Finite Group Databasebeta

← #35323071

Subgroups — #35323071

Order 21248.

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

normal subgroups only

This group has 488 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#24907455C211yesmaximal, characteristic
2C4×#14581707C211yesmaximal, characteristic
3#24905581C212yesmaximal
4#24905581C212yesmaximal
5C83×#2579C211yesmaximal, characteristic
6#19127725—21no—
7C2×#14581709C2212yes—
8C2×#14581707C2212yes—
9C4×C22⋊Dic83—21no—
10#19127470—21no—
11#19127468—21no—
12C2×C4×D4×C83C2211yescharacteristic
13C22×C4⋊C4×C83C411yescharacteristic, complemented
14#398×C83—21no—
15#396×C83—21no—
16#387×C83—21no—
17#386×C83C2211yescharacteristic
18#386×C83C2211yescharacteristic
19#385×C83—21no—
20#385×C83C411yescharacteristic, complemented
21#385×C83C411yescharacteristic, complemented
22#380×C83C411yescharacteristic, complemented
23C2×#13050574—22no—
24C2×C22⋊Dic83—22no—
25C2×C4×Dic83—41no—
26C4⋊(C83⋊C8)—41no—
27C4⋊(C83⋊C8)—41no—
28#14581709—24no—
29#14581707—22no—
30#14581707Q814yes—
31C83⋊#95—41no—
32C22×D4×C83C2311yescharacteristic
33C23×C4×C83C2×C411yescharacteristic
34C23×C4×C83C2×C411yescharacteristic
35C23×C4×C83C2×C411yescharacteristic
36C23×C4×C83D411yescharacteristic
37C4×D4×C83—41no—
38C4×D4×C83—41no—
39C2×C4⋊C4×C83—21no—
40C2×C4⋊C4×C83—21no—
41C2×C4⋊C4×C83—21no—
42C2×C4⋊C4×C83—21no—
43C2×C4⋊C4×C83—21no—
44C2×C4⋊C4×C83—41no—
45C2×C4⋊C4×C83—41no—
46C2×C4⋊C4×C83D411yescharacteristic
47C2×C22⋊C4×C83—21no—
48C2×C22⋊C4×C83—21no—
49C2×C22⋊C4×C83C2×C411yescharacteristic
50C2×C22⋊C4×C83C2×C411yescharacteristic
51C2×C22⋊C4×C83C2×C411yescharacteristic
52C2×C22⋊C4×C83D411yescharacteristic
53C2×C42×C83D411yescharacteristic
54#95×C83—21no—
55#95×C83—21no—
56#95×C83—21no—
57#95×C83—21no—
58#95×C83—21no—
59C22×Dic83—42no—
60#13050574—42no—
61#13050574—42no—
62C2×C83⋊C8—42no—
63C22⋊Dic83—24no—
64C22⋊Dic83—42no—
65C4×Dic83—82no—
66C24×C83C22×C411yescharacteristic
67C24×C83C2×D411yescharacteristic
68C2×D4×C83—41no—
69C2×D4×C83—41no—
70C2×D4×C83C2×Q812yes—
71C2×D4×C83#4112yes—
72C22×C4×C83—21no—
73C22×C4×C83—21no—
74C22×C4×C83—21no—
75C22×C4×C83—21no—
76C22×C4×C83—21no—
77C22×C4×C83—21no—
78C22×C4×C83—21no—
79C22×C4×C83—21no—
80C22×C4×C83—42no—
81C22×C4×C83—41no—
82C22×C4×C83—41no—
83C22×C4×C83—81no—
84C22×C4×C83—81no—
85C22×C4×C83—81no—
86C22×C4×C83C22⋊C411yescharacteristic
87C22×C4×C83C22⋊C411yescharacteristic
88C22×C4×C83C22⋊C411yescharacteristic
89C22×C4×C83C22⋊C411yescharacteristic
90C22×C4×C83C2×D411yescharacteristic
91C4⋊C4×C83—41no—
92C4⋊C4×C83—42no—
93C4⋊C4×C83—42no—
94C4⋊C4×C83—41no—
95C4⋊C4×C83—41no—
96C4⋊C4×C83—81no—
97C4⋊C4×C83—81no—
98C22⋊C4×C83—41no—
99C22⋊C4×C83—41no—
100C22⋊C4×C83—42no—
101C22⋊C4×C83—41no—
102C22⋊C4×C83—41no—
103C22⋊C4×C83—81no—
104C42×C83—21no—
105C42×C83—21no—
106C23×C83—21no—
107C23×C83—21no—
108C23×C83—21no—
109C23×C83—21no—
110C23×C83—41no—
111C23×C83—41no—
112C23×C83—41no—
113C23×C83—41no—
114C23×C83C2×C22⋊C411yescharacteristic
115C23×C83C4×Q812yes—
116C23×C83C22⋊Q812yes—
117D4×C83—42no—
118D4×C83—42no—
119C2×C4×C83—21no—
120C2×C4×C83—21no—
121C2×C4×C83—22no—
122C2×C4×C83—21no—
123C2×C4×C83—21no—
124C2×C4×C83—21no—
125C2×C4×C83—41no—
126C2×C4×C83—42no—
127C2×C4×C83—41no—
128C2×C4×C83—42no—
129C2×C4×C83—41no—
130C2×C4×C83—41no—
131C2×C4×C83—41no—
132C2×C4×C83—41no—
133C2×C4×C83—42no—
134C2×C4×C83—41no—
135C2×C4×C83—41no—
136C2×C4×C83—41no—
137C2×C4×C83—41no—
138C2×C4×C83—41no—
139C2×C4×C83—41no—
140C2×C4×C83—41no—
141C2×C4×C83—41no—
142C2×C4×C83—41no—
143C2×C4×C83—81no—
144C2×C4×C83—81no—
145C2×C4×C83—81no—
146C2×C4×C83—82no—
147C2×C4×C83#9911yescharacteristic
148C2×C4×C83#9911yescharacteristic
149C2×C4×C83#12411yescharacteristic
150C2×C4×C83C4⋊Q811yescharacteristic
151C2×Dic83—44no—
152C2×Dic83—82no—
153C2×Dic83—82no—
154C83⋊C8—82no—
155C22×C83—21no—
156C22×C83—21no—
157C22×C83—22no—
158C22×C83—22no—
159C22×C83—42no—
160C22×C83—42no—
161C22×C83—42no—
162C22×C83—41no—
163C22×C83—41no—
164C22×C83—41no—
165C22×C83—41no—
166C22×C83—81no—
167C22×C83#39111yescharacteristic
168C22×C83C2×#9911yescharacteristic
169C22×C83#41011yescharacteristic
170C4×C83—21no—
171C4×C83—21no—
172C4×C83—21no—
173C4×C83—21no—
174C4×C83—41no—
175C4×C83—42no—
176C4×C83—41no—
177C4×C83—41no—
178C4×C83—81no—
179C4×C83—81no—
180C4×C83—81no—
181Dic83—84no—
182#13297—831nomaximal, Sylow
183C2×C83—21no—
184C2×C83—21no—
185C2×C83—42no—
186C2×C83—42no—
187C2×C83#199611yescharacteristic
188C2×C83#212011yescharacteristic
189C2×C83#212111yescharacteristic
190#2579D8311yescharacteristic
191#1800—831no—
192C4×#99—831no—
193#1390—832no—
194#1390—832no—
195C83#1329711yescharacteristic, complemented, Sylow
196C2×C4×D4C2×D8311yescharacteristic
197C22×C4⋊C4Dic8311yescharacteristic, complemented
198#423—1661no—
199C2×#100—832no—
200C2×#99—832no—
201#398—21no—
202#396—21no—
203#387—21no—
204#386C2×D8311yescharacteristic
205#386C2×D8311yescharacteristic
206#385—21no—
207#385Dic8311yescharacteristic, complemented
208#385Dic8311yescharacteristic, complemented
209#380Dic8311yescharacteristic, complemented
210C4×C22⋊C4—1661no—
211#343—1661no—
212#342—1661no—
213C22×D4C22×D8311yescharacteristic
214C23×C4C2×Dic8311yescharacteristic
215C23×C4C2×Dic8311yescharacteristic
216C23×C4C2×Dic8311yescharacteristic
217C23×C4C22⋊D8311yescharacteristic
218C2×M16—1662no—
219C4×D4—41no—
220C4×D4—41no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4C22⋊D8311yescharacteristic
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—1662no—
232C2×C22⋊C4C2×Dic8311yescharacteristic
233C2×C22⋊C4C2×Dic8311yescharacteristic
234C2×C22⋊C4C2×Dic8311yescharacteristic
235C2×C22⋊C4C22⋊D8311yescharacteristic
236C2×C42—3321no—
237C2×C42C22⋊D8311yescharacteristic
238C4⋊C8—3321no—
239C4⋊C8—3321no—
240#100—1664no—
241#99—834no—
242#99—1662no—
243#95—21no—
244#95—21no—
245#95—21no—
246#95—21no—
247#95—21no—
248#95—3321no—
249C24C22×Dic8311yescharacteristic
250C24C2×C22⋊D8311yescharacteristic
251C2×D4—41no—
252C2×D4—41no—
253C2×D4Q8×D8312yes—
254C2×D4#1305059112yes—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—42no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—81no—
267C22×C4—81no—
268C22×C4—81no—
269C22×C4—3322no—
270C22×C4C22⋊Dic8311yescharacteristic
271C22×C4C22⋊Dic8311yescharacteristic
272C22×C4C22⋊Dic8311yescharacteristic
273C22×C4C22⋊Dic8311yescharacteristic
274C22×C4C2×C22⋊D8311yescharacteristic
275M16—3322no—
276M16—3322no—
277C2×C8—3322no—
278C4⋊C4—41no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—81no—
284C4⋊C4—81no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—42no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—81no—
291C22⋊C4—3322no—
292C22⋊C4—1664no—
293C42—21no—
294C42—21no—
295C42—6642no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23C2×C22⋊Dic8311yescharacteristic
305C23Q8×Dic8312yes—
306C23#1458176812yes—
307D4—42no—
308D4—42no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—42no—
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—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—82no—
337C2×C4—3324no—
338C2×C4—6642no—
339C2×C4—6642no—
340C2×C4#1458170711yescharacteristic
341C2×C4#1458170711yescharacteristic
342C2×C4#1458178111yescharacteristic
343C2×C4#1458179011yescharacteristic
344C8—6642no—
345C22—21no—
346C22—21no—
347C22—22no—
348C22—22no—
349C22—42no—
350C22—42no—
351C22—42no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—81no—
357C22#1912760811yescharacteristic
358C22C2×#1458170711yescharacteristic
359C22#1912767511yescharacteristic
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—42no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—81no—
369C4—81no—
370C4—81no—
371C4—6644no—
372C2—21no—
373C2—21no—
374C2—42no—
375C2—42no—
376C2#2490844911yescharacteristic
377C2#2490910411yescharacteristic
378C2#2490910911yescharacteristic
subgroupG/H
centerC22C2×#14581707
commutatorC24×C83C22×C4
FrattiniC22×D4C22×D83
FittingC83×#2579C2
socleC22×C83C2×#99
prime radical, cosocleC22×D4×C83C23
hypercenter#2579D83
hypocenterC83#13297
LCSG ▷ C24×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2579

Complements

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

NH ≅ G/Ntype
13. C22×C4⋊C4×C83C4semidirect
20. #385×C83C4semidirect
21. #385×C83C4semidirect
22. #380×C83C4semidirect
195. C83#13297semidirect
197. C22×C4⋊C4Dic83semidirect
207. #385Dic83semidirect
208. #385Dic83semidirect
209. #380Dic83semidirect

Maximal subgroups

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

subgroupindex
#249074552
C4×#145817072
#249055812
C83×#25792
#1329783