Finite Group Databasebeta

← #34003545

Subgroups — #34003545

Order 18688.

486 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22625717C211yesmaximal, characteristic
2#22625645C211yesmaximal, characteristic
3C2×Q8⋊#10596833C212yesmaximal
4#22624767C212yesmaximal
5#22612406C211yesmaximal, characteristic
6C2×C4⋊#10596833C2212yes—
7C2×#14556646—21no—
8C2×C4×#10596833C2212yes—
9#17989453—21no—
10#17989453—21no—
11#17989453—21no—
12Q8⋊#10596833C418yescomplemented
13C2×Q8×Dic73C2211yescharacteristic
14#17988494C411yescharacteristic
15#17988485C411yescharacteristic
16#17988483C411yescharacteristic
17#17988472—21no—
18#17988470—21no—
19C4⋊C4×Dic73C2211yescharacteristic
20#17988438—21no—
21#17988432—21no—
22C42×Dic73C2211yescharacteristic
23#391×C73C411yescharacteristic
24C22×#10596833—22no—
25C22×#10596833—22no—
26C4⋊#10596833—22no—
27C4⋊#10596833C2×C414yes—
28#14556646—42no—
29C4×#10596833—22no—
30C4×#10596833C2×C414yes—
31Q8×Dic73—22no—
32Q8×Dic73C2×C414yes—
33C2×C4⋊Dic73—21no—
34C2×C4⋊Dic73C2×C411yescharacteristic
35C2×C4⋊Dic73C2×C411yescharacteristic
36C2×#11924007—21no—
37C2×#11924007—21no—
38C2×C4×Dic73—21no—
39C2×C4×Dic73—21no—
40C2×C4×Dic73C2×C411yescharacteristic
41C2×C4×Dic73D411yescharacteristic
42C2×C4×Dic73D411yescharacteristic
43C2×C4×Dic73D411yescharacteristic
44C2×C4×Dic73D411yescharacteristic
45C2×C4×Dic73C2311yescharacteristic
46#14556498—21no—
47#14556498—21no—
48#14556498—21no—
49#14556498—21no—
50#14556498—21no—
51C22×Q8×C73C2×C411yescharacteristic
52C2×C4⋊C4×C73C2×C411yescharacteristic
53C2×C42×C73C2×C411yescharacteristic
54#95×C73—21no—
55#95×C73—21no—
56C2×#10596833—24no—
57C2×#10596833—24no—
58C2×#10596833—42no—
59C2×#10596833—42no—
60C2×#10596833—42no—
61C2×#10596833—42no—
62C22×Dic73—21no—
63C22×Dic73—21no—
64C22×Dic73C22⋊C411yescharacteristic
65C22×Dic73C22⋊C411yescharacteristic
66C22×Dic73C2×D411yescharacteristic
67C22×Dic73C2×D411yescharacteristic
68C4⋊Dic73—21no—
69C4⋊Dic73—24no—
70C4⋊Dic73—22no—
71C4⋊Dic73C4212yes—
72#11924007—42no—
73#11924007—42no—
74C4×Dic73—22no—
75C4×Dic73—24no—
76C4×Dic73—21no—
77C4×Dic73—21no—
78C4×Dic73—22no—
79C4×Dic73—41no—
80C4×Dic73—41no—
81C4×Dic73—41no—
82C4×Dic73—41no—
83C4×Dic73C22×C412yes—
84C4×Dic73#4112yes—
85C2×Q8×C73—21no—
86C2×Q8×C73—22no—
87C2×Q8×C73—22no—
88C2×Q8×C73C4212yes—
89C22×C4×C73—21no—
90C22×C4×C73—21no—
91C22×C4×C73C22⋊C411yescharacteristic
92C22×C4×C73C22⋊C411yescharacteristic
93C22×C4×C73C22×C411yescharacteristic
94C4⋊C4×C73—22no—
95C42×C73—22no—
96C23×C73C2×C22⋊C411yescharacteristic
97Q8×C73—24no—
98Q8×C73—42no—
99C2×C4×C73—22no—
100C2×C4×C73—21no—
101C2×C4×C73—21no—
102C2×C4×C73—21no—
103C2×C4×C73—21no—
104C2×C4×C73—21no—
105C2×C4×C73—21no—
106C2×C4×C73—41no—
107C2×C4×C73—41no—
108C2×C4×C73—41no—
109C2×C4×C73—41no—
110C2×C4×C73—41no—
111C2×C4×C73C2×C4211yescharacteristic
112C2×C4×C73#11711yescharacteristic
113C2×C4×C73#13112yes—
114C2×C4×C73#13112yes—
115C2×Dic73—21no—
116C2×Dic73—21no—
117C2×Dic73—22no—
118C2×Dic73—21no—
119C2×Dic73—21no—
120C2×Dic73—21no—
121C2×Dic73—21no—
122C2×Dic73—21no—
123C2×Dic73—21no—
124C2×Dic73—41no—
125C2×Dic73—41no—
126C2×Dic73—41no—
127C2×Dic73—41no—
128C2×Dic73—41no—
129C2×Dic73C4×D412yes—
130C2×Dic73C4×D412yes—
131C2×Dic73AGL(1,ℤ/8)11yescharacteristic
132C2×Dic73#13712yes—
133C2×Dic73#13711yescharacteristic
134#10596833—44no—
135#10596833—44no—
136C22×C73C4×C22⋊C411yescharacteristic
137C22×C73#40811yescharacteristic
138C22×C73#40811yescharacteristic
139C22×C73#41711yescharacteristic
140C22×C73#41911yescharacteristic
141C22×C73#42111yescharacteristic
142C22×C73#42111yescharacteristic
143C4×C73—42no—
144C4×C73—41no—
145C4×C73—41no—
146C4×C73—81no—
147C4×C73#40512yes—
148C4×C73#43312yes—
149Dic73—22no—
150Dic73—41no—
151Dic73—41no—
152Dic73—42no—
153Dic73—81no—
154Dic73#44012yes—
155Dic73#44112yes—
156#8178—731nomaximal, Sylow
157C2×C73#149511yescharacteristic
158C2×C73#149511yescharacteristic
159C2×C73#149711yescharacteristic
160C2×C73#149711yescharacteristic
161C2×C73#175011yescharacteristic
162C2×C73#175911yescharacteristic
163C2×C73#176311yescharacteristic
164#2295—731no—
165#1767—731no—
166#1744—731no—
167C2×Q8⋊C8—732no—
168#1279—732no—
169C73#817811yescharacteristic, complemented, Sylow
170C2×C4×Q8—731no—
171C2×C4⋊C8—732no—
172C2×#97—1461no—
173C2×C4×C8—732no—
174#391—731no—
175#391#6440811yescharacteristic
176#389—731no—
177#389—731no—
178#387—1461no—
179#387—1461no—
180C4×C4⋊C4—731no—
181C4×C4⋊C4—1461no—
182#376—1461no—
183C43—731no—
184#336—1461no—
185#336—1461no—
186#336—1461no—
187Q8⋊C8—738no—
188C22×Q8C2×#6440811yescharacteristic
189C22×C8—1462no—
190C22×C8—1462no—
191C4×Q8—734no—
192C4×Q8—1462no—
193C2×C4⋊C4—731no—
194C2×C4⋊C4—731no—
195C2×C4⋊C4—1461no—
196C2×C4⋊C4—1461no—
197C2×C4⋊C4—1461no—
198C2×C4⋊C4C2×#6440811yescharacteristic
199C2×C42—731no—
200C2×C42—731no—
201C2×C42—731no—
202C2×C42—731no—
203C2×C42—731no—
204C2×C42—731no—
205C2×C42—1461no—
206C2×C42—1461no—
207C2×C42C2×#6440811yescharacteristic
208C4⋊C8—734no—
209C4⋊C8—1462no—
210#97—2922no—
211C4×C8—734no—
212C4×C8—1462no—
213#95—21no—
214#95—21no—
215#95—1461no—
216#95—1461no—
217#95—1461no—
218#95—1461no—
219#95—1461no—
220C2×Q8—21no—
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8C4×#6440812yes—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—731no—
227C22×C4—731no—
228C22×C4—731no—
229C22×C4—731no—
230C22×C4—1461no—
231C22×C4—1461no—
232C22×C4C22⋊#6440811yescharacteristic
233C22×C4C22⋊#6440811yescharacteristic
234C22×C4C22×#6440811yescharacteristic
235C2×C8—2922no—
236C2×C8—2922no—
237C2×C8—2922no—
238C2×C8—2922no—
239C2×C8—1464no—
240C2×C8—1464no—
241C4⋊C4—22no—
242C4⋊C4—2922no—
243C4⋊C4—2922no—
244C4⋊C4—732no—
245C4⋊C4—1464no—
246C4⋊C4—1462no—
247C4⋊C4—1461no—
248C42—22no—
249C42—2921no—
250C42—2921no—
251C42—2921no—
252C42—2921no—
253C42—732no—
254C42—732no—
255C42—1461no—
256C42—1461no—
257C42—1462no—
258C42—1464no—
259C42—1462no—
260C23C2×C22⋊#6440811yescharacteristic
261Q8—24no—
262Q8—42no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—2921no—
276C2×C4—2921no—
277C2×C4—2921no—
278C2×C4—2921no—
279C2×C4—2921no—
280C2×C4—732no—
281C2×C4—731no—
282C2×C4—731no—
283C2×C4—732no—
284C2×C4—732no—
285C2×C4—1462no—
286C2×C4—1461no—
287C2×C4—1461no—
288C2×C4—1461no—
289C2×C4—1461no—
290C2×C4—1461no—
291C2×C4—1461no—
292C2×C4—1461no—
293C2×C4—1461no—
294C2×C4C2×C4×#6440811yescharacteristic
295C2×C4#1455667011yescharacteristic
296C2×C4#1455667812yes—
297C2×C4#1455667812yes—
298C8—2924no—
299C8—2924no—
300C22C4×C22⋊#6440811yescharacteristic
301C22#1798956011yescharacteristic
302C22#1798956111yescharacteristic
303C22#1798957811yescharacteristic
304C22#1798958011yescharacteristic
305C22#1798958411yescharacteristic
306C22#1798958611yescharacteristic
307C4—41no—
308C4—41no—
309C4—42no—
310C4—81no—
311C4—2921no—
312C4—2921no—
313C4—2922no—
314C4—5841no—
315C4—732no—
316C4—732no—
317C4—1462no—
318C4#1798955112yes—
319C4#1798961112yes—
320C2#2262528711yescharacteristic
321C2#2262528811yescharacteristic
322C2#2262529311yescharacteristic
323C2#2262529411yescharacteristic
324C2#2262566111yescharacteristic
325C2#2262569511yescharacteristic
326C2#2262570311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#64408
commutatorC2×C4×C73C2×C42
FrattiniC22×C4C22×#64408
Fitting#391×C73C4
socle, prime radicalC23×C73C2×C22⋊C4
cosocleC2×C4×Dic73C23
hypercenter#391#64408
hypocenterC73#8178
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
12. Q8⋊#10596833C4semidirect
169. C73#8178semidirect

Maximal subgroups

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

subgroupindex
#226257172
#226256452
C2×Q8⋊#105968332
#226247672
#226124062
#817873