Finite Group Databasebeta

← #34006237

Subgroups — #34006237

Order 18688.

498 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 110 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#22626826C211yesmaximal, characteristic
2#22625233C212yesmaximal
3#22625175C211yesmaximal, characteristic
4#22624755C212yesmaximal
5#22612649C211yesmaximal, characteristic
6#17989636—24no—
7#17989604—22no—
8#17989603C2212yes—
9#17989603C2212yes—
10Q8⋊#10596833—24no—
11#17989433—22no—
12C2×#14556589C411yescharacteristic
13C2×Q8×Dic73C2211yescharacteristic
14C2×#14556556C2211yescharacteristic
15#17988493C411yescharacteristic
16#17988474—22no—
17#17988469—22no—
18#17988452C411yescharacteristic
19#17988433C2211yescharacteristic
20#391×C73C411yescharacteristic
21#14556658—24no—
22C4⋊#10596833—22no—
23C4⋊#10596833—22no—
24C22⋊#10596833—22no—
25C22⋊#10596833—22no—
26C4×#10596833—24no—
27C4×#10596833—22no—
28C4×#10596833D414yes—
29C22×Dic146C2×C411yescharacteristic
30#14556589—22no—
31#14556589C814yescomplemented
32Q8×Dic73—22no—
33Q8×Dic73—22no—
34#14556556—22no—
35#14556556—22no—
36C2×C4⋊Dic73—21no—
37C2×C4⋊Dic73C2×C411yescharacteristic
38C2×#11924007—22no—
39C2×#11924007D412yes—
40C2×C4×Dic73—21no—
41C2×C4×Dic73C2×C411yescharacteristic
42C2×C4×Dic73D412yes—
43C2×C4×Dic73C2311yescharacteristic
44#14556498—22no—
45#14556498—21no—
46#14556498—21no—
47#14556498—22no—
48C22×Q8×C73C2×C411yescharacteristic
49C2×C4⋊C4×C73C2×C411yescharacteristic
50C2×C42×C73C2×C411yescharacteristic
51#95×C73—22no—
52C2×#10596833—24no—
53C2×#10596833—42no—
54C2×Dic146—22no—
55C2×Dic146—22no—
56C2×Dic146C2×C812yes—
57C2×Dic146M1612yes—
58C22×Dic73—21no—
59C22×Dic73—21no—
60C22×Dic73C22⋊C412yes—
61C22×Dic73C2×D411yescharacteristic
62C22×Dic73C2×D411yescharacteristic
63C4⋊Dic73—22no—
64C4⋊Dic73—41no—
65C4⋊Dic73—41no—
66#11924007—28no—
67#11924007—24no—
68C4×Dic73—21no—
69C4×Dic73—22no—
70C4×Dic73—22no—
71C4×Dic73—42no—
72C4×Dic73C2×C812yes—
73C4×Dic73C2×D412yes—
74C4×Dic73#4112yes—
75C2×Q8×C73—21no—
76C2×Q8×C73—22no—
77C2×Q8×C73—22no—
78C2×Q8×C73C2×C812yes—
79C22×C4×C73—21no—
80C22×C4×C73—21no—
81C22×C4×C73C22⋊C412yes—
82C22×C4×C73C22×C411yescharacteristic
83C4⋊C4×C73—22no—
84C42×C73—22no—
85C23×C73C2×C22⋊C411yescharacteristic
86Q8×C73—24no—
87Q8×C73—42no—
88C2×C4×C73—22no—
89C2×C4×C73—22no—
90C2×C4×C73—41no—
91C2×C4×C73—41no—
92C2×C4×C73—42no—
93C2×C4×C73—41no—
94C2×C4×C73—41no—
95C2×C4×C73—41no—
96C2×C4×C73C4×D412yes—
97C2×C4×C73C22×C811yescharacteristic
98C2×C4×C73C2×M1611yescharacteristic
99C2×C4×C73#13112yes—
100C2×Dic73—22no—
101C2×Dic73—22no—
102C2×Dic73—22no—
103C2×Dic73—41no—
104C2×Dic73—41no—
105C2×Dic73—41no—
106C2×Dic73—41no—
107C2×Dic73—42no—
108C2×Dic73—41no—
109C2×Dic73C22⋊C814yes—
110C2×Dic73#12011yescharacteristic
111C2×Dic73#12112yes—
112C2×Dic73#12311yescharacteristic
113Dic146—24no—
114Dic146—24no—
115#10596833—44no—
116#10596833—82no—
117C22×C73—21no—
118C22×C73—21no—
119C22×C73#38611yescharacteristic
120C22×C73C2×C22⋊C811yescharacteristic
121C22×C73#40811yescharacteristic
122C4×C73—42no—
123C4×C73—42no—
124C4×C73—81no—
125C4×C73C8×D412yes—
126C4×C73#43512yes—
127Dic73—24no—
128Dic73—22no—
129Dic73—21no—
130Dic73—21no—
131Dic73—42no—
132Dic73—81no—
133#9100—731nomaximal, Sylow
134C2×C73—21no—
135C2×C73—21no—
136C2×C73#186911yescharacteristic
137C2×C73#190911yescharacteristic
138C2×C73#191211yescharacteristic
139#2982—731no—
140#2327—731no—
141#1479—732no—
142#1461—731no—
143#1274—732no—
144C73#910011yescharacteristic, complemented, Sylow
145C2×C4⋊Q8—731no—
146C2×C4×Q8—731no—
147C2×C4×Q8—731no—
148C8×Q8—1464no—
149#431—732no—
150#431—732no—
151#431—1462no—
152#391—731no—
153#391#6440811yescharacteristic
154#387—1462no—
155#387—1462no—
156#382—731no—
157#376—731no—
158Q8⋊C8—1464no—
159#324—1462no—
160C22×Q8—731no—
161C22×Q8C2×#6440811yescharacteristic
162C4⋊Q8—734no—
163C4⋊Q8—1462no—
164C4×Q8—1462no—
165C4×Q8—1462no—
166C4×Q8—1462no—
167C4×Q8—1462no—
168C2×C4⋊C4—732no—
169C2×C4⋊C4—731no—
170C2×C4⋊C4—1461no—
171C2×C4⋊C4—1462no—
172C2×C4⋊C4C2×#6440811yescharacteristic
173C2×C42—731no—
174C2×C42—731no—
175C2×C42—732no—
176C2×C42—1461no—
177C2×C42C2×#6440811yescharacteristic
178C4⋊C8—1462no—
179C4⋊C8—1462no—
180C4⋊C8—1464no—
181C22⋊C8—1462no—
182C22⋊C8—1462no—
183C4×C8—734no—
184C4×C8—1462no—
185C4×C8—1464no—
186#95—22no—
187#95—1461no—
188#95—1461no—
189#95—1462no—
190#95—1462no—
191C2×Q8—21no—
192C2×Q8—22no—
193C2×Q8—22no—
194C2×Q8—732no—
195C2×Q8—732no—
196C2×Q8—1462no—
197C2×Q8—1462no—
198C2×Q8#1192403612yes—
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—22no—
211C4⋊C4—2921no—
212C4⋊C4—2921no—
213C4⋊C4—1464no—
214C4⋊C4—1468no—
215C4⋊C4—1462no—
216C42—22no—
217C42—2922no—
218C42—732no—
219C42—732no—
220C42—732no—
221C42—1462no—
222C42—1462no—
223C42—1461no—
224C23C2×C22⋊#6440811yescharacteristic
225Q8—24no—
226Q8—42no—
227Q8—1464no—
228Q8—1464no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—42no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—2921no—
238C2×C4—2921no—
239C2×C4—2922no—
240C2×C4—2921no—
241C2×C4—2921no—
242C2×C4—2921no—
243C2×C4—732no—
244C2×C4—731no—
245C2×C4—731no—
246C2×C4—734no—
247C2×C4—1462no—
248C2×C4—1462no—
249C2×C4—1462no—
250C2×C4D4×#6440812yes—
251C2×C4C2×#1192403611yescharacteristic
252C2×C4D73⋊M1611yescharacteristic
253C2×C4#1455667812yes—
254C8—2924no—
255C8—5842no—
256C22—21no—
257C22—21no—
258C22#1798953511yescharacteristic
259C22#1798955511yescharacteristic
260C22#1798956111yescharacteristic
261C4—42no—
262C4—42no—
263C4—81no—
264C4—2922no—
265C4—5841no—
266C4—1464no—
267C4—1462no—
268C4—1461no—
269C4—1461no—
270C4#1798961412yes—
271C4#1798961912yes—
272C2—21no—
273C2—21no—
274C2#2262590511yescharacteristic
275C2#2262597211yescharacteristic
276C2#2262597811yescharacteristic
subgroupG/H
centerC22#17989535
commutatorC2×C4×C73C22×C8
FrattiniC22×C4C22×#64408
Fitting#391×C73C4
socleC22×C73#386
prime radicalC23×C73C2×C22⋊C4
cosocleC2×C4×Dic73C23
hypercenter#391#64408
hypocenterC73#9100
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ #391

Complements

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

NH ≅ G/Ntype
31. #14556589C8semidirect
144. C73#9100semidirect

Maximal subgroups

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

subgroupindex
#226268262
#226252332
#226251752
#226247552
#226126492
#910073