Finite Group Databasebeta

← #10686752

Subgroups — #10686752

Order 768.

686 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 686 conjugacy classes of subgroups — too many to draw. Only its 143 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#85803C211yesmaximal, characteristic
2C3×#1801C211yesmaximal, characteristic
3#71634C211yesmaximal, characteristic
4#70509C212yesmaximal
5#70501C212yesmaximal
6#8356—31nomaximal, Sylow
7C2×C3×C4×Q8C2211yescharacteristic
8#5812—21no—
9#5570—22no—
10#5569—24no—
11Q8×Dic6—22no—
12#5500—21no—
13#5500C414yes—
14#5470—22no—
15#5470C2212yes—
16C3×#423C2212yes—
17C2×C3×#101—21no—
18C2×C4⋊(C3⋊C8)C2211yescharacteristic
19C2×C4×C3⋊C8C2211yescharacteristic
20C3×#343—21no—
21#4524—22no—
22#4486—22no—
23#4484—24no—
24#4481—22no—
25#3502—31no—
26#1984—31no—
27#1801D311yescharacteristic
28#1550—32no—
29#1493—32no—
30C22×C3×Q8D411yescharacteristic
31C2×C3×M16—22no—
32C3×C4×Q8—22no—
33C3×C4×Q8—22no—
34C2×C3×C4⋊C4C2×C412yes—
35C2×C3×C4⋊C4D411yescharacteristic
36C2×C3×C42D412yes—
37C2×C3×C42C2311yescharacteristic
38Q8×Dic3—24no—
39Q8×Dic3—24no—
40#926—24no—
41#926—22no—
42#924—24no—
43#924—22no—
44#924C2×C414yes—
45C22×C3⋊C8—21no—
46C22×C3⋊C8—21no—
47#890—24no—
48#890—28no—
49#888—24no—
50#888—28no—
51#870—24no—
52#870—22no—
53#869—22no—
54#869—22no—
55#869D412yes—
56#869D412yes—
57C4×Dic6—24no—
58C4×Dic6—24no—
59C4×Dic6—22no—
60C4×Dic6D414yes—
61C3×C4⋊C8—22no—
62C3×C4⋊C8—22no—
63C3×#101—42no—
64C4⋊(C3⋊C8)—22no—
65C4×C3⋊C8—22no—
66Q82—62no—
67#557—64no—
68#557—62no—
69#544—34no—
70#544—61no—
71#533—32no—
72#533—62no—
73#519—61no—
74C2×C4×Q8C2×D311yescharacteristic
75#423C2×D312yes—
76C2×C4⋊C8—31no—
77C2×#101—21no—
78C2×C4×C8—31no—
79#343—21no—
80#336—62no—
81#333—62no—
82#332—62no—
83Q8⋊C8—64no—
84C2×C3×Q8—22no—
85C2×C3×Q8—22no—
86C2×C3×Q8SD1612yes—
87C2×C3×Q8Q1612yes—
88C22×C3×C4C22⋊C412yes—
89C22×C3×C4C22⋊C412yes—
90C22×C3×C4C22×C411yescharacteristic
91C22×C3×C4C2×D411yescharacteristic
92C22×C3×C4C2×D411yescharacteristic
93C2×Dic6—22no—
94C2×Dic6—24no—
95C2×Dic6C22⋊C414yes—
96C3×M16—42no—
97C3×M16—42no—
98C2×C3×C8—42no—
99C3×C4⋊C4—24no—
100C3×C4⋊C4—22no—
101C3×C42—22no—
102C3×C42—22no—
103C3×C42C2×D412yes—
104C3×C42C2×D412yes—
105#217—24no—
106C4⋊Dic3—24no—
107C4⋊Dic3—24no—
108C4⋊Dic3—24no—
109C4⋊Dic3—22no—
110C4⋊Dic3—22no—
111C4⋊Dic3—22no—
112C4⋊Dic3C22⋊C414yes—
113Dic3⋊C4—28no—
114Dic3⋊C4—24no—
115Dic3⋊C4—24no—
116C4×Dic3—28no—
117C4×Dic3—24no—
118C2×C3⋊C8—22no—
119C2×C3⋊C8—41no—
120C2×C3⋊C8—41no—
121C2×C3⋊C8—42no—
122C2×C3⋊C8—41no—
123C22×Q8#6711yescharacteristic
124C2×M16—22no—
125C22×C8—61no—
126C22×C8—61no—
127C4⋊Q8—32no—
128C4⋊Q8—32no—
129C4⋊Q8—62no—
130C4⋊Q8—62no—
131C4⋊Q8—68no—
132C4⋊Q8—64no—
133#125—64no—
134#125—68no—
135#125—64no—
136#125—62no—
137C22⋊Q8—34no—
138C22⋊Q8—64no—
139C22⋊Q8—62no—
140C4×Q8—22no—
141C4×Q8—22no—
142C4×Q8—34no—
143C4×Q8—64no—
144C4×Q8—64no—
145C4×Q8—64no—
146C4×Q8—64no—
147C4×Q8—62no—
148#117—64no—
149#117—62no—
150C2×C4⋊C4C4×D312yes—
151C2×C4⋊C4#6711yescharacteristic
152C2×C42D1212yes—
153C2×C42C22×D311yescharacteristic
154C4⋊C8—22no—
155C4⋊C8—22no—
156C4⋊C8—62no—
157#101—42no—
158C4×C8—62no—
159C23×C3C2×C22⋊C411yescharacteristic
160C3×Q8—24no—
161C3×Q8—24no—
162C2×C3×C4—22no—
163C2×C3×C4—22no—
164C2×C3×C4—21no—
165C2×C3×C4—21no—
166C2×C3×C4—22no—
167C2×C3×C4—41no—
168C2×C3×C4—42no—
169C2×C3×C4#10314yes—
170C2×C3×C4C2×C22⋊C412yes—
171C2×C3×C4#12011yescharacteristic
172C2×C3×C4#12011yescharacteristic
173C2×C3×C4#12012yes—
174C2×C3×C4C2×SD1611yescharacteristic
175C2×C3×C4C2×Q1611yescharacteristic
176C2×C3×C4#13712yes—
177C2×Dic3—24no—
178C2×Dic3—24no—
179Dic6—42no—
180Dic6—42no—
181Dic6—44no—
182Dic6—84no—
183C3×C8—82no—
184C3⋊C8—42no—
185C3⋊C8—81no—
186C2×Q8—22no—
187C2×Q8—22no—
188C2×Q8—34no—
189C2×Q8—64no—
190C2×Q8—62no—
191C2×Q8#21412yes—
192C2×Q8#21612yes—
193C22×C4(C2×D3)⋊C412yes—
194C22×C4(C2×D3)⋊C412yes—
195C22×C4C2×C4×D311yescharacteristic
196C22×C4C2×D1211yescharacteristic
197C22×C4C2×#6711yescharacteristic
198M16—42no—
199M16—42no—
200C2×C8—42no—
201C2×C8—62no—
202C2×C8—121no—
203C2×C8—121no—
204C2×C8—122no—
205C2×C8—121no—
206C4⋊C4—24no—
207C4⋊C4—22no—
208C4⋊C4—34no—
209C4⋊C4—64no—
210C4⋊C4—64no—
211C4⋊C4—64no—
212C4⋊C4—68no—
213C4⋊C4—64no—
214C4⋊C4—64no—
215C4⋊C4—62no—
216C4⋊C4—62no—
217C4⋊C4—62no—
218C22⋊C4—64no—
219C42—22no—
220C42—22no—
221C42—68no—
222C42—64no—
223C42D3×D412yes—
224C42D3×D412yes—
225C22×C3—21no—
226C22×C3—21no—
227C22×C3#37911yescharacteristic
228C22×C3C2×#10311yescharacteristic
229C22×C3#41911yescharacteristic
230C3×C4—21no—
231C3×C4—24no—
232C3×C4—22no—
233C3×C4—42no—
234C3×C4#44812yes—
235C3×C4#45112yes—
236C3×C4#45411yescharacteristic
237C3×C4#45611yescharacteristic
238Dic3—44no—
239Dic3—44no—
240C23C2×(C2×D3)⋊C411yescharacteristic
241Q8—24no—
242Q8—24no—
243Q8—124no—
244Q8—122no—
245Q8—122no—
246Q8—244no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—64no—
255C2×C4—64no—
256C2×C4#81014yes—
257C2×C4D3×C22⋊C412yes—
258C2×C4#88211yescharacteristic
259C2×C4#88211yescharacteristic
260C2×C4C2×#21411yescharacteristic
261C2×C4#93712yes—
262C2×C4C2×#21611yescharacteristic
263C2×C4#95112yes—
264C8—82no—
265C8—122no—
266C8—241no—
267C2×C3—21no—
268C2×C3—21no—
269C2×C3#178711yescharacteristic
270C2×C3#179111yescharacteristic
271C2×C3#179311yescharacteristic
272C22—21no—
273C22—21no—
274C22#495511yescharacteristic
275C22C2×#81011yescharacteristic
276C22#500411yescharacteristic
277C4—24no—
278C4—21no—
279C4—22no—
280C4—42no—
281C4—124no—
282C4—124no—
283C4#475211yescharacteristic
284C4#475311yescharacteristic
285C4#504012yes—
286C4#505012yes—
287C3#835611yescharacteristic, complemented, Sylow
288C2—21no—
289C2—21no—
290C2#6952511yescharacteristic
291C2#7012111yescharacteristic
292C2#7014311yescharacteristic
subgroupG/H
centerC22#4955
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1801C2
socleC22×C3#379
prime radicalC23×C3C2×C22⋊C4
cosocleC2×C3×C42C23
hypercenter#1801D3
hypocenterC3#8356
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1801

Complements

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

NH ≅ G/Ntype
287. C3#8356semidirect

Maximal subgroups

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

subgroupindex
#858032
C3×#18012
#716342
#705092
#705012
#83563