Finite Group Databasebeta

← #31906869

Subgroups — #31906869

Order 18176.

750 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 750 conjugacy classes of subgroups — too many to draw. Only its 107 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#22586667C211yesmaximal, characteristic
2#22586667C211yesmaximal, characteristic
3#22586288C214yesmaximal, complemented
4C2×#385×C71C211yesmaximal, characteristic
5#17985963—21no—
6#17985963—21no—
7#17985723—21no—
8#17985723—21no—
9#17985723C2212yes—
10#17985723C2212yes—
11C24×C4×C71C411yescharacteristic
12C22×C4⋊C4×C71—21no—
13C22×C22⋊C4×C71C411yescharacteristic
14C22×C22⋊C4×C71C2211yescharacteristic
15#385×C71—21no—
16#385×C71—21no—
17#385×C71C412yes—
18#385×C71C2212yes—
19C2×#95×C71—21no—
20C2×#11765849—41no—
21C2×#11765849—41no—
22C22⋊(C71⋊C8)—22no—
23C22⋊(C71⋊C8)—22no—
24C22⋊(C71⋊C8)—41no—
25C22⋊(C71⋊C8)—41no—
26C25×C71C2×C411yescharacteristic
27C23×C4×C71—21no—
28C23×C4×C71—21no—
29C23×C4×C71—21no—
30C23×C4×C71—44no—
31C23×C4×C71—41no—
32C23×C4×C71—41no—
33C23×C4×C71C2×C412yes—
34C23×C4×C71D411yescharacteristic
35C23×C4×C71D411yescharacteristic
36C2×C4⋊C4×C71—22no—
37C2×C4⋊C4×C71—22no—
38C2×C4⋊C4×C71—44no—
39C2×C22⋊C4×C71—21no—
40C2×C22⋊C4×C71—24no—
41C2×C22⋊C4×C71—41no—
42C2×C22⋊C4×C71—41no—
43C2×C22⋊C4×C71C2×C412yes—
44C2×C22⋊C4×C71C2×C411yescharacteristic
45C2×C22⋊C4×C71D411yescharacteristic
46C2×C22⋊C4×C71D411yescharacteristic
47C2×C22⋊C4×C71C2311yescharacteristic
48#95×C71—22no—
49#95×C71—22no—
50#11765849—82no—
51#11765849—82no—
52C2×C71⋊C8—42no—
53C2×C71⋊C8—42no—
54C24×C71—21no—
55C24×C71—21no—
56C24×C71—22no—
57C24×C71—22no—
58C24×C71—42no—
59C24×C71—42no—
60C24×C71C22⋊C411yescharacteristic
61C24×C71C22⋊C411yescharacteristic
62C24×C71C22×C411yescharacteristic
63C22×C4×C71—24no—
64C22×C4×C71—24no—
65C22×C4×C71—21no—
66C22×C4×C71—21no—
67C22×C4×C71—22no—
68C22×C4×C71—22no—
69C22×C4×C71—21no—
70C22×C4×C71—21no—
71C22×C4×C71—44no—
72C22×C4×C71—44no—
73C22×C4×C71—42no—
74C22×C4×C71—42no—
75C22×C4×C71—42no—
76C22×C4×C71—42no—
77C22×C4×C71—48no—
78C22×C4×C71—44no—
79C22×C4×C71—41no—
80C22×C4×C71—41no—
81C22×C4×C71—42no—
82C22×C4×C71—44no—
83C22×C4×C71—42no—
84C22×C4×C71—42no—
85C22×C4×C71C22⋊C411yescharacteristic
86C22×C4×C71C22⋊C411yescharacteristic
87C22×C4×C71C2×D411yescharacteristic
88C22×C4×C71C2×D411yescharacteristic
89C4⋊C4×C71—44no—
90C4⋊C4×C71—44no—
91C22⋊C4×C71—24no—
92C22⋊C4×C71—22no—
93C22⋊C4×C71—41no—
94C22⋊C4×C71—41no—
95C22⋊C4×C71—41no—
96C22⋊C4×C71—41no—
97C22⋊C4×C71M1614yescomplemented
98C23×C71—22no—
99C23×C71—22no—
100C23×C71—21no—
101C23×C71—21no—
102C23×C71—24no—
103C23×C71—24no—
104C23×C71—21no—
105C23×C71—21no—
106C23×C71—21no—
107C23×C71—21no—
108C23×C71—41no—
109C23×C71—41no—
110C23×C71—41no—
111C23×C71—41no—
112C23×C71—42no—
113C23×C71—48no—
114C23×C71—48no—
115C23×C71—42no—
116C23×C71—42no—
117C23×C71—42no—
118C23×C71#9911yescharacteristic
119C23×C71#9911yescharacteristic
120C23×C71#10112yes—
121C23×C71C2×C22⋊C411yescharacteristic
122C23×C71C2×M1612yes—
123C2×C4×C71—24no—
124C2×C4×C71—24no—
125C2×C4×C71—21no—
126C2×C4×C71—21no—
127C2×C4×C71—21no—
128C2×C4×C71—21no—
129C2×C4×C71—42no—
130C2×C4×C71—48no—
131C2×C4×C71—42no—
132C2×C4×C71—42no—
133C2×C4×C71—44no—
134C2×C4×C71—48no—
135C2×C4×C71—44no—
136C2×C4×C71—44no—
137C2×C4×C71—42no—
138C2×C4×C71—42no—
139C2×C4×C71—42no—
140C2×C4×C71—41no—
141C2×C4×C71—41no—
142C2×C4×C71—41no—
143C2×C4×C71—41no—
144C2×C4×C71—44no—
145C2×C4×C71—81no—
146C2×C4×C71—81no—
147C71⋊C8—82no—
148C71⋊C8—82no—
149C22×C71—24no—
150C22×C71—24no—
151C22×C71—21no—
152C22×C71—21no—
153C22×C71—21no—
154C22×C71—21no—
155C22×C71—21no—
156C22×C71—21no—
157C22×C71—22no—
158C22×C71—22no—
159C22×C71—42no—
160C22×C71—42no—
161C22×C71—42no—
162C22×C71—48no—
163C22×C71—48no—
164C22×C71—42no—
165C22×C71—41no—
166C22×C71—41no—
167C22×C71—41no—
168C22×C71—41no—
169C22×C71AΣL(1,𝔽16)12yes—
170C22×C71#35212yes—
171C22×C71#40711yescharacteristic
172C22×C71C2×#9911yescharacteristic
173C22×C71C2×#10111yescharacteristic
174C4×C71—44no—
175C4×C71—41no—
176C4×C71—44no—
177C4×C71—41no—
178C4×C71—41no—
179C4×C71—41no—
180#11517—711nomaximal, Sylow
181C2×C71—21no—
182C2×C71—22no—
183C2×C71—22no—
184C2×C71—21no—
185C2×C71—42no—
186C2×C71—42no—
187C2×C71#146311yescharacteristic
188C2×C71C2×AΣL(1,𝔽16)11yescharacteristic
189C2×C71C2×#35211yescharacteristic
190C2×#385D7111yescharacteristic
191#1462—711no—
192#1462—711no—
193#1326—714no—
194C71#1151711yescharacteristic, complemented, Sylow
195C24×C4Dic7111yescharacteristic
196C22×C4⋊C4—21no—
197C22×C22⋊C4Dic7111yescharacteristic
198C22×C22⋊C4C2×D7111yescharacteristic
199#407—1421no—
200#407—1421no—
201#385—21no—
202#385—21no—
203#385Dic7112yes—
204#385C2×D7112yes—
205C2×#95—21no—
206#323—712no—
207#323—712no—
208#323—1421no—
209#323—1421no—
210C25C2×Dic7111yescharacteristic
211C23×C4—21no—
212C23×C4—21no—
213C23×C4—21no—
214C23×C4—44no—
215C23×C4—41no—
216C23×C4—41no—
217C23×C4C2×Dic7112yes—
218C23×C4C22⋊D7111yescharacteristic
219C23×C4C22⋊D7111yescharacteristic
220C2×M16—2841no—
221C2×M16—2841no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—44no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—41no—
228C2×C22⋊C4—41no—
229C2×C22⋊C4C2×Dic7112yes—
230C2×C22⋊C4C2×Dic7111yescharacteristic
231C2×C22⋊C4C22⋊D7111yescharacteristic
232C2×C22⋊C4C22⋊D7111yescharacteristic
233C2×C22⋊C4C22×D7111yescharacteristic
234C22⋊C8—2841no—
235C22⋊C8—2841no—
236C22⋊C8—1422no—
237C22⋊C8—1422no—
238#95—22no—
239#95—22no—
240C24—21no—
241C24—21no—
242C24—22no—
243C24—22no—
244C24—42no—
245C24—42no—
246C24C22⋊Dic7111yescharacteristic
247C24C22⋊Dic7111yescharacteristic
248C24C22×Dic7111yescharacteristic
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—44no—
258C22×C4—44no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—48no—
264C22×C4—44no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—42no—
268C22×C4—44no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4C22⋊Dic7111yescharacteristic
272C22×C4C22⋊Dic7111yescharacteristic
273C22×C4C2×C22⋊D7111yescharacteristic
274C22×C4C2×C22⋊D7111yescharacteristic
275M16—5682no—
276M16—5682no—
277C2×C8—2842no—
278C2×C8—2842no—
279C4⋊C4—44no—
280C4⋊C4—44no—
281C22⋊C4—24no—
282C22⋊C4—22no—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4#1176584914yescomplemented
288C23—21no—
289C23—21no—
290C23—22no—
291C23—22no—
292C23—24no—
293C23—24no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—42no—
303C23—48no—
304C23—48no—
305C23—42no—
306C23—42no—
307C23—42no—
308C23#1455604311yescharacteristic
309C23#1455604311yescharacteristic
310C23#1455604712yes—
311C23C2×C22⋊Dic7111yescharacteristic
312C23C2×#1176584912yes—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—44no—
322C2×C4—48no—
323C2×C4—42no—
324C2×C4—48no—
325C2×C4—44no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—44no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—44no—
335C2×C4—81no—
336C2×C4—81no—
337C8—5682no—
338C8—5682no—
339C22—24no—
340C22—24no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—22no—
348C22—22no—
349C22—42no—
350C22—42no—
351C22—42no—
352C22—42no—
353C22—48no—
354C22—48no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22#1798578612yes—
360C22#1798578912yes—
361C22#1798596311yescharacteristic
362C22C2×#1455604311yescharacteristic
363C22C2×#1455604711yescharacteristic
364C4—44no—
365C4—41no—
366C4—44no—
367C4—41no—
368C4—41no—
369C4—41no—
370C2—21no—
371C2—21no—
372C2—22no—
373C2—22no—
374C2—42no—
375C2—42no—
376C2#2258667011yescharacteristic
377C2C2×#1798578611yescharacteristic
378C2C2×#1798578911yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC24×C71C22×C4
FrattiniC2×C22⋊C4C22×D71
FittingC2×#385×C71C2
socleC22×C71C2×#99
prime radicalC25×C71C2×C4
cosocleC2×C22⋊C4×C71C23
hypercenterC2×#385D71
hypocenterC71#11517
LCSG ▷ C24×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ C2×#385

Complements

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

NH ≅ G/Ntype
3. #22586288C2semidirect
97. C22⋊C4×C71M16semidirect
194. C71#11517semidirect
287. C22⋊C4#11765849semidirect

Maximal subgroups

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

subgroupindex
#225866672
#225862882
C2×#385×C712
#1151771