—31nomaximal, Sylow
9#5500C2211yescharacteristic
10C2×#889—21no—
11C3×#398C2211yescharacteristic
12#5121—21no—
13#4982—21no—
14#4982—21no—
15#4981C2211yescharacteristic
16#4977C2211yescharacteristic
17#4971—21no—
18#4721—21no—
19#4660—21no—
20#4658—21no—
21#4657—21no—
22C3×#343C2211yescharacteristic
23C3×#324C2211yescharacteristic
24#4555—21no—
25#4528C2211yescharacteristic
26#4467—21no—
27#4452—21no—
28#2822—31no—
29#2061—31no—
30#1611—31no—
31#1611—31no—
32#1346—31no—
33#1346D311yescharacteristic
34#1324—31no—
35C2×C3×M16—21no—
36C2×C3×C4⋊C4—21no—
37C2×C3×C4⋊C4D411yescharacteristic
38C2×C3×C4⋊C4C2311yescharacteristic
39#926—21no—
40C2×C4⋊Dic3—21no—
41C2×C4⋊Dic3D411yescharacteristic
42C2×C4⋊Dic3D411yescharacteristic
43#924—21no—
44#924—21no—
45C2×Dic3⋊C4—21no—
46C2×Dic3⋊C4—21no—
47C2×Dic3⋊C4—21no—
48C2×Dic3⋊C4—21no—
49C2×Dic3⋊C4—21no—
50C2×C4×Dic3D411yescharacteristic
51#890—41no—
52#889—42no—
53#889—42no—
54#888—41no—
55C3×C22⋊C8—21no—
56C3×#95—21no—
57C3×#95D411yescharacteristic
58C3⋊#95—21no—
59C3⋊#95—21no—
60C3⋊#95—21no—
61C3⋊#95—21no—
62C3⋊#95—21no—
63C3⋊#95—21no—
64C3⋊#95D411yescharacteristic
65#830—21no—
66#818—41no—
67#817—41no—
68#817—41no—
69#816—41no—
70#810—41no—
71#808—41no—
72#544—31no—
73C2×#125—61no—
74#483—61no—
75#483—61no—
76#428—61no—
77#400—61no—
78#400—61no—
79#398—31no—
80#398—61no—
81#398—61no—
82#398—61no—
83#398C2×D311yescharacteristic
84#382—31no—
85#343C2×D311yescharacteristic
86#339—61no—
87#328—61no—
88#328—61no—
89#324—31no—
90#324C2×D311yescharacteristic
91C22×C3×C4—21no—
92C22×C3×C4—21no—
93C22×C3×C4C2×D411yescharacteristic
94C22×C3×C4C2×D411yescharacteristic
95C22×C3×C4C2×D411yescharacteristic
96C22×Dic3—21no—
97C22×Dic3—21no—
98C22×Dic3—21no—
99C22×Dic3—21no—
100C2×Dic6—21no—
101C2×Dic6—21no—
102C3×M16—22no—
103C3×M16—41no—
104C2×C3×C8—41no—
105C2×C3×C8—41no—
106C3×C4⋊C4—41no—
107C3×C4⋊C4—41no—
108C3×C4⋊C4—41no—
109C3×C4⋊C4—41no—
110#217—41no—
111C4⋊Dic3—21no—
112C4⋊Dic3—21no—
113C4⋊Dic3—21no—
114C4⋊Dic3—21no—
115C4⋊Dic3—42no—
116C4⋊Dic3—41no—
117C4⋊Dic3—41no—
118Dic3⋊C4—42no—
119Dic3⋊C4—42no—
120Dic3⋊C4—42no—
121Dic3⋊C4—41no—
122Dic3⋊C4—41no—
123Dic3⋊C4—41no—
124Dic3⋊C4—41no—
125Dic3⋊C4—81no—
126C4×Dic3—21no—
127C4×Dic3—41no—
128C4×Dic3SD1612yes—
129C2×C3⋊C8—41no—
130C2×M16—21no—
131C4⋊Q8—121no—
132#125—122no—
133#125—122no—
134#125—121no—
135C22⋊Q8—61no—
136C22⋊Q8—61no—
137#117—61no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—31no—
140C2×C4⋊C4—31no—
141C2×C4⋊C4—61no—
142C2×C4⋊C4—61no—
143C2×C4⋊C4—61no—
144C2×C4⋊C4—61no—
145C2×C4⋊C4—61no—
146C2×C4⋊C4—61no—
147C2×C4⋊C4D1211yescharacteristic
148C2×C4⋊C4C22×D311yescharacteristic
149C2×C42—31no—
150#107—121no—
151#106—121no—
152#106—121no—
153#106—121no—
154#103—121no—
155#103—121no—
156C22⋊C8—21no—
157C22⋊C8—61no—
158#95—21no—
159#95—31no—
160#95—61no—
161#95—61no—
162#95—61no—
163#95—61no—
164#95—61no—
165#95—61no—
166#95D1211yescharacteristic
167C23×C3#12011yescharacteristic
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—41no—
173C2×C3×C4—41no—
174C2×C3×C4—41no—
175C2×C3×C4—41no—
176C2×C3×C4—41no—
177C2×C3×C4—41no—
178C2×C3×C4—41no—
179C2×C3×C4—41no—
180C2×C3×C4C2×SD1611yescharacteristic
181C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
182C2×Dic3—22no—
183C2×Dic3—41no—
184C2×Dic3—41no—
185C2×Dic3—41no—
186C2×Dic3—41no—
187C2×Dic3—42no—
188C2×Dic3—41no—
189C2×Dic3—41no—
190C2×Dic3—41no—
191C2×Dic3—41no—
192C2×Dic3—41no—
193C2×Dic3—41no—
194C2×Dic3—41no—
195Dic6—81no—
196Dic6—81no—
197C3×C8—42no—
198C3×C8—81no—
199C3⋊C8—81no—
200C2×Q8—61no—
201C2×Q8—61no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—61no—
205C22×C4—61no—
206C22×C4—61no—
207C22×C4—61no—
208C22×C4C2×D1211yescharacteristic
209C22×C4D3×D411yescharacteristic
210C22×C4D3×D411yescharacteristic
211M16—22no—
212M16—41no—
213C2×C8—41no—
214C2×C8—41no—
215C2×C8—121no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—61no—
221C4⋊C4—61no—
222C4⋊C4—61no—
223C4⋊C4—61no—
224C4⋊C4—121no—
225C4⋊C4—121no—
226C4⋊C4—121no—
227C4⋊C4—121no—
228C4⋊C4—122no—
229C4⋊C4—122no—
230C4⋊C4—122no—
231C4⋊C4—122no—
232C4⋊C4—121no—
233C4⋊C4—121no—
234C4⋊C4—241no—
235C22⋊C4—121no—
236C42—32no—
237C42—61no—
238C42—121no—
239C22×C3—21no—
240C22×C3—21no—
241C22×C3#45011yescharacteristic
242C22×C3#45611yescharacteristic
243C22×C3B(4,𝔽2)11yescharacteristic
244C3×C4—21no—
245C3×C4—21no—
246C3×C4—41no—
247C3×C4—41no—
248C3×C4—81no—
249C3×C4—81no—
250Dic3—42no—
251Dic3—81no—
252Dic3—81no—
253Dic3—81no—
254Dic3—81no—
255Dic3—81no—
256C23#88211yescharacteristic
257Q8—241no—
258Q8—241no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—62no—
272C2×C4—121no—
273C2×C4—121no—
274C2×C4—121no—
275C2×C4—121no—
276C2×C4—122no—
277C2×C4—121no—
278C2×C4—121no—
279C2×C4—121no—
280C2×C4—121no—
281C2×C4—121no—
282C2×C4—121no—
283C2×C4—121no—
284C2×C4#91111yescharacteristic
285C2×C4D3×SD1611yescharacteristic
286C8—42no—
287C8—81no—
288C8—241no—
289C2×C3—21no—
290C2×C3—21no—
291C2×C3#160411yescharacteristic
292C2×C3#220111yescharacteristic
293C2×C3#220311yescharacteristic
294C22—21no—
295C22—21no—
296C22#474111yescharacteristic
297C22#477611yescharacteristic
298C22#482611yescharacteristic
299C4—21no—
300C4—21no—
301C4—41no—
302C4—41no—
303C4—81no—
304C4—81no—
305C4—122no—
306C4—241no—
307C4—241no—
308C4—241no—
309C4—241no—
310C4—241no—
311C3#1225111yescharacteristic, complemented, Sylow
312C2—21no—
313C2—21no—
314C2#6829511yescharacteristic
315C2#6914111yescharacteristic
316C2#6916711yescharacteristic
subgroupG/H
centerC22#4741
commutator, cosocleC2×C3×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1346C2
socleC22×C3B(4,𝔽2)
prime radicalC23×C3#120
hypercenter#1346D3
hypocenterC3#12251
LCSG ▷ C2×C3×C4⋊C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1346

Complements

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

NH ≅ G/Ntype
311. C3#12251semidirect

Maximal subgroups

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

subgroupindex
#779272
#697132
#689792
#681402
C3×#13462
#674952
#674522
#122513