Finite Group Databasebeta

← #31893003

Subgroups — #31893003

Order 18176.

560 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22597369C211yesmaximal, characteristic
2#22586658C211yesmaximal, characteristic
3C2×#17985725C212yesmaximal, complemented
4#22586188C212yesmaximal, complemented
5C71×#2322C211yesmaximal, characteristic
6#17986071—24no—
7#17986056—21no—
8#17985963—21no—
9C2×C22⋊(C71⋊C8)C2212yes—
10C2×C22⋊(C71⋊C8)C2212yes—
11#17985725—24no—
12#17985725—21no—
13#17985723—21no—
14C2×#123×C71C411yescharacteristic
15C2×C4×D4×C71C2211yescharacteristic
16C22×C4⋊C4×C71C2211yescharacteristic
17#389×C71—21no—
18#388×C71—21no—
19#387×C71—21no—
20#386×C71C411yescharacteristic
21#385×C71—21no—
22#385×C71C411yescharacteristic
23#381×C71C411yescharacteristic
24C4×C22⋊C4×C71C2211yescharacteristic
25C2×#11765849—22no—
26C22×C71⋊C8—22no—
27C22×C71⋊C8—22no—
28C4⋊(C71⋊C8)—22no—
29C22⋊(C71⋊C8)—22no—
30C22⋊(C71⋊C8)—22no—
31C22⋊(C71⋊C8)—22no—
32C22⋊(C71⋊C8)—22no—
33C22⋊(C71⋊C8)D414yes—
34#14556039—22no—
35C22×D4×C71C2×C411yescharacteristic
36C23×C4×C71C2×C411yescharacteristic
37C23×C4×C71C2×C411yescharacteristic
38C23×C4×C71D411yescharacteristic
39C23×C4×C71C2311yescharacteristic
40#123×C71—22no—
41#123×C71—22no—
42C4×D4×C71—22no—
43C4×D4×C71—22no—
44C2×C4⋊C4×C71—21no—
45C2×C4⋊C4×C71—22no—
46C2×C4⋊C4×C71—21no—
47C2×C4⋊C4×C71—21no—
48C2×C4⋊C4×C71—22no—
49C2×C4⋊C4×C71—21no—
50C2×C4⋊C4×C71D411yescharacteristic
51C2×C22⋊C4×C71—21no—
52C2×C22⋊C4×C71—21no—
53C2×C22⋊C4×C71C2×C411yescharacteristic
54C2×C22⋊C4×C71C2×C411yescharacteristic
55C2×C22⋊C4×C71C2×C411yescharacteristic
56C2×C22⋊C4×C71D411yescharacteristic
57C2×C42×C71—21no—
58C2×C42×C71D411yescharacteristic
59#95×C71—21no—
60#95×C71—21no—
61#95×C71—21no—
62#95×C71—21no—
63#95×C71—21no—
64#11765849—82no—
65C2×C71⋊C8—24no—
66C2×C71⋊C8—42no—
67C2×C71⋊C8—82no—
68C2×C71⋊C8—82no—
69C24×C71C22⋊C411yescharacteristic
70C24×C71C22×C411yescharacteristic
71C2×D4×C71—22no—
72C2×D4×C71—21no—
73C2×D4×C71—21no—
74C2×D4×C71—22no—
75C22×C4×C71—21no—
76C22×C4×C71—21no—
77C22×C4×C71—22no—
78C22×C4×C71—21no—
79C22×C4×C71—22no—
80C22×C4×C71—21no—
81C22×C4×C71—21no—
82C22×C4×C71—21no—
83C22×C4×C71—21no—
84C22×C4×C71—22no—
85C22×C4×C71—81no—
86C22×C4×C71—81no—
87C22×C4×C71—81no—
88C22×C4×C71C22⋊C411yescharacteristic
89C22×C4×C71C22⋊C411yescharacteristic
90C22×C4×C71C22⋊C411yescharacteristic
91C22×C4×C71M1612yescomplemented
92C22×C4×C71C2×D412yes—
93C22×C4×C71C2×D411yescharacteristic
94C22×C4×C71C2×D411yescharacteristic
95C22×C4×C71#4112yes—
96C4⋊C4×C71—24no—
97C4⋊C4×C71—22no—
98C4⋊C4×C71—41no—
99C4⋊C4×C71—42no—
100C4⋊C4×C71—41no—
101C4⋊C4×C71—81no—
102C22⋊C4×C71—24no—
103C22⋊C4×C71—22no—
104C22⋊C4×C71—21no—
105C22⋊C4×C71—22no—
106C22⋊C4×C71—22no—
107C22⋊C4×C71—81no—
108C22⋊C4×C71M1612yescomplemented
109C42×C71—21no—
110C42×C71—21no—
111C42×C71—81no—
112C23×C71—21no—
113C23×C71—21no—
114C23×C71—21no—
115C23×C71—41no—
116C23×C71—41no—
117C23×C71—41no—
118C23×C71—41no—
119C23×C71C2×C22⋊C411yescharacteristic
120C23×C71C4×D412yes—
121C23×C71C2×M1611yescharacteristic
122C23×C71C2×M1611yescharacteristic
123C23×C71#13112yes—
124D4×C71—42no—
125D4×C71—42no—
126C2×C4×C71—21no—
127C2×C4×C71—21no—
128C2×C4×C71—21no—
129C2×C4×C71—22no—
130C2×C4×C71—22no—
131C2×C4×C71—21no—
132C2×C4×C71—21no—
133C2×C4×C71—24no—
134C2×C4×C71—41no—
135C2×C4×C71—41no—
136C2×C4×C71—42no—
137C2×C4×C71—42no—
138C2×C4×C71—42no—
139C2×C4×C71—41no—
140C2×C4×C71—41no—
141C2×C4×C71—41no—
142C2×C4×C71—41no—
143C2×C4×C71—42no—
144C2×C4×C71—41no—
145C2×C4×C71—41no—
146C2×C4×C71—41no—
147C2×C4×C71—41no—
148C2×C4×C71—41no—
149C2×C4×C71—41no—
150C2×C4×C71—41no—
151C2×C4×C71—41no—
152C2×C4×C71—81no—
153C2×C4×C71—82no—
154C2×C4×C71#12011yescharacteristic
155C2×C4×C71#12112yes—
156C2×C4×C71#12311yescharacteristic
157C71⋊C8—44no—
158C71⋊C8—82no—
159C22×C71—21no—
160C22×C71—22no—
161C22×C71—21no—
162C22×C71—22no—
163C22×C71—41no—
164C22×C71—41no—
165C22×C71—42no—
166C22×C71—42no—
167C22×C71—41no—
168C22×C71—41no—
169C22×C71—41no—
170C22×C71—81no—
171C22×C71#38611yescharacteristic
172C22×C71#40711yescharacteristic
173C22×C71#40811yescharacteristic
174C22×C71#43512yes—
175C22×C71#43612yes—
176C4×C71—21no—
177C4×C71—22no—
178C4×C71—21no—
179C4×C71—42no—
180C4×C71—42no—
181C4×C71—41no—
182C4×C71—41no—
183C4×C71—41no—
184C4×C71—81no—
185C4×C71—81no—
186#9110—711nomaximal, Sylow
187C2×C71—21no—
188C2×C71—21no—
189C2×C71—22no—
190C2×C71—22no—
191C2×C71—81no—
192C2×C71#187011yescharacteristic
193C2×C71#191311yescharacteristic
194C2×C71#191511yescharacteristic
195#2973—711no—
196#2322D7111yescharacteristic
197#1459—711no—
198C2×#324—712no—
199#1282—712no—
200C71#911011yescharacteristic, complemented, Sylow
201C2×#123Dic7111yescharacteristic
202C2×C4×D4C2×D7111yescharacteristic
203C22×C4⋊C4C2×D7111yescharacteristic
204#435—1424no—
205#432—1421no—
206#407—1421no—
207C2×C22⋊C8—712no—
208C2×C22⋊C8—712no—
209#389—21no—
210#388—21no—
211#387—21no—
212#386Dic7111yescharacteristic
213#385—21no—
214#385Dic7111yescharacteristic
215#381Dic7111yescharacteristic
216C4×C22⋊C4C2×D7111yescharacteristic
217#324—1424no—
218#324—1421no—
219#323—1421no—
220C22×D4C2×Dic7111yescharacteristic
221C23×C4C2×Dic7111yescharacteristic
222C23×C4C2×Dic7111yescharacteristic
223C23×C4C22⋊D7111yescharacteristic
224C23×C4C22×D7111yescharacteristic
225C2×M16—1422no—
226C22×C8—1422no—
227C22×C8—1422no—
228#123—22no—
229#123—22no—
230C4×D4—22no—
231C4×D4—22no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4C22⋊D7111yescharacteristic
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4C2×Dic7111yescharacteristic
242C2×C22⋊C4C2×Dic7111yescharacteristic
243C2×C22⋊C4C2×Dic7111yescharacteristic
244C2×C22⋊C4C22⋊D7111yescharacteristic
245C2×C42—21no—
246C2×C42C22⋊D7111yescharacteristic
247C4⋊C8—1422no—
248C22⋊C8—714no—
249C22⋊C8—1422no—
250C22⋊C8—1422no—
251C22⋊C8—1422no—
252C22⋊C8—1422no—
253#97—1422no—
254#95—21no—
255#95—21no—
256#95—21no—
257#95—21no—
258#95—21no—
259C24C22⋊Dic7111yescharacteristic
260C24C22×Dic7111yescharacteristic
261C2×D4—22no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—22no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—81no—
276C22×C4—81no—
277C22×C4—81no—
278C22×C4C22⋊Dic7111yescharacteristic
279C22×C4C22⋊Dic7111yescharacteristic
280C22×C4C22⋊Dic7111yescharacteristic
281C22×C4#1176584912yescomplemented
282C22×C4D4×D7112yes—
283C22×C4C2×C22⋊D7111yescharacteristic
284C22×C4C2×C22⋊D7111yescharacteristic
285C22×C4#1176586412yes—
286M16—5682no—
287C2×C8—2842no—
288C2×C8—5682no—
289C2×C8—5682no—
290C2×C8—1424no—
291C4⋊C4—24no—
292C4⋊C4—22no—
293C4⋊C4—41no—
294C4⋊C4—42no—
295C4⋊C4—41no—
296C4⋊C4—81no—
297C22⋊C4—24no—
298C22⋊C4—22no—
299C22⋊C4—21no—
300C22⋊C4—22no—
301C22⋊C4—22no—
302C22⋊C4—81no—
303C22⋊C4#1176584912yescomplemented
304C42—21no—
305C42—21no—
306C42—81no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23C2×C22⋊Dic7111yescharacteristic
315C23D4×Dic7112yes—
316C23C2×#1176584911yescharacteristic
317C23C2×#1176584911yescharacteristic
318C23#1455613512yes—
319D4—42no—
320D4—42no—
321C2×C4—24no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—81no—
348C2×C4—82no—
349C2×C4#1455609611yescharacteristic
350C2×C4#1455610012yes—
351C2×C4#1455610911yescharacteristic
352C8—2844no—
353C8—5682no—
354C22—21no—
355C22—22no—
356C22—21no—
357C22—22no—
358C22—41no—
359C22—41no—
360C22—42no—
361C22—42no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—81no—
366C22#1798587911yescharacteristic
367C22#1798596311yescharacteristic
368C22#1798596611yescharacteristic
369C22#1798607112yes—
370C22#1798607712yes—
371C4—21no—
372C4—22no—
373C4—21no—
374C4—42no—
375C4—41no—
376C4—41no—
377C4—42no—
378C4—41no—
379C4—81no—
380C4—81no—
381C2—21no—
382C2—21no—
383C2—22no—
384C2—22no—
385C2—81no—
386C2#2258863811yescharacteristic
387C2#2258888311yescharacteristic
388C2#2258889311yescharacteristic
subgroupG/H
centerC22#17985879
commutatorC24×C71C22×C4
FrattiniC23×C4C22×D71
FittingC71×#2322C2
socleC22×C71#386
prime radicalC22×D4×C71C2×C4
cosocleC23×C4×C71C23
hypercenter#2322D71
hypocenterC71#9110
LCSG ▷ C24×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2322

Complements

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

NH ≅ G/Ntype
3. C2×#17985725C2semidirect
4. #22586188C2semidirect
91. C22×C4×C71M16semidirect
108. C22⋊C4×C71M16semidirect
200. C71#9110semidirect
281. C22×C4#11765849semidirect
303. C22⋊C4#11765849semidirect

Maximal subgroups

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

subgroupindex
#225973692
#225866582
C2×#179857252
#225861882
C71×#23222
#911071