Finite Group Databasebeta

← #31907983

Subgroups — #31907983

Order 18176.

462 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C2×#17985768C211yesmaximal, characteristic
2#22586670C211yesmaximal, characteristic
3#22586370C212yesmaximal, complemented
4#22586346C212yesmaximal, complemented
5C71×#2583C211yesmaximal, characteristic
6C22×#11765849—21no—
7#17985963—21no—
8#17985768—21no—
9#17985768C2212yes—
10#17985725—21no—
11#17985725C2212yes—
12C2×C22⋊Q8×C71C411yescharacteristic
13C22×C4⋊C4×C71C2211yescharacteristic
14#398×C71C411yescharacteristic
15#398×C71C2211yescharacteristic
16#395×C71C411yescharacteristic
17#395×C71C2211yescharacteristic
18#389×C71—21no—
19#389×C71—21no—
20#385×C71—21no—
21#385×C71—21no—
22C4×C22⋊C4×C71C411yescharacteristic
23C2×#11765849—24no—
24C2×#11765849—22no—
25C2×#11765849—41no—
26C2×#11765849—42no—
27C2×#11765849—41no—
28C22×C71⋊C8—41no—
29C22⋊(C71⋊C8)—22no—
30C22⋊(C71⋊C8)—41no—
31C22×Q8×C71C2×C411yescharacteristic
32C23×C4×C71C2×C411yescharacteristic
33C23×C4×C71D411yescharacteristic
34C23×C4×C71D411yescharacteristic
35C22⋊Q8×C71—41no—
36C22⋊Q8×C71—41no—
37C2×C4⋊C4×C71—21no—
38C2×C4⋊C4×C71—21no—
39C2×C4⋊C4×C71—21no—
40C2×C4⋊C4×C71—21no—
41C2×C4⋊C4×C71—21no—
42C2×C4⋊C4×C71—41no—
43C2×C4⋊C4×C71—41no—
44C2×C4⋊C4×C71C2×C411yescharacteristic
45C2×C4⋊C4×C71C2×C411yescharacteristic
46C2×C4⋊C4×C71D411yescharacteristic
47C2×C4⋊C4×C71D411yescharacteristic
48C2×C4⋊C4×C71C2311yescharacteristic
49C2×C22⋊C4×C71—21no—
50C2×C22⋊C4×C71—21no—
51C2×C42×C71—21no—
52#95×C71—21no—
53#95×C71—21no—
54#95×C71—21no—
55#95×C71—21no—
56#95×C71C2×C411yescharacteristic
57#95×C71C2×C411yescharacteristic
58#11765849—24no—
59#11765849—44no—
60#11765849—42no—
61#11765849—82no—
62C2×C71⋊C8—44no—
63C2×C71⋊C8—42no—
64C2×C71⋊C8—42no—
65C24×C71C22⋊C411yescharacteristic
66C2×Q8×C71—41no—
67C2×Q8×C71—81no—
68C22×C4×C71—21no—
69C22×C4×C71—21no—
70C22×C4×C71—22no—
71C22×C4×C71—21no—
72C22×C4×C71—21no—
73C22×C4×C71—21no—
74C22×C4×C71—21no—
75C22×C4×C71—21no—
76C22×C4×C71—21no—
77C22×C4×C71—21no—
78C22×C4×C71—21no—
79C22×C4×C71—21no—
80C22×C4×C71—21no—
81C22×C4×C71—81no—
82C22×C4×C71—81no—
83C22×C4×C71C22⋊C411yescharacteristic
84C22×C4×C71C22⋊C411yescharacteristic
85C22×C4×C71C22⋊C411yescharacteristic
86C22×C4×C71C22×C411yescharacteristic
87C22×C4×C71C2×D411yescharacteristic
88C22×C4×C71C2×D411yescharacteristic
89C4⋊C4×C71—41no—
90C4⋊C4×C71—41no—
91C4⋊C4×C71—41no—
92C4⋊C4×C71—41no—
93C4⋊C4×C71—41no—
94C4⋊C4×C71—42no—
95C4⋊C4×C71—42no—
96C4⋊C4×C71—41no—
97C4⋊C4×C71—41no—
98C4⋊C4×C71—81no—
99C4⋊C4×C71—81no—
100C22⋊C4×C71—24no—
101C22⋊C4×C71—41no—
102C22⋊C4×C71—41no—
103C42×C71—81no—
104C23×C71—22no—
105C23×C71—21no—
106C23×C71—21no—
107C23×C71—41no—
108C23×C71#9911yescharacteristic
109C23×C71#9911yescharacteristic
110C23×C71C2×C22⋊C411yescharacteristic
111Q8×C71—82no—
112C2×C4×C71—22no—
113C2×C4×C71—22no—
114C2×C4×C71—22no—
115C2×C4×C71—22no—
116C2×C4×C71—21no—
117C2×C4×C71—21no—
118C2×C4×C71—21no—
119C2×C4×C71—21no—
120C2×C4×C71—22no—
121C2×C4×C71—22no—
122C2×C4×C71—21no—
123C2×C4×C71—21no—
124C2×C4×C71—21no—
125C2×C4×C71—21no—
126C2×C4×C71—41no—
127C2×C4×C71—41no—
128C2×C4×C71—42no—
129C2×C4×C71—41no—
130C2×C4×C71—41no—
131C2×C4×C71—41no—
132C2×C4×C71—41no—
133C2×C4×C71—41no—
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—81no—
142C2×C4×C71—81no—
143C2×C4×C71—81no—
144C2×C4×C71#10412yes—
145C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
146C2×C4×C71#13711yescharacteristic
147C71⋊C8—44no—
148C71⋊C8—82no—
149C22×C71—21no—
150C22×C71—21no—
151C22×C71—21no—
152C22×C71—21no—
153C22×C71—21no—
154C22×C71—21no—
155C22×C71—21no—
156C22×C71—42no—
157C22×C71—42no—
158C22×C71#35612yes—
159C22×C71C2×#9911yescharacteristic
160C22×C71#41711yescharacteristic
161C22×C71C2×#10411yescharacteristic
162C4×C71—21no—
163C4×C71—22no—
164C4×C71—21no—
165C4×C71—42no—
166C4×C71—42no—
167C4×C71—41no—
168C4×C71—41no—
169C4×C71—41no—
170C4×C71—41no—
171C4×C71—81no—
172C4×C71—81no—
173#11768—711nomaximal, Sylow
174C2×C71—21no—
175C2×C71—21no—
176C2×C71—22no—
177C2×C71—41no—
178C2×C71#150411yescharacteristic
179C2×C71#212111yescharacteristic
180C2×C71C2×#35611yescharacteristic
181#2583D7111yescharacteristic
182C2×#343—711no—
183#1463—711no—
184#1354—712no—
185#1346—712no—
186C71#1176811yescharacteristic, complemented, Sylow
187C22×M16—1421no—
188C2×C22⋊Q8Dic7111yescharacteristic
189C22×C4⋊C4C2×D7111yescharacteristic
190#407—1421no—
191#398Dic7111yescharacteristic
192#398C2×D7111yescharacteristic
193#395Dic7111yescharacteristic
194#395C2×D7111yescharacteristic
195#389—21no—
196#389—21no—
197#385—21no—
198#385—21no—
199C4×C22⋊C4Dic7111yescharacteristic
200#343—712no—
201#343—1421no—
202#324—712no—
203#324—1421no—
204C22×Q8C2×Dic7111yescharacteristic
205C23×C4C2×Dic7111yescharacteristic
206C23×C4C22⋊D7111yescharacteristic
207C23×C4C22⋊D7111yescharacteristic
208C2×M16—2841no—
209C2×M16—2842no—
210C2×M16—2841no—
211C2×M16—1424no—
212C2×M16—1422no—
213C22×C8—2841no—
214C22⋊Q8—41no—
215C22⋊Q8—41no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4C2×Dic7111yescharacteristic
224C2×C4⋊C4C2×Dic7111yescharacteristic
225C2×C4⋊C4C22⋊D7111yescharacteristic
226C2×C4⋊C4C22⋊D7111yescharacteristic
227C2×C4⋊C4C22×D7111yescharacteristic
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C42—21no—
231C22⋊C8—2841no—
232C22⋊C8—1422no—
233#95—21no—
234#95—21no—
235#95—21no—
236#95—21no—
237#95C2×Dic7111yescharacteristic
238#95C2×Dic7111yescharacteristic
239C24C22⋊Dic7111yescharacteristic
240C2×Q8—41no—
241C2×Q8—81no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—81no—
256C22×C4—81no—
257C22×C4C22⋊Dic7111yescharacteristic
258C22×C4C22⋊Dic7111yescharacteristic
259C22×C4C22⋊Dic7111yescharacteristic
260C22×C4C22×Dic7111yescharacteristic
261C22×C4C2×C22⋊D7111yescharacteristic
262C22×C4C2×C22⋊D7111yescharacteristic
263M16—2844no—
264M16—2842no—
265M16—5682no—
266M16—1424no—
267C2×C8—2844no—
268C2×C8—2842no—
269C2×C8—2842no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—81no—
280C4⋊C4—81no—
281C22⋊C4—24no—
282C22⋊C4—41no—
283C22⋊C4—41no—
284C42—81no—
285C23—22no—
286C23—21no—
287C23—21no—
288C23—41no—
289C23#1455604311yescharacteristic
290C23#1455604311yescharacteristic
291C23C2×C22⋊Dic7111yescharacteristic
292Q8—82no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4#1455605512yes—
326C2×C4#1455615311yescharacteristic
327C2×C4#1455615811yescharacteristic
328C8—2844no—
329C8—5682no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—42no—
338C22—42no—
339C22#1798579812yes—
340C22C2×#1455604311yescharacteristic
341C22#1798600011yescharacteristic
342C22C2×#1455605511yescharacteristic
343C4—21no—
344C4—22no—
345C4—21no—
346C4—42no—
347C4—42no—
348C4—41no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—81no—
353C4—81no—
354C2—21no—
355C2—21no—
356C2—22no—
357C2—41no—
358C2#2258684411yescharacteristic
359C2#2258998611yescharacteristic
360C2C2×#1798579811yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC22×C4×C71C22×C4
FrattiniC2×C4⋊C4C22×D71
FittingC71×#2583C2
socleC22×C71C2×#99
prime radicalC24×C71C22⋊C4
cosocleC2×C4⋊C4×C71C23
hypercenter#2583D71
hypocenterC71#11768
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #2583

Complements

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

NH ≅ G/Ntype
3. #22586370C2semidirect
4. #22586346C2semidirect
186. C71#11768semidirect

Maximal subgroups

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

subgroupindex
C2×#179857682
#225866702
#225863702
#225863462
C71×#25832
#1176871