Finite Group Databasebeta

← #31893058

Subgroups — #31893058

Order 18176.

626 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556039C211yesmaximal, characteristic
2#22588102C211yesmaximal, characteristic
3C2×#17985741C212yesmaximal, complemented
4#22586221C212yesmaximal, complemented
5C71×#2851C211yesmaximal, characteristic
6C23×C71⋊C8—21no—
7C2×C22⋊(C71⋊C8)—21no—
8C2×#14556039—24no—
9C2×#14556039—21no—
10C2×#14556039C2212yes—
11C2×#14556039C2212yes—
12#17985754—21no—
13#17985741—24no—
14C2×#125×C71C2211yescharacteristic
15C22×C42×C71C2211yescharacteristic
16#389×C71C411yescharacteristic
17#389×C71C411yescharacteristic
18#387×C71—21no—
19#387×C71—21no—
20#385×C71—21no—
21#385×C71—21no—
22#382×C71C2211yescharacteristic
23#380×C71C411yescharacteristic
24#380×C71C411yescharacteristic
25C22×C71⋊C8—24no—
26C22×C71⋊C8—24no—
27C22×C71⋊C8—24no—
28C22×C71⋊C8—22no—
29C22×C71⋊C8—22no—
30C22⋊(C71⋊C8)—24no—
31#14556039—24no—
32#14556039—22no—
33#14556039—22no—
34#14556039—22no—
35#14556039D414yes—
36C23×C4×C71C2×C411yescharacteristic
37C23×C4×C71D411yescharacteristic
38C23×C4×C71D411yescharacteristic
39#125×C71—22no—
40#125×C71—22no—
41C2×C4⋊C4×C71—21no—
42C2×C4⋊C4×C71—21no—
43C2×C4⋊C4×C71—21no—
44C2×C4⋊C4×C71—21no—
45C2×C4⋊C4×C71C2×C411yescharacteristic
46C2×C4⋊C4×C71C2×C411yescharacteristic
47C2×C4⋊C4×C71C2×C411yescharacteristic
48C2×C4⋊C4×C71C2×C411yescharacteristic
49C2×C22⋊C4×C71—21no—
50C2×C22⋊C4×C71—21no—
51C2×C42×C71—22no—
52C2×C42×C71—22no—
53C2×C42×C71C2×C411yescharacteristic
54C2×C42×C71D411yescharacteristic
55C2×C42×C71D411yescharacteristic
56C2×C42×C71C2311yescharacteristic
57#95×C71—21no—
58#95×C71—21no—
59#95×C71—21no—
60#95×C71—21no—
61#95×C71—21no—
62#95×C71—21no—
63C2×C71⋊C8—216no—
64C2×C71⋊C8—24no—
65C2×C71⋊C8—24no—
66C2×C71⋊C8—24no—
67C2×C71⋊C8—24no—
68C2×C71⋊C8—42no—
69C2×C71⋊C8—42no—
70C24×C71C22⋊C411yescharacteristic
71C22×C4×C71—22no—
72C22×C4×C71—22no—
73C22×C4×C71—21no—
74C22×C4×C71—21no—
75C22×C4×C71—22no—
76C22×C4×C71—22no—
77C22×C4×C71—22no—
78C22×C4×C71—22no—
79C22×C4×C71—21no—
80C22×C4×C71—21no—
81C22×C4×C71—21no—
82C22×C4×C71—21no—
83C22×C4×C71—41no—
84C22×C4×C71—41no—
85C22×C4×C71C22⋊C411yescharacteristic
86C22×C4×C71C22⋊C411yescharacteristic
87C22×C4×C71C22⋊C411yescharacteristic
88C22×C4×C71C22×C411yescharacteristic
89C22×C4×C71C2×D411yescharacteristic
90C22×C4×C71C2×D411yescharacteristic
91C4⋊C4×C71—22no—
92C4⋊C4×C71—22no—
93C4⋊C4×C71—41no—
94C4⋊C4×C71—41no—
95C4⋊C4×C71—42no—
96C4⋊C4×C71—42no—
97C4⋊C4×C71—41no—
98C4⋊C4×C71—42no—
99C4⋊C4×C71—41no—
100C22⋊C4×C71—42no—
101C22⋊C4×C71—42no—
102C42×C71—22no—
103C42×C71—21no—
104C42×C71—22no—
105C42×C71—21no—
106C42×C71C2×D412yes—
107C42×C71#4112yes—
108C23×C71—21no—
109C23×C71—21no—
110C23×C71—21no—
111C23×C71—21no—
112C23×C71#10112yes—
113C23×C71#10412yes—
114C23×C71#10412yes—
115C23×C71C2×C22⋊C411yescharacteristic
116C2×C4×C71—24no—
117C2×C4×C71—24no—
118C2×C4×C71—21no—
119C2×C4×C71—21no—
120C2×C4×C71—21no—
121C2×C4×C71—21no—
122C2×C4×C71—21no—
123C2×C4×C71—21no—
124C2×C4×C71—21no—
125C2×C4×C71—21no—
126C2×C4×C71—21no—
127C2×C4×C71—21no—
128C2×C4×C71—21no—
129C2×C4×C71—21no—
130C2×C4×C71—21no—
131C2×C4×C71—21no—
132C2×C4×C71—44no—
133C2×C4×C71—44no—
134C2×C4×C71—41no—
135C2×C4×C71—41no—
136C2×C4×C71—41no—
137C2×C4×C71—41no—
138C2×C4×C71—41no—
139C2×C4×C71—41no—
140C2×C4×C71—41no—
141C2×C4×C71—41no—
142C2×C4×C71—41no—
143C2×C4×C71—41no—
144C2×C4×C71—41no—
145C2×C4×C71—41no—
146C2×C4×C71#10412yes—
147C2×C4×C71C4×D412yes—
148C2×C4×C71#12011yescharacteristic
149C2×C4×C71#12112yes—
150C2×C4×C71#12311yescharacteristic
151C71⋊C8—28no—
152C71⋊C8—44no—
153C22×C71—22no—
154C22×C71—22no—
155C22×C71—22no—
156C22×C71—22no—
157C22×C71—22no—
158C22×C71—22no—
159C22×C71#33014yescomplemented
160C22×C71#38611yescharacteristic
161C22×C71C2×#10111yescharacteristic
162C22×C71#41311yescharacteristic
163C22×C71C2×#10411yescharacteristic
164C22×C71C2×#10411yescharacteristic
165C22×C71C2×#10411yescharacteristic
166C22×C71#42111yescharacteristic
167C4×C71—22no—
168C4×C71—21no—
169C4×C71—22no—
170C4×C71—21no—
171C4×C71—22no—
172C4×C71—41no—
173C4×C71—41no—
174C4×C71—81no—
175C4×C71—81no—
176C4×C71—81no—
177C4×C71—81no—
178#9121—711nomaximal, Sylow
179C2×C71—24no—
180C2×C71C2×#33011yescharacteristic
181C2×C71#152411yescharacteristic
182C2×C71#153111yescharacteristic
183C2×C71#153211yescharacteristic
184C2×C71#187111yescharacteristic
185C2×C71#189611yescharacteristic
186C2×C71#190111yescharacteristic
187C22×#97—711no—
188#2851D7111yescharacteristic
189#1749—711no—
190C2×#330—712no—
191#1295—712no—
192C71#912111yescharacteristic, complemented, Sylow
193C23×C8—1421no—
194C2×#125C2×D7111yescharacteristic
195C22×C42C2×D7111yescharacteristic
196C2×C22⋊C8—1421no—
197C2×#97—712no—
198C2×#97—712no—
199C2×#97—1424no—
200C2×#97—1421no—
201#389Dic7111yescharacteristic
202#389Dic7111yescharacteristic
203#387—21no—
204#387—21no—
205#385—21no—
206#385—21no—
207#382C2×D7111yescharacteristic
208#380Dic7111yescharacteristic
209#380Dic7111yescharacteristic
210#336—1421no—
211#330—1424no—
212C23×C4C2×Dic7111yescharacteristic
213C23×C4C22⋊D7111yescharacteristic
214C23×C4C22⋊D7111yescharacteristic
215C22×C8—1422no—
216C22×C8—1424no—
217C22×C8—1424no—
218C22×C8—1424no—
219C22×C8—1422no—
220#125—22no—
221#125—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4C2×Dic7111yescharacteristic
227C2×C4⋊C4C2×Dic7111yescharacteristic
228C2×C4⋊C4C2×Dic7111yescharacteristic
229C2×C4⋊C4C2×Dic7111yescharacteristic
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42C2×Dic7111yescharacteristic
235C2×C42C22⋊D7111yescharacteristic
236C2×C42C22⋊D7111yescharacteristic
237C2×C42C22×D7111yescharacteristic
238C22⋊C8—1424no—
239#97—714no—
240#97—1424no—
241#97—1422no—
242#97—1422no—
243#97—1422no—
244#95—21no—
245#95—21no—
246#95—21no—
247#95—21no—
248#95—21no—
249#95—21no—
250C24C22⋊Dic7111yescharacteristic
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4C22⋊Dic7111yescharacteristic
266C22×C4C22⋊Dic7111yescharacteristic
267C22×C4C22⋊Dic7111yescharacteristic
268C22×C4C22×Dic7111yescharacteristic
269C22×C4C2×C22⋊D7111yescharacteristic
270C22×C4C2×C22⋊D7111yescharacteristic
271C2×C8—2842no—
272C2×C8—2842no—
273C2×C8—1424no—
274C2×C8—14216no—
275C2×C8—1424no—
276C2×C8—1424no—
277C2×C8—1424no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—41no—
285C4⋊C4—42no—
286C4⋊C4—41no—
287C22⋊C4—42no—
288C22⋊C4—42no—
289C42—22no—
290C42—21no—
291C42—22no—
292C42—21no—
293C42D4×D7112yes—
294C42#1176586412yes—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23#1455604712yes—
300C23#1455605512yes—
301C23#1455605512yes—
302C23C2×C22⋊Dic7111yescharacteristic
303C2×C4—24no—
304C2×C4—24no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4#1455605512yes—
334C2×C4D4×Dic7112yes—
335C2×C4#1455609611yescharacteristic
336C2×C4#1455610012yes—
337C2×C4#1455610911yescharacteristic
338C8—2844no—
339C8—1428no—
340C22—22no—
341C22—22no—
342C22—22no—
343C22—22no—
344C22—22no—
345C22—22no—
346C22#1798574114yescomplemented
347C22#1798587911yescharacteristic
348C22C2×#1455604711yescharacteristic
349C22#1798598311yescharacteristic
350C22C2×#1455605511yescharacteristic
351C22C2×#1455605511yescharacteristic
352C22C2×#1455605511yescharacteristic
353C22#1798601611yescharacteristic
354C4—21no—
355C4—22no—
356C4—21no—
357C4—22no—
358C4—22no—
359C4—41no—
360C4—41no—
361C4—81no—
362C4—81no—
363C4—81no—
364C4—81no—
365C2—24no—
366C2C2×#1798574111yescharacteristic
367C2#2258692811yescharacteristic
368C2#2258695511yescharacteristic
369C2#2258696011yescharacteristic
370C2#2258864311yescharacteristic
371C2#2258878411yescharacteristic
372C2#2258880911yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic71
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#2851C2
socleC23×C71C2×C22⋊C4
prime radicalC24×C71C22⋊C4
cosocleC2×C42×C71C23
hypercenter#2851D71
hypocenterC71#9121
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2851

Complements

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

NH ≅ G/Ntype
3. C2×#17985741C2semidirect
4. #22586221C2semidirect
159. C22×C71#330semidirect
192. C71#9121semidirect
346. C22#17985741semidirect

Maximal subgroups

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

subgroupindex
C22×#145560392
#225881022
C2×#179857412
#225862212
C71×#28512
#912171