Finite Group Databasebeta

← #31997955

Subgroups — #31997955

Order 18176.

854 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986060C212yesmaximal
2#22588492C212yesmaximal, complemented
3#22588460C212yesmaximal, complemented
4#22588324C212yesmaximal
5#22588320C212yesmaximal
6#22588197C212yesmaximal, complemented
7#22588193C211yesmaximal, characteristic, complemented
8C4×C4⋊(C71⋊C8)C211yesmaximal, characteristic, complemented
9C4×#117×C71C211yesmaximal, characteristic
10#17986060—28no—
11C2×C4⋊(C71⋊C8)C2212yes—
12C2×C4⋊(C71⋊C8)C2212yes—
13C2×C4⋊(C71⋊C8)C2214yes—
14C2×C4⋊(C71⋊C8)C2214yes—
15C2×C22⋊(C71⋊C8)C2214yes—
16C2×#14556039C2212yes—
17C2×C4×C71⋊C8C2212yes—
18#17985754C2212yes—
19#17985754C2212yes—
20#17985754C2214yes—
21C2×#117×C71C2211yescharacteristic
22C2×#117×C71C2211yescharacteristic
23C22×C42×C71C2211yescharacteristic
24C4×C4⋊C4×C71C412yes—
25C4×C4⋊C4×C71C412yes—
26C4×C22⋊C4×C71C412yes—
27C4×C22⋊C4×C71C412yes—
28#376×C71C2211yescharacteristic
29#376×C71C2211yescharacteristic
30C43×C71C2211yescharacteristic
31C43×C71C2211yescharacteristic
32C22×C71⋊C8C2314yes—
33C22×C71⋊C8C2314yes—
34C4⋊(C71⋊C8)—28no—
35C4⋊(C71⋊C8)—24no—
36C4⋊(C71⋊C8)—24no—
37C4⋊(C71⋊C8)—28no—
38C22⋊(C71⋊C8)—28no—
39#14556039—24no—
40C4×C71⋊C8—24no—
41C23×C4×C71C2×C411yescharacteristic
42C23×C4×C71C2×C411yescharacteristic
43C23×C4×C71C2311yescharacteristic
44#117×C71—24no—
45#117×C71D414yes—
46#117×C71Q814yes—
47C2×C4⋊C4×C71C2×C412yes—
48C2×C4⋊C4×C71C2×C412yes—
49C2×C22⋊C4×C71C2×C412yes—
50C2×C22⋊C4×C71C2×C412yes—
51C2×C42×C71—24no—
52C2×C42×C71C2×C411yescharacteristic
53C2×C42×C71C2×C411yescharacteristic
54C2×C42×C71C2×C412yes—
55C2×C42×C71C2×C412yes—
56C2×C42×C71C2×C412yes—
57C2×C42×C71C2×C412yes—
58C2×C42×C71C2311yescharacteristic
59C2×C42×C71C2311yescharacteristic
60C2×C42×C71C2311yescharacteristic
61C2×C42×C71C2311yescharacteristic
62C2×C42×C71C2311yescharacteristic
63C2×C42×C71C2311yescharacteristic
64#95×C71C2×C412yes—
65#95×C71C2×C412yes—
66#95×C71C2×C412yes—
67#95×C71C2×C412yes—
68C2×C71⋊C8—24no—
69C2×C71⋊C8—24no—
70C2×C71⋊C8—24no—
71C2×C71⋊C8—24no—
72C2×C71⋊C8—24no—
73C2×C71⋊C8—24no—
74C24×C71C22×C411yescharacteristic
75C22×C4×C71—22no—
76C22×C4×C71—22no—
77C22×C4×C71—22no—
78C22×C4×C71—22no—
79C22×C4×C71—24no—
80C22×C4×C71—42no—
81C22×C4×C71C22×C411yescharacteristic
82C22×C4×C71C22×C411yescharacteristic
83C22×C4×C71C22×C412yes—
84C22×C4×C71C22×C412yes—
85C22×C4×C71C22×C412yes—
86C22×C4×C71C22×C411yescharacteristic
87C22×C4×C71C22×C412yes—
88C22×C4×C71C22×C411yescharacteristic
89C22×C4×C71C22×C411yescharacteristic
90C22×C4×C71C2×D412yes—
91C22×C4×C71C2×Q812yes—
92C22×C4×C71C2411yescharacteristic
93C4⋊C4×C71—24no—
94C4⋊C4×C71C4⋊C418yes—
95C22⋊C4×C71—24no—
96C22⋊C4×C71C4⋊C418yes—
97C42×C71—22no—
98C42×C71—22no—
99C42×C71—24no—
100C42×C71—24no—
101C42×C71—22no—
102C42×C71—22no—
103C42×C71—42no—
104C42×C71—42no—
105C42×C71C2×D412yes—
106C42×C71C2×D412yes—
107C42×C71C2×Q812yes—
108C42×C71C2×Q812yes—
109C42×C71#4114yes—
110C42×C71#4114yes—
111C23×C71—21no—
112C23×C71—21no—
113C23×C71—22no—
114C23×C71—22no—
115C23×C71C2×C4⋊C412yes—
116C23×C71C23×C411yescharacteristic
117C2×C4×C71—21no—
118C2×C4×C71—21no—
119C2×C4×C71—22no—
120C2×C4×C71—22no—
121C2×C4×C71—22no—
122C2×C4×C71—22no—
123C2×C4×C71—22no—
124C2×C4×C71—22no—
125C2×C4×C71—22no—
126C2×C4×C71—24no—
127C2×C4×C71—24no—
128C2×C4×C71—21no—
129C2×C4×C71—21no—
130C2×C4×C71—22no—
131C2×C4×C71—22no—
132C2×C4×C71—22no—
133C2×C4×C71—22no—
134C2×C4×C71—22no—
135C2×C4×C71—44no—
136C2×C4×C71—44no—
137C2×C4×C71C2×C4⋊C414yes—
138C2×C4×C71C2×C4⋊C414yes—
139C2×C4×C71C2×C4⋊C412yes—
140C2×C4×C71#11712yes—
141C2×C4×C71#11714yes—
142C2×C4×C71#11712yes—
143C2×C4×C71#13114yes—
144C2×C4×C71C22×D411yescharacteristic
145C2×C4×C71C22×Q811yescharacteristic
146C2×C4×C71C2×#4112yes—
147C2×C4×C71C2×#4112yes—
148C71⋊C8—44no—
149C71⋊C8—44no—
150C22×C71—24no—
151C22×C71—22no—
152C22×C71—41no—
153C22×C71—41no—
154C22×C71—42no—
155C22×C71C22×C4⋊C411yescharacteristic
156C22×C71C2×#11711yescharacteristic
157C22×C71C2×#11711yescharacteristic
158C22×C71C2×#13112yes—
159C22×C71#56812yes—
160C4×C71—24no—
161C4×C71—24no—
162C4×C71—22no—
163C4×C71—24no—
164C4×C71—22no—
165C4×C71—41no—
166C4×C71—41no—
167C4×C71—42no—
168C4×C71#43314yescomplemented
169C4×C71#53212yes—
170C4×C71#53312yes—
171#20812—711nomaximal, Sylow
172C2×C71—42no—
173C2×C71#228511yescharacteristic
174C2×C71#290612yes—
175C2×C71C2×#43311yescharacteristic
176C2×C71#292111yescharacteristic
177C2×C71#292512yes—
178C2×#433—712no—
179C4×#117D7111yescharacteristic
180#1842—712no—
181#1832—712no—
182#1798—712no—
183#1797—712no—
184#1768—712no—
185#1767—711no—
186C4×C4⋊C8—711no—
187C71#2081211yescharacteristic, complemented, Sylow
188C2×#117C2×D7111yescharacteristic
189C2×#117C2×D7111yescharacteristic
190C22×C42C2×D7111yescharacteristic
191#433—1428no—
192C2×C4⋊C8—712no—
193C2×C4⋊C8—712no—
194C2×C4⋊C8—714no—
195C2×C4⋊C8—714no—
196C2×C22⋊C8—714no—
197C2×#97—712no—
198C2×C4×C8—712no—
199C4×C4⋊C4Dic7112yes—
200C4×C4⋊C4Dic7112yes—
201C4×C22⋊C4Dic7112yes—
202C4×C22⋊C4Dic7112yes—
203#376C2×D7111yescharacteristic
204#376C2×D7111yescharacteristic
205C43C2×D7111yescharacteristic
206C43C2×D7111yescharacteristic
207#336—714no—
208#336—712no—
209#336—712no—
210C23×C4C2×Dic7111yescharacteristic
211C23×C4C2×Dic7111yescharacteristic
212C23×C4C22×D7111yescharacteristic
213C22×C8—714no—
214C22×C8—714no—
215#117—24no—
216#117Dic14214yes—
217#117D28414yes—
218C2×C4⋊C4C2×Dic7112yes—
219C2×C4⋊C4C2×Dic7112yes—
220C2×C22⋊C4C2×Dic7112yes—
221C2×C22⋊C4C2×Dic7112yes—
222C2×C42—24no—
223C2×C42C2×Dic7112yes—
224C2×C42C2×Dic7112yes—
225C2×C42C2×Dic7111yescharacteristic
226C2×C42C2×Dic7111yescharacteristic
227C2×C42C2×Dic7112yes—
228C2×C42C2×Dic7112yes—
229C2×C42C22×D7111yescharacteristic
230C2×C42C22×D7111yescharacteristic
231C2×C42C22×D7111yescharacteristic
232C2×C42C22×D7111yescharacteristic
233C2×C42C22×D7111yescharacteristic
234C2×C42C22×D7111yescharacteristic
235C4⋊C8—1428no—
236C4⋊C8—1424no—
237C4⋊C8—1424no—
238C4⋊C8—1428no—
239C22⋊C8—1428no—
240#97—1424no—
241C4×C8—1424no—
242#95C2×Dic7112yes—
243#95C2×Dic7112yes—
244#95C2×Dic7112yes—
245#95C2×Dic7112yes—
246C24C22×Dic7111yescharacteristic
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—24no—
252C22×C4—42no—
253C22×C4C22×Dic7111yescharacteristic
254C22×C4C22×Dic7111yescharacteristic
255C22×C4C22×Dic7112yes—
256C22×C4C22×Dic7112yes—
257C22×C4C22×Dic7112yes—
258C22×C4C22×Dic7111yescharacteristic
259C22×C4C22×Dic7112yes—
260C22×C4C22×Dic7111yescharacteristic
261C22×C4C22×Dic7111yescharacteristic
262C22×C4C2×D28412yes—
263C22×C4C2×Dic14212yes—
264C22×C4C23×D7111yescharacteristic
265C2×C8—1424no—
266C2×C8—1424no—
267C2×C8—1424no—
268C2×C8—1424no—
269C2×C8—1424no—
270C2×C8—1424no—
271C4⋊C4—24no—
272C4⋊C4C4⋊Dic7118yes—
273C22⋊C4—24no—
274C22⋊C4C4⋊Dic7118yes—
275C42—22no—
276C42—22no—
277C42—22no—
278C42—22no—
279C42—24no—
280C42—24no—
281C42—42no—
282C42—42no—
283C42C2×D28412yes—
284C42C2×D28412yes—
285C42C2×Dic14212yes—
286C42C2×Dic14212yes—
287C42#1176586514yes—
288C42#1176586514yes—
289C23—21no—
290C23—21no—
291C23—22no—
292C23—22no—
293C23C2×C4⋊Dic7112yes—
294C23C23×Dic7111yescharacteristic
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—24no—
305C2×C4—24no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4C2×C4⋊Dic7114yes—
316C2×C4C2×C4⋊Dic7114yes—
317C2×C4C2×C4⋊Dic7112yes—
318C2×C4#1455608712yes—
319C2×C4#1455608714yes—
320C2×C4#1455608712yes—
321C2×C4#1455613514yes—
322C2×C4C22×D28411yescharacteristic
323C2×C4C22×Dic14211yescharacteristic
324C2×C4C2×#1176586512yes—
325C2×C4C2×#1176586512yes—
326C8—2844no—
327C8—2844no—
328C22—22no—
329C22—24no—
330C22—41no—
331C22—41no—
332C22—42no—
333C22C22×C4⋊Dic7111yescharacteristic
334C22C2×#1455608711yescharacteristic
335C22C2×#1455608711yescharacteristic
336C22C2×#1455613512yes—
337C22#1798676612yes—
338C4—22no—
339C4—24no—
340C4—24no—
341C4—22no—
342C4—24no—
343C4—41no—
344C4—41no—
345C4—42no—
346C4#1798606014yescomplemented
347C4#1798654912yes—
348C4#1798655612yes—
349C2—42no—
350C2#2259074711yescharacteristic
351C2#2259690812yes—
352C2C2×#1798606011yescharacteristic
353C2#2259697811yescharacteristic
354C2#2259700612yes—
subgroupG/H
center, FrattiniC22×C4C23×D71
commutator, socleC23×C71C23×C4
FittingC4×#117×C71C2
prime radicalC24×C71C22×C4
cosocleC22×C4×C71C24
hypercenterC4×#117D71
hypocenterC71#20812
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C22×C4 ◁ C4×#117

Complements

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

NH ≅ G/Ntype
2. #22588492C2semidirect
3. #22588460C2semidirect
6. #22588197C2semidirect
7. #22588193C2semidirect
8. C4×C4⋊(C71⋊C8)C2semidirect
168. C4×C71#433semidirect
187. C71#20812semidirect
346. C4#17986060semidirect

Maximal subgroups

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

subgroupindex
C2×#179860602
#225884922
#225884602
#225883242
#225883202
#225881972
#225881932
C4×C4⋊(C71⋊C8)2
C4×#117×C712
#2081271