Finite Group Databasebeta

← #35296246

Subgroups — #35296246

Order 21248.

692 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24907447C211yesmaximal, characteristic
2#24906120C212yesmaximal, complemented
3#24905866C212yesmaximal, complemented
4#24905777C211yesmaximal, characteristic
5C83×#3458C211yesmaximal, characteristic
6#19127754C2212yes—
7C2×C4⋊(C83⋊C8)C2211yescharacteristic
8C2×C4⋊(C83⋊C8)C2211yescharacteristic
9C2×C22⋊(C83⋊C8)—21no—
10#19127456—21no—
11#19127450—22no—
12#19127447—22no—
13Q8⋊(C83⋊C8)—24no—
14#19127427—21no—
15#19127425—21no—
16#544×C83—22no—
17#544×C83—22no—
18#544×C83—21no—
19#544×C83C412yes—
20#543×C83—21no—
21#543×C83C412yes—
22#533×C83—24no—
23#533×C83—21no—
24#533×C83C2212yes—
25C2×#117×C83—21no—
26C2×#117×C83C2211yescharacteristic
27C22×C83⋊C8—21no—
28C22×C83⋊C8—21no—
29C4⋊(C83⋊C8)—22no—
30C4⋊(C83⋊C8)—22no—
31C4⋊(C83⋊C8)—22no—
32C22⋊(C83⋊C8)—22no—
33C22⋊(C83⋊C8)—42no—
34C4×C83⋊C8—22no—
35C23×C4×C83D411yescharacteristic
36C4⋊Q8×C83—24no—
37C4⋊Q8×C83—21no—
38C4⋊Q8×C83—21no—
39C4⋊Q8×C83—24no—
40C4⋊Q8×C83D412yes—
41C4⋊Q8×C83D412yes—
42#125×C83—24no—
43#125×C83—22no—
44#125×C83—24no—
45#125×C83—22no—
46C22⋊Q8×C83—24no—
47C22⋊Q8×C83—24no—
48C22⋊Q8×C83—24no—
49C22⋊Q8×C83—22no—
50C22⋊Q8×C83—22no—
51C22⋊Q8×C83—22no—
52C22⋊Q8×C83—24no—
53C22⋊Q8×C83C2×C414yes—
54C4×Q8×C83—22no—
55C4×Q8×C83—22no—
56C4×Q8×C83—22no—
57C4×Q8×C83D414yes—
58#117×C83—22no—
59#117×C83—22no—
60#117×C83—22no—
61#117×C83—22no—
62#117×C83—22no—
63#117×C83—22no—
64C2×C4⋊C4×C83—21no—
65C2×C4⋊C4×C83—21no—
66C2×C4⋊C4×C83C2×C411yescharacteristic
67C2×C4⋊C4×C83D411yescharacteristic
68C2×C22⋊C4×C83—22no—
69C2×C22⋊C4×C83C2×C411yescharacteristic
70C2×C22⋊C4×C83D411yescharacteristic
71C2×C42×C83—22no—
72C2×C42×C83D411yescharacteristic
73C2×C42×C83C2311yescharacteristic
74C2×C83⋊C8—41no—
75C2×C83⋊C8—41no—
76C2×C83⋊C8—42no—
77C2×C83⋊C8—41no—
78C2×C83⋊C8—41no—
79C2×C83⋊C8—41no—
80C2×C83⋊C8—41no—
81C24×C83C22⋊C411yescharacteristic
82C2×Q8×C83—22no—
83C2×Q8×C83—22no—
84C2×Q8×C83—22no—
85C2×Q8×C83C22⋊C414yes—
86C22×C4×C83—21no—
87C22×C4×C83—21no—
88C22×C4×C83—21no—
89C22×C4×C83—21no—
90C22×C4×C83—22no—
91C22×C4×C83—22no—
92C22×C4×C83C22⋊C411yescharacteristic
93C22×C4×C83C22⋊C411yescharacteristic
94C22×C4×C83C22⋊C411yescharacteristic
95C22×C4×C83SD1612yes—
96C22×C4×C83Q1612yes—
97C22×C4×C83C22×C411yescharacteristic
98C22×C4×C83C2×D411yescharacteristic
99C22×C4×C83C2×D411yescharacteristic
100C4⋊C4×C83—22no—
101C4⋊C4×C83—22no—
102C4⋊C4×C83—22no—
103C4⋊C4×C83—24no—
104C4⋊C4×C83—24no—
105C4⋊C4×C83—22no—
106C4⋊C4×C83—22no—
107C4⋊C4×C83—24no—
108C4⋊C4×C83—22no—
109C4⋊C4×C83—22no—
110C4⋊C4×C83—22no—
111C4⋊C4×C83—22no—
112C4⋊C4×C83—24no—
113C4⋊C4×C83C22⋊C414yes—
114C22⋊C4×C83—24no—
115C22⋊C4×C83—24no—
116C22⋊C4×C83—22no—
117C22⋊C4×C83—22no—
118C42×C83—24no—
119C42×C83—24no—
120C42×C83—21no—
121C42×C83—21no—
122C42×C83C2×D412yes—
123C42×C83C2×D412yes—
124C23×C83—21no—
125C23×C83—21no—
126C23×C83—41no—
127C23×C83—41no—
128C23×C83#10312yes—
129C23×C83C2×C22⋊C411yescharacteristic
130Q8×C83—44no—
131Q8×C83—82no—
132Q8×C83—82no—
133Q8×C83—82no—
134C2×C4×C83—21no—
135C2×C4×C83—21no—
136C2×C4×C83—22no—
137C2×C4×C83—22no—
138C2×C4×C83—22no—
139C2×C4×C83—22no—
140C2×C4×C83—24no—
141C2×C4×C83—21no—
142C2×C4×C83—21no—
143C2×C4×C83—24no—
144C2×C4×C83—41no—
145C2×C4×C83—41no—
146C2×C4×C83—41no—
147C2×C4×C83—41no—
148C2×C4×C83—82no—
149C2×C4×C83—82no—
150C2×C4×C83#10312yes—
151C2×C4×C83C2×C22⋊C412yes—
152C2×C4×C83#12012yes—
153C2×C4×C83#12011yescharacteristic
154C2×C4×C83#12011yescharacteristic
155C2×C4×C83C2×SD1611yescharacteristic
156C2×C4×C83C2×Q1611yescharacteristic
157C2×C4×C83#13711yescharacteristic
158C2×C4×C83#13711yescharacteristic
159C83⋊C8—82no—
160C83⋊C8—81no—
161C83⋊C8—81no—
162C22×C83—21no—
163C22×C83—21no—
164C22×C83—22no—
165C22×C83—42no—
166C22×C83—41no—
167C22×C83—41no—
168C22×C83—81no—
169C22×C83#37911yescharacteristic
170C22×C83C2×#10311yescharacteristic
171C22×C83#41911yescharacteristic
172C4×C83—22no—
173C4×C83—22no—
174C4×C83—22no—
175C4×C83—41no—
176C4×C83—44no—
177C4×C83—41no—
178C4×C83—44no—
179C4×C83#44811yescharacteristic
180C4×C83#44811yescharacteristic
181C4×C83#45111yescharacteristic
182C4×C83#45111yescharacteristic
183#8410—831nomaximal, Sylow
184C2×C83—21no—
185C2×C83—21no—
186C2×C83—42no—
187C2×C83#178711yescharacteristic
188C2×C83#179111yescharacteristic
189C2×C83#179411yescharacteristic
190#3458D8311yescharacteristic
191#1798—831no—
192#1542—832no—
193#1481—832no—
194#1458—831no—
195C83#841011yescharacteristic, complemented, Sylow
196#544—22no—
197#544—22no—
198#544—21no—
199#544Dic8312yes—
200#543—21no—
201#543Dic8312yes—
202#533—24no—
203#533—21no—
204#533C2×D8312yes—
205C2×#117—21no—
206C2×#117C2×D8311yescharacteristic
207#431—832no—
208C2×C4⋊C8—831no—
209C2×C4⋊C8—831no—
210C2×C22⋊C8—1661no—
211#336—1661no—
212#333—1662no—
213#332—1662no—
214Q8⋊C8—1664no—
215#324—1661no—
216#323—1661no—
217C23×C4C22⋊D8311yescharacteristic
218C22×C8—1661no—
219C22×C8—1661no—
220C4⋊Q8—24no—
221C4⋊Q8—21no—
222C4⋊Q8—21no—
223C4⋊Q8—24no—
224C4⋊Q8C22⋊D8312yes—
225C4⋊Q8C22⋊D8312yes—
226#125—24no—
227#125—22no—
228#125—24no—
229#125—22no—
230C22⋊Q8—24no—
231C22⋊Q8—24no—
232C22⋊Q8—24no—
233C22⋊Q8—22no—
234C22⋊Q8—22no—
235C22⋊Q8—22no—
236C22⋊Q8—24no—
237C22⋊Q8C2×Dic8314yes—
238C4×Q8—22no—
239C4×Q8—22no—
240C4×Q8—22no—
241C4×Q8C22⋊D8314yes—
242#117—22no—
243#117—22no—
244#117—22no—
245#117—22no—
246#117—22no—
247#117—22no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4C2×Dic8311yescharacteristic
251C2×C4⋊C4C22⋊D8311yescharacteristic
252C2×C22⋊C4—22no—
253C2×C22⋊C4C2×Dic8311yescharacteristic
254C2×C22⋊C4C22⋊D8311yescharacteristic
255C2×C42—22no—
256C2×C42C22⋊D8311yescharacteristic
257C2×C42C22×D8311yescharacteristic
258C4⋊C8—1662no—
259C4⋊C8—1662no—
260C4⋊C8—1662no—
261C22⋊C8—3322no—
262C22⋊C8—1662no—
263C4×C8—1662no—
264C24C22⋊Dic8311yescharacteristic
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8C22⋊Dic8314yes—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4C22⋊Dic8311yescharacteristic
276C22×C4C22⋊Dic8311yescharacteristic
277C22×C4C22⋊Dic8311yescharacteristic
278C22×C4#1305057712yes—
279C22×C4#1305058012yes—
280C22×C4C22×Dic8311yescharacteristic
281C22×C4C2×C22⋊D8311yescharacteristic
282C22×C4C2×C22⋊D8311yescharacteristic
283C2×C8—3322no—
284C2×C8—3321no—
285C2×C8—3321no—
286C2×C8—3321no—
287C2×C8—3321no—
288C2×C8—3321no—
289C2×C8—3321no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—24no—
301C4⋊C4—22no—
302C4⋊C4—24no—
303C4⋊C4C22⋊Dic8314yes—
304C22⋊C4—24no—
305C22⋊C4—24no—
306C22⋊C4—22no—
307C22⋊C4—22no—
308C42—24no—
309C42—21no—
310C42—24no—
311C42—21no—
312C42C2×C22⋊D8312yes—
313C42C2×C22⋊D8312yes—
314C23—21no—
315C23—21no—
316C23—41no—
317C23—41no—
318C23#1458171612yes—
319C23C2×C22⋊Dic8311yescharacteristic
320Q8—44no—
321Q8—82no—
322Q8—82no—
323Q8—82no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—24no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—24no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4#1458171612yes—
341C2×C4C2×C22⋊Dic8312yes—
342C2×C4#1458176112yes—
343C2×C4#1458176111yescharacteristic
344C2×C4#1458176111yescharacteristic
345C2×C4C2×#1305057711yescharacteristic
346C2×C4C2×#1305058011yescharacteristic
347C2×C4#1458182311yescharacteristic
348C2×C4#1458182311yescharacteristic
349C8—6642no—
350C8—6641no—
351C8—6641no—
352C22—21no—
353C22—21no—
354C22—22no—
355C22—42no—
356C22—41no—
357C22—41no—
358C22—81no—
359C22#1912755711yescharacteristic
360C22C2×#1458171611yescharacteristic
361C22#1912771011yescharacteristic
362C4—22no—
363C4—22no—
364C4—22no—
365C4—41no—
366C4—44no—
367C4—41no—
368C4—44no—
369C4#1912784011yescharacteristic
370C4#1912784011yescharacteristic
371C4#1912785711yescharacteristic
372C4#1912785711yescharacteristic
373C2—21no—
374C2—21no—
375C2—42no—
376C2#2490739811yescharacteristic
377C2#2490741411yescharacteristic
378C2#2490742711yescharacteristic
subgroupG/H
centerC2×C4#14581761
commutatorC22×C4×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#3458C2
socleC22×C83#379
prime radicalC24×C83C22⋊C4
cosocleC2×C42×C83C23
hypercenter#3458D83
hypocenterC83#8410
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3458

Complements

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

NH ≅ G/Ntype
2. #24906120C2semidirect
3. #24905866C2semidirect
195. C83#8410semidirect

Maximal subgroups

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

subgroupindex
#249074472
#249061202
#249058662
#249057772
C83×#34582
#841083