Finite Group Databasebeta

← #34006252

Subgroups — #34006252

Order 18688.

474 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 100 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#22626827C211yesmaximal, characteristic
2#22625233C212yesmaximal
3#22625174C211yesmaximal, characteristic
4#22624755C212yesmaximal
5#22612651C211yesmaximal, characteristic
6#17989638—24no—
7#17989607—22no—
8#17989603C2212yes—
9#17989603C2212yes—
10Q8⋊#10596833—24no—
11#17989432—22no—
12C2×Q8×Dic73C2211yescharacteristic
13C2×Q8×Dic73C2211yescharacteristic
14#17988494C411yescharacteristic
15#17988494C411yescharacteristic
16#17988473—22no—
17#17988473—22no—
18#17988453C411yescharacteristic
19#17988433C2211yescharacteristic
20C2×C4⋊Q8×C73C411yescharacteristic
21#14556658—24no—
22C4⋊#10596833—22no—
23C4⋊#10596833—22no—
24C22⋊#10596833—22no—
25C22⋊#10596833—22no—
26#14556646—24no—
27C4×#10596833—22no—
28C4×#10596833D414yes—
29Q8×Dic73—22no—
30Q8×Dic73—22no—
31Q8×Dic73—22no—
32Q8×Dic73—22no—
33C2×C4⋊Dic73—21no—
34C2×C4⋊Dic73C2×C411yescharacteristic
35C2×C4⋊Dic73C2×C411yescharacteristic
36C2×C4⋊Dic73D412yes—
37C2×C4×Dic73—21no—
38C2×C4×Dic73C2×C411yescharacteristic
39C2×C4×Dic73D412yes—
40C2×C4×Dic73C2311yescharacteristic
41#14556498—22no—
42#14556498—22no—
43#14556498—21no—
44#14556498—21no—
45#14556498—22no—
46C22×Q8×C73C2×C411yescharacteristic
47C22×Q8×C73C2×C411yescharacteristic
48C4⋊Q8×C73—22no—
49C4⋊Q8×C73—22no—
50C2×C4⋊C4×C73—22no—
51C2×C42×C73C2×C411yescharacteristic
52C2×#10596833—24no—
53C2×#10596833—42no—
54C22×Dic73—21no—
55C22×Dic73—21no—
56C22×Dic73C22⋊C412yes—
57C22×Dic73C2×D411yescharacteristic
58C22×Dic73C2×D411yescharacteristic
59C4⋊Dic73—22no—
60C4⋊Dic73—24no—
61C4⋊Dic73—22no—
62C4⋊Dic73—41no—
63C4⋊Dic73—41no—
64C4×Dic73—22no—
65C4×Dic73—22no—
66C4×Dic73—22no—
67C4×Dic73—42no—
68C4×Dic73C2×D412yes—
69C4×Dic73#4112yes—
70C2×Q8×C73—22no—
71C2×Q8×C73—21no—
72C2×Q8×C73—21no—
73C2×Q8×C73—21no—
74C2×Q8×C73—22no—
75C2×Q8×C73—22no—
76C2×Q8×C73—22no—
77C2×Q8×C73M1612yes—
78C22×C4×C73—21no—
79C22×C4×C73—21no—
80C22×C4×C73C22⋊C412yes—
81C22×C4×C73C22×C411yescharacteristic
82C4⋊C4×C73—28no—
83C42×C73—21no—
84C42×C73M1612yes—
85C23×C73C2×C22⋊C411yescharacteristic
86Q8×C73—24no—
87Q8×C73—42no—
88Q8×C73—42no—
89Q8×C73—42no—
90C2×C4×C73—22no—
91C2×C4×C73—22no—
92C2×C4×C73—22no—
93C2×C4×C73—22no—
94C2×C4×C73—22no—
95C2×C4×C73—41no—
96C2×C4×C73—41no—
97C2×C4×C73—41no—
98C2×C4×C73—41no—
99C2×C4×C73C4×D412yes—
100C2×C4×C73C2×M1611yescharacteristic
101C2×C4×C73C2×M1611yescharacteristic
102C2×C4×C73#13112yes—
103C2×Dic73—22no—
104C2×Dic73—41no—
105C2×Dic73—41no—
106C2×Dic73—42no—
107C2×Dic73—41no—
108C2×Dic73—41no—
109C2×Dic73—42no—
110C2×Dic73—41no—
111C2×Dic73—41no—
112C2×Dic73#12011yescharacteristic
113C2×Dic73#12112yes—
114C2×Dic73#12311yescharacteristic
115#10596833—44no—
116#10596833—82no—
117C22×C73—21no—
118C22×C73—21no—
119C22×C73#38611yescharacteristic
120C22×C73#40711yescharacteristic
121C22×C73#40811yescharacteristic
122C4×C73—42no—
123C4×C73—42no—
124C4×C73—42no—
125C4×C73#43512yes—
126C4×C73#43612yes—
127Dic73—22no—
128Dic73—21no—
129Dic73—21no—
130Dic73—42no—
131Dic73—81no—
132Dic73—81no—
133#9104—731nomaximal, Sylow
134C2×C73—21no—
135C2×C73—21no—
136C2×C73#187011yescharacteristic
137C2×C73#190911yescharacteristic
138C2×C73#191211yescharacteristic
139#2983—731no—
140#2327—731no—
141#1479—732no—
142#1461—731no—
143#1274—732no—
144C73#910411yescharacteristic, complemented, Sylow
145C2×C4⋊Q8#6440811yescharacteristic
146C2×C4×Q8—731no—
147C2×C4×Q8—731no—
148#446—1464no—
149#432—1462no—
150#431—732no—
151#431—732no—
152#391—731no—
153#391—731no—
154#387—1462no—
155#387—1462no—
156#382—731no—
157#376—731no—
158Q8⋊C8—1464no—
159#324—1462no—
160C22×Q8C2×#6440811yescharacteristic
161C22×Q8C2×#6440811yescharacteristic
162C4⋊Q8—22no—
163C4⋊Q8—22no—
164C4×Q8—1462no—
165C4×Q8—1462no—
166C4×Q8—1462no—
167C4×Q8—1462no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—732no—
170C2×C4⋊C4—731no—
171C2×C4⋊C4—731no—
172C2×C4⋊C4—1461no—
173C2×C42—731no—
174C2×C42—732no—
175C2×C42—731no—
176C2×C42—1461no—
177C2×C42C2×#6440811yescharacteristic
178C4⋊C8—1462no—
179C4⋊C8—1462no—
180C4⋊C8—1464no—
181C22⋊C8—1462no—
182C22⋊C8—1462no—
183#97—1464no—
184C4×C8—734no—
185C4×C8—1462no—
186#95—1461no—
187#95—1461no—
188#95—1462no—
189#95—1462no—
190#95—1462no—
191C2×Q8—21no—
192C2×Q8—21no—
193C2×Q8—21no—
194C2×Q8—22no—
195C2×Q8—22no—
196C2×Q8—22no—
197C2×Q8—22no—
198C2×Q8#1192403712yes—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—731no—
202C22×C4—731no—
203C22×C4—732no—
204C22×C4—1461no—
205C22×C4—1461no—
206C22×C4C22⋊#6440812yes—
207C22×C4C22×#6440811yescharacteristic
208C2×C8—2922no—
209C2×C8—1464no—
210C4⋊C4—28no—
211C4⋊C4—2921no—
212C4⋊C4—2921no—
213C4⋊C4—1464no—
214C4⋊C4—1462no—
215C4⋊C4—1462no—
216C42—21no—
217C42—2922no—
218C42—732no—
219C42—732no—
220C42—1462no—
221C42—1462no—
222C42—1462no—
223C42#1192403712yes—
224C23C2×C22⋊#6440811yescharacteristic
225Q8—24no—
226Q8—42no—
227Q8—42no—
228Q8—42no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—2922no—
239C2×C4—2921no—
240C2×C4—2921no—
241C2×C4—2922no—
242C2×C4—2921no—
243C2×C4—2921no—
244C2×C4—2921no—
245C2×C4—2921no—
246C2×C4—732no—
247C2×C4—731no—
248C2×C4—731no—
249C2×C4—1462no—
250C2×C4D4×#6440812yes—
251C2×C4C2×#1192403711yescharacteristic
252C2×C4C2×#1192403711yescharacteristic
253C2×C4#1455667812yes—
254C8—2924no—
255C8—5842no—
256C22—21no—
257C22—21no—
258C22#1798953511yescharacteristic
259C22#1798955711yescharacteristic
260C22#1798956011yescharacteristic
261C4—42no—
262C4—42no—
263C4—42no—
264C4—2922no—
265C4—5841no—
266C4—5841no—
267C4—1462no—
268C4—1461no—
269C4—1461no—
270C4#1798961812yes—
271C4#1798962212yes—
272C2—21no—
273C2—21no—
274C2#2262590811yescharacteristic
275C2#2262597311yescharacteristic
276C2#2262597811yescharacteristic
subgroupG/H
centerC22#17989535
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
FittingC2×C4⋊Q8×C73C4
socleC22×C73#386
prime radicalC23×C73C2×C22⋊C4
cosocleC2×C4×Dic73C23
hypercenterC2×C4⋊Q8#64408
hypocenterC73#9104
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
144. C73#9104semidirect

Maximal subgroups

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

subgroupindex
#226268272
#226252332
#226251742
#226247552
#226126512
#910473