Finite Group Databasebeta

← #34007604

Subgroups — #34007604

Order 18688.

558 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 122 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#22625846C211yesmaximal, characteristic, complemented
2#22625646C211yesmaximal, characteristic, complemented
3C2×#17989437C212yesmaximal, complemented
4C2×D4⋊#10596833C212yesmaximal
5#22612478C211yesmaximal, characteristic
6C2×C4⋊#10596833C2212yes—
7C2×C4×#10596833C2212yes—
8#17989453—22no—
9#17989453—22no—
10#17989437—24no—
11D4⋊#10596833—24no—
12C2×#14556578C411yescharacteristic
13C2×#14556556C2211yescharacteristic
14C2×D4×Dic73C2211yescharacteristic
15#17988493C411yescharacteristic
16#17988475—22no—
17#17988452C411yescharacteristic
18C22⋊C4×Dic73—22no—
19C42×Dic73C2211yescharacteristic
20#390×C73C411yescharacteristic
21C22×#10596833—22no—
22C22×#10596833—22no—
23C4⋊#10596833—22no—
24C4⋊#10596833—22no—
25C4×#10596833—22no—
26C4×#10596833D414yes—
27C22×Dic146C2×C411yescharacteristic
28C23×Dic73—21no—
29#14556578—22no—
30#14556578—22no—
31#14556556—22no—
32#14556556—22no—
33D4×Dic73—22no—
34D4×Dic73—22no—
35C2×C4⋊Dic73C2×C411yescharacteristic
36C2×#11924007—21no—
37C2×C22⋊Dic73—22no—
38C2×C22⋊Dic73—21no—
39C2×C4×Dic73—21no—
40C2×C4×Dic73C2×C411yescharacteristic
41C2×C4×Dic73D412yes—
42C2×C4×Dic73D412yes—
43C2×C4×Dic73C2311yescharacteristic
44#14556498—22no—
45#14556498—22no—
46C22×D4×C73C2×C411yescharacteristic
47C2×C4⋊C4×C73C2×C411yescharacteristic
48C2×C22⋊C4×C73—22no—
49C2×C42×C73C2×C411yescharacteristic
50C2×#10596833—24no—
51C2×#10596833—44no—
52C2×#10596833—42no—
53C2×#10596833—44no—
54C2×Dic146—22no—
55C2×Dic146—22no—
56C2×Dic146—21no—
57C2×Dic146M1612yes—
58C22×Dic73—21no—
59C22×Dic73—21no—
60C22×Dic73—41no—
61C22×Dic73—41no—
62C22×Dic73—44no—
63C22×Dic73C22⋊C412yes—
64C22×Dic73C2×D411yescharacteristic
65C22×Dic73C2×D411yescharacteristic
66C4⋊Dic73—22no—
67#11924007—42no—
68C22⋊Dic73—28no—
69C22⋊Dic73—42no—
70C4×Dic73—21no—
71C4×Dic73—21no—
72C4×Dic73—24no—
73C4×Dic73—42no—
74C4×Dic73—41no—
75C4×Dic73—41no—
76C4×Dic73C2×D412yes—
77C4×Dic73#4112yes—
78C24×C73—21no—
79C2×D4×C73—22no—
80C2×D4×C73—21no—
81C2×D4×C73—22no—
82C2×D4×C73M1612yes—
83C22×C4×C73—21no—
84C22×C4×C73C22⋊C412yes—
85C22×C4×C73C22×C411yescharacteristic
86C4⋊C4×C73—22no—
87C22⋊C4×C73—44no—
88C42×C73—22no—
89C23×C73—22no—
90C23×C73—41no—
91C23×C73—41no—
92C23×C73—42no—
93C23×C73—42no—
94C23×C73C2×C22⋊C411yescharacteristic
95D4×C73—24no—
96D4×C73—42no—
97C2×C4×C73—22no—
98C2×C4×C73—22no—
99C2×C4×C73—22no—
100C2×C4×C73—41no—
101C2×C4×C73—42no—
102C2×C4×C73C4×D412yes—
103C2×C4×C73C2×M1611yescharacteristic
104C2×C4×C73C2×M1611yescharacteristic
105C2×C4×C73#13112yes—
106C2×Dic73—22no—
107C2×Dic73—22no—
108C2×Dic73—22no—
109C2×Dic73—22no—
110C2×Dic73—22no—
111C2×Dic73—41no—
112C2×Dic73—41no—
113C2×Dic73—41no—
114C2×Dic73—42no—
115C2×Dic73—82no—
116C2×Dic73—82no—
117C2×Dic73#12112yes—
118C2×Dic73#12411yescharacteristic
119C2×Dic73#12711yescharacteristic
120C2×Dic73#13514yes—
121Dic146—24no—
122Dic146—42no—
123#10596833—44no—
124#10596833—82no—
125C22×C73—42no—
126C22×C73—44no—
127C22×C73—81no—
128C22×C73—81no—
129C22×C73—82no—
130C22×C73#39011yescharacteristic
131C22×C73#40711yescharacteristic
132C22×C73#40811yescharacteristic
133C22×C73#41612yes—
134C22×C73#42112yes—
135C4×C73—42no—
136C4×C73—81no—
137C4×C73#43512yes—
138C4×C73#43512yes—
139Dic73—22no—
140Dic73—42no—
141Dic73—42no—
142Dic73—81no—
143Dic73#48812yes—
144Dic73#49512yes—
145#9645—731nomaximal, Sylow
146C2×C73—82no—
147C2×C73#148212yes—
148C2×C73#148812yes—
149C2×C73#195511yescharacteristic
150C2×C73#195811yescharacteristic
151C2×C73#196311yescharacteristic
152C4×#124—731no—
153#1831—731no—
154#1744—731no—
155C2×Q8⋊C8—732no—
156C2×D4⋊C8—732no—
157C73#964511yescharacteristic, complemented, Sylow
158C2×#124—731no—
159C2×C4×Q8—731no—
160C2×C4×D4—731no—
161C2×C4⋊C8—732no—
162C2×C4×C8—732no—
163#391—731no—
164#390#6440811yescharacteristic
165#388—1462no—
166#382—731no—
167C4×C22⋊C4—1462no—
168C43—731no—
169#336—1462no—
170#336—1462no—
171Q8⋊C8—1464no—
172D4⋊C8—1464no—
173C22×Q8—731no—
174C22×D4C2×#6440811yescharacteristic
175C23×C4—1461no—
176C22×C8—1462no—
177C22×C8—1462no—
178#124—1462no—
179#124—1462no—
180C4×Q8—1462no—
181C4×Q8—1462no—
182C4×D4—1462no—
183C4×D4—1462no—
184C2×C4⋊C4—731no—
185C2×C4⋊C4—1461no—
186C2×C4⋊C4C2×#6440811yescharacteristic
187C2×C22⋊C4—22no—
188C2×C22⋊C4—1461no—
189C2×C22⋊C4—1462no—
190C2×C42—732no—
191C2×C42—731no—
192C2×C42—732no—
193C2×C42—731no—
194C2×C42—1461no—
195C2×C42C2×#6440811yescharacteristic
196C4⋊C8—1462no—
197C4⋊C8—1462no—
198C4×C8—734no—
199C4×C8—1462no—
200#95—1462no—
201#95—1462no—
202C24—21no—
203C2×Q8—732no—
204C2×Q8—1462no—
205C2×Q8—1462no—
206C2×Q8—1461no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—22no—
210C2×D4#1192403812yes—
211C22×C4—21no—
212C22×C4—2921no—
213C22×C4—2921no—
214C22×C4—2924no—
215C22×C4—731no—
216C22×C4—731no—
217C22×C4—732no—
218C22×C4—1461no—
219C22×C4—1461no—
220C22×C4C22⋊#6440812yes—
221C22×C4C22×#6440811yescharacteristic
222C2×C8—2924no—
223C2×C8—2922no—
224C2×C8—2924no—
225C2×C8—1464no—
226C4⋊C4—22no—
227C4⋊C4—2922no—
228C4⋊C4—1462no—
229C22⋊C4—44no—
230C22⋊C4—2922no—
231C22⋊C4—1468no—
232C42—22no—
233C42—2922no—
234C42—2921no—
235C42—2921no—
236C42—732no—
237C42—732no—
238C42—1464no—
239C42—1461no—
240C42—1461no—
241C23—22no—
242C23—41no—
243C23—41no—
244C23—42no—
245C23—42no—
246C23C2×C22⋊#6440811yescharacteristic
247Q8—2922no—
248Q8—1464no—
249D4—24no—
250D4—42no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—2921no—
257C2×C4—2921no—
258C2×C4—2921no—
259C2×C4—2922no—
260C2×C4—5842no—
261C2×C4—5842no—
262C2×C4—731no—
263C2×C4—731no—
264C2×C4—734no—
265C2×C4—732no—
266C2×C4—1462no—
267C2×C4—1462no—
268C2×C4—1462no—
269C2×C4—1462no—
270C2×C4—1462no—
271C2×C4D4×#6440812yes—
272C2×C4D73⋊M1611yescharacteristic
273C2×C4C2×#1192403811yescharacteristic
274C2×C4#1455667812yes—
275C8—2924no—
276C8—5842no—
277C22—42no—
278C22—44no—
279C22—81no—
280C22—81no—
281C22—82no—
282C22#1798953911yescharacteristic
283C22#1798955811yescharacteristic
284C22#1798956011yescharacteristic
285C22#1798957712yes—
286C22#1798958512yes—
287C4—42no—
288C4—81no—
289C4—2922no—
290C4—2922no—
291C4—5841no—
292C4—732no—
293C4—732no—
294C4—1462no—
295C4#1798961612yes—
296C4#1798961912yes—
297C2—82no—
298C2#2262524212yes—
299C2#2262526312yes—
300C2#2262605611yescharacteristic
301C2#2262606011yescharacteristic
302C2#2262606911yescharacteristic
subgroupG/H
centerC23C2×C22⋊#64408
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
Fitting#390×C73C4
socleC23×C73C2×C22⋊C4
prime radicalC22×D4×C73C2×C4
cosocleC2×C4×Dic73C23
hypercenter#390#64408
hypocenterC73#9645
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #390

Complements

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

NH ≅ G/Ntype
1. #22625846C2semidirect
2. #22625646C2semidirect
3. C2×#17989437C2semidirect
157. C73#9645semidirect

Maximal subgroups

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

subgroupindex
#226258462
#226256462
C2×#179894372
C2×D4⋊#105968332
#226124782
#964573