Finite Group Databasebeta

← #35296157

Subgroups — #35296157

Order 21248.

998 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 998 conjugacy classes of subgroups — too many to draw. Only its 159 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#24907455C211yesmaximal, characteristic, complemented
2#24906144C212yesmaximal, complemented
3#24905888C212yesmaximal, complemented
4#24905784C211yesmaximal, characteristic, complemented
5C2×#547×C83C211yesmaximal, characteristic
6#19127754C2212yes—
7#19127725C2212yes—
8C2×#14581709—21no—
9#19127470—21no—
10#19127450—24no—
11D4⋊(C83⋊C8)—24no—
12#19127425—22no—
13C22×#41×C83C412yes—
14#547×C83—24no—
15#547×C83—24no—
16C2×C4⋊Q8×C83C2211yescharacteristic
17C2×#123×C83—22no—
18C2×C22⋊Q8×C83C412yes—
19C2×#121×C83—21no—
20C2×C4×D4×C83C2211yescharacteristic
21C2×C4×D4×C83C2211yescharacteristic
22C22×C4⋊C4×C83—21no—
23C2×#13050574—22no—
24C4⋊(C83⋊C8)—22no—
25C4⋊(C83⋊C8)—22no—
26C4⋊(C83⋊C8)—22no—
27#14581709—42no—
28C22⋊(C83⋊C8)—22no—
29C4×C83⋊C8—22no—
30C2×#41×C83—24no—
31C2×#41×C83—24no—
32C2×#41×C83—48no—
33C22×Q8×C83C2×C412yes—
34C22×D4×C83—22no—
35C22×D4×C83C2×C411yescharacteristic
36C22×D4×C83D411yescharacteristic
37C23×C4×C83—22no—
38C23×C4×C83—21no—
39C23×C4×C83C2×C412yes—
40C4⋊Q8×C83—22no—
41C4⋊Q8×C83D414yes—
42#123×C83—216no—
43C22⋊Q8×C83—24no—
44C22⋊Q8×C83—24no—
45#121×C83—28no—
46C4×D4×C83—22no—
47C4×D4×C83—22no—
48C4×D4×C83—22no—
49C4×D4×C83D414yes—
50C2×C4⋊C4×C83—28no—
51C2×C4⋊C4×C83—22no—
52C2×C4⋊C4×C83—21no—
53C2×C4⋊C4×C83—21no—
54C2×C4⋊C4×C83C2×C411yescharacteristic
55C2×C4⋊C4×C83D411yescharacteristic
56C2×C22⋊C4×C83—22no—
57C2×C22⋊C4×C83—21no—
58C2×C22⋊C4×C83D412yes—
59C2×C42×C83C2311yescharacteristic
60#13050574—42no—
61#13050574—42no—
62C2×C83⋊C8—42no—
63C2×C83⋊C8—42no—
64C24×C83—21no—
65C24×C83C22⋊C412yes—
66#41×C83—416no—
67#41×C83—48no—
68#41×C83—48no—
69C2×Q8×C83—22no—
70C2×Q8×C83—44no—
71C2×Q8×C83C22⋊C414yes—
72C2×D4×C83—24no—
73C2×D4×C83—28no—
74C2×D4×C83—21no—
75C2×D4×C83—22no—
76C2×D4×C83—22no—
77C2×D4×C83—21no—
78C2×D4×C83—21no—
79C2×D4×C83—48no—
80C2×D4×C83C22⋊C412yes—
81C22×C4×C83—28no—
82C22×C4×C83—21no—
83C22×C4×C83—22no—
84C22×C4×C83—22no—
85C22×C4×C83—22no—
86C22×C4×C83—24no—
87C22×C4×C83—44no—
88C22×C4×C83—48no—
89C22×C4×C83—42no—
90C22×C4×C83—41no—
91C22×C4×C83—41no—
92C22×C4×C83C22⋊C412yes—
93C22×C4×C83C22×C411yescharacteristic
94C22×C4×C83C2×D411yescharacteristic
95C22×C4×C83C2×D411yescharacteristic
96C4⋊C4×C83—28no—
97C4⋊C4×C83—21no—
98C4⋊C4×C83—22no—
99C4⋊C4×C83—24no—
100C4⋊C4×C83—24no—
101C4⋊C4×C83C22⋊C412yes—
102C22⋊C4×C83—28no—
103C22⋊C4×C83—24no—
104C22⋊C4×C83—24no—
105C42×C83C2×D412yes—
106C42×C83C2×D412yes—
107C23×C83—24no—
108C23×C83—22no—
109C23×C83—22no—
110C23×C83—22no—
111C23×C83—22no—
112C23×C83—42no—
113C23×C83—42no—
114C23×C83—42no—
115C23×C83—41no—
116C23×C83—41no—
117C23×C83C2×C22⋊C411yescharacteristic
118Q8×C83—44no—
119Q8×C83—44no—
120D4×C83—24no—
121D4×C83—22no—
122D4×C83—416no—
123D4×C83—42no—
124D4×C83—42no—
125D4×C83#10014yescomplemented
126C2×C4×C83—22no—
127C2×C4×C83—22no—
128C2×C4×C83—22no—
129C2×C4×C83—24no—
130C2×C4×C83—41no—
131C2×C4×C83—44no—
132C2×C4×C83—44no—
133C2×C4×C83—416no—
134C2×C4×C83—41no—
135C2×C4×C83—41no—
136C2×C4×C83—44no—
137C2×C4×C83—44no—
138C2×C4×C83—44no—
139C2×C4×C83—42no—
140C2×C4×C83C2×C22⋊C411yescharacteristic
141C2×C4×C83C2×C22⋊C411yescharacteristic
142C2×C4×C83#12012yes—
143C2×C4×C83#12012yes—
144C2×C4×C83#13512yes—
145C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
146C2×C4×C83#13711yescharacteristic
147C83⋊C8—82no—
148C83⋊C8—82no—
149C22×C83—21no—
150C22×C83—21no—
151C22×C83—24no—
152C22×C83—44no—
153C22×C83—44no—
154C22×C83—42no—
155C22×C83—42no—
156C22×C83—42no—
157C22×C83—42no—
158C22×C83—42no—
159C22×C83—82no—
160C22×C83—82no—
161C22×C83#37911yescharacteristic
162C22×C83#41611yescharacteristic
163C22×C83#41711yescharacteristic
164C4×C83—22no—
165C4×C83—21no—
166C4×C83—42no—
167C4×C83—44no—
168C4×C83—42no—
169C4×C83C2×#10012yes—
170C4×C83#44912yes—
171C4×C83#45212yes—
172#8393—831nomaximal, Sylow
173C2×C83—21no—
174C2×C83—21no—
175C2×C83—44no—
176C2×C83—82no—
177C2×C83#178311yescharacteristic
178C2×C83#178811yescharacteristic
179C2×C83#179611yescharacteristic
180C2×#547D8311yescharacteristic
181#1800—831no—
182#1548—832no—
183#1486—832no—
184#1460—831no—
185C83#839311yescharacteristic, complemented, Sylow
186C22×#41Dic8312yes—
187#547—24no—
188#547—24no—
189C2×C4⋊Q8C2×D8311yescharacteristic
190C2×#123—22no—
191C2×C22⋊Q8Dic8312yes—
192C2×#121—21no—
193C2×C4×D4C2×D8311yescharacteristic
194C2×C4×D4C2×D8311yescharacteristic
195C22×C4⋊C4—21no—
196#431—832no—
197#423—832no—
198C2×#100—1661no—
199#343—1661no—
200#333—1664no—
201D4⋊C8—1664no—
202#323—1662no—
203C2×#41—24no—
204C2×#41—24no—
205C2×#41—48no—
206C22×Q8C2×Dic8312yes—
207C22×D4—22no—
208C22×D4C2×Dic8311yescharacteristic
209C22×D4C22⋊D8311yescharacteristic
210C23×C4—22no—
211C23×C4—21no—
212C23×C4C2×Dic8312yes—
213C2×M16—1662no—
214C4⋊Q8—22no—
215C4⋊Q8C22⋊D8314yes—
216#123—216no—
217C22⋊Q8—24no—
218C22⋊Q8—24no—
219#121—28no—
220C4×D4—22no—
221C4×D4—22no—
222C4×D4—22no—
223C4×D4C22⋊D8314yes—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4C2×Dic8311yescharacteristic
229C2×C4⋊C4C22⋊D8311yescharacteristic
230C2×C22⋊C4—22no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4C22⋊D8312yes—
233C2×C42C22×D8311yescharacteristic
234C4⋊C8—1662no—
235C4⋊C8—1662no—
236C4⋊C8—1662no—
237#100—3322no—
238C22⋊C8—1662no—
239C4×C8—1662no—
240C24—21no—
241C24C22⋊Dic8312yes—
242#41—416no—
243#41—48no—
244#41—48no—
245C2×Q8—22no—
246C2×Q8—44no—
247C2×Q8C22⋊Dic8314yes—
248C2×D4—24no—
249C2×D4—28no—
250C2×D4—22no—
251C2×D4—22no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—48no—
256C2×D4C22⋊Dic8312yes—
257C22×C4—28no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—44no—
264C22×C4—48no—
265C22×C4—42no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4C22⋊Dic8312yes—
269C22×C4C22×Dic8311yescharacteristic
270C22×C4C2×C22⋊D8311yescharacteristic
271C22×C4C2×C22⋊D8311yescharacteristic
272M16—3322no—
273M16—3322no—
274C2×C8—3322no—
275C2×C8—3322no—
276C4⋊C4—28no—
277C4⋊C4—22no—
278C4⋊C4—24no—
279C4⋊C4—24no—
280C4⋊C4—21no—
281C4⋊C4C22⋊Dic8312yes—
282C22⋊C4—28no—
283C22⋊C4—24no—
284C22⋊C4—24no—
285C42C2×C22⋊D8312yes—
286C42C2×C22⋊D8312yes—
287C23—24no—
288C23—22no—
289C23—22no—
290C23—22no—
291C23—22no—
292C23—42no—
293C23—42no—
294C23—42no—
295C23—41no—
296C23—41no—
297C23C2×C22⋊Dic8311yescharacteristic
298Q8—44no—
299Q8—44no—
300D4—24no—
301D4—22no—
302D4—416no—
303D4—42no—
304D4—42no—
305D4#1458170914yescomplemented
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—24no—
310C2×C4—41no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—416no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—42no—
320C2×C4C2×C22⋊Dic8311yescharacteristic
321C2×C4C2×C22⋊Dic8311yescharacteristic
322C2×C4#1458176112yes—
323C2×C4#1458176112yes—
324C2×C4#1458181212yes—
325C2×C4#1458181711yescharacteristic
326C2×C4#1458182211yescharacteristic
327C8—6642no—
328C8—6642no—
329C22—21no—
330C22—21no—
331C22—24no—
332C22—44no—
333C22—44no—
334C22—42no—
335C22—42no—
336C22—42no—
337C22—42no—
338C22—42no—
339C22—82no—
340C22—82no—
341C22#1912755711yescharacteristic
342C22#1912769711yescharacteristic
343C22#1912770211yescharacteristic
344C4—21no—
345C4—22no—
346C4—42no—
347C4—44no—
348C4—42no—
349C4C2×#1458170912yes—
350C4#1912784612yes—
351C4#1912786312yes—
352C2—21no—
353C2—21no—
354C2—44no—
355C2—82no—
356C2#2490738411yescharacteristic
357C2#2490740311yescharacteristic
358C2#2490743811yescharacteristic
subgroupG/H
centerC22#19127557
commutatorC22×C4×C83C22×C4
FrattiniC2×C42C22×D83
Fitting, prime radicalC2×#547×C83C2
socleC22×C83#379
cosocleC2×C42×C83C23
hypercenterC2×#547D83
hypocenterC83#8393
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. #24907455C2semidirect
2. #24906144C2semidirect
3. #24905888C2semidirect
4. #24905784C2semidirect
125. D4×C83#100semidirect
185. C83#8393semidirect
305. D4#14581709semidirect

Maximal subgroups

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

subgroupindex
#249074552
#249061442
#249058882
#249057842
C2×#547×C832
#839383