Finite Group Databasebeta

← #35389686

Subgroups — #35389686

Order 21248.

656 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24918388C211yesmaximal, characteristic
2#24916935C211yesmaximal, characteristic
3#24916544C211yesmaximal, characteristic, complemented
4#24906891C214yesmaximal, complemented
5#24906813C214yesmaximal, complemented
6#24906233C211yesmaximal, characteristic, complemented
7#24906148C211yesmaximal, characteristic, complemented
8#24906113C211yesmaximal, characteristic
9C2×#475×C83C211yesmaximal, characteristic, complemented
10C2×#14581823—21no—
11#19128251C2211yescharacteristic
12#19128075C2212yes—
13#19128051C2212yes—
14#19127923—22no—
15#19127911C2212yes—
16#19127903—22no—
17#19127901C2212yes—
18#19127840—22no—
19Q8×C83⋊C8—22no—
20Q8×C83⋊C8C2214yes—
21#19127754—22no—
22#19127754C2211yescharacteristic
23#19127754C2211yescharacteristic
24#19127725C2211yescharacteristic
25#19127700—21no—
26#19127451C2214yes—
27#19127448C2212yes—
28#19127447C2214yes—
29#19127444C2212yes—
30C2×C4⋊Q8×C83C2211yescharacteristic
31C2×C4×Q8×C83C2211yescharacteristic
32#475×C83—22no—
33#475×C83C2214yes—
34C2×#106×C83—21no—
35C2×C4⋊C8×C83C2211yescharacteristic
36C2×#103×C83—21no—
37C2×#13050590—21no—
38#14581823—44no—
39C2×#13050580—22no—
40C2×#13050577—24no—
41C2×#13050577—22no—
42C2×#13050574—21no—
43#14581792C2311yescharacteristic
44#14581789C2311yescharacteristic
45#14581780—21no—
46#14581766—21no—
47#14581766—21no—
48C4×D332—21no—
49C4×D332—21no—
50C4⋊(C83⋊C8)—24no—
51C4⋊(C83⋊C8)—21no—
52C4⋊(C83⋊C8)D412yes—
53C4⋊(C83⋊C8)C2312yes—
54C4⋊(C83⋊C8)C2312yes—
55#14581716—24no—
56#14581715—22no—
57#14581712—22no—
58C22⋊(C83⋊C8)—21no—
59C22⋊(C83⋊C8)D412yes—
60C4×C83⋊C8—24no—
61C4×C83⋊C8—21no—
62C4×C83⋊C8C2312yes—
63C22×Q8×C83—21no—
64C22×Q8×C83D411yescharacteristic
65C22×C8×C83—21no—
66C4⋊Q8×C83—21no—
67C4⋊Q8×C83—21no—
68C4⋊Q8×C83—21no—
69C4⋊Q8×C83C2312yes—
70C4×Q8×C83—21no—
71C4×Q8×C83—21no—
72C4×Q8×C83D412yes—
73C4×Q8×C83C2312yes—
74C2×C4⋊C4×C83—21no—
75C2×C4⋊C4×C83—21no—
76C2×C4⋊C4×C83—21no—
77C2×C4⋊C4×C83D411yescharacteristic
78C2×C42×C83—21no—
79C2×C42×C83C2311yescharacteristic
80#106×C83—24no—
81C4⋊C8×C83—21no—
82C4⋊C8×C83C2312yes—
83#103×C83—24no—
84#13050590—42no—
85#13050590—42no—
86C2×Dic166—21no—
87C2×D332—21no—
88C2×D332—21no—
89C2×D332—21no—
90C2×C22⋊D83—41no—
91C2×C4×D83—41no—
92#13050580—44no—
93#13050577—44no—
94#13050577—84no—
95#13050574—42no—
96C2×C83⋊C8—22no—
97C2×C83⋊C8—22no—
98C2×C83⋊C8C2×D414yes—
99C4⋊Dic83—21no—
100C4⋊Dic83—21no—
101(C2×D83)⋊C4—41no—
102(C2×D83)⋊C4—41no—
103C2×Q8×C83—21no—
104C2×Q8×C83—22no—
105C2×Q8×C83—22no—
106C2×Q8×C83—41no—
107C2×Q8×C83—82no—
108C2×Q8×C83SD1614yescomplemented
109C2×Q8×C83C2×D412yes—
110C22×C4×C83—21no—
111C22×C4×C83—21no—
112C22×C4×C83—41no—
113C22×C4×C83C2×D411yescharacteristic
114C22×C4×C83C2×D411yescharacteristic
115C22×C4×C83#4111yescharacteristic
116C2×C8×C83—22no—
117C2×C8×C83—42no—
118C4⋊C4×C83—22no—
119C4⋊C4×C83—21no—
120C4⋊C4×C83—22no—
121C4⋊C4×C83—22no—
122C4⋊C4×C83—22no—
123C4⋊C4×C83—41no—
124C4⋊C4×C83—41no—
125C4⋊C4×C83C2×D412yes—
126C42×C83—22no—
127C42×C83—22no—
128C42×C83C2×D411yescharacteristic
129C42×C83C2×D411yescharacteristic
130C42×C83#4111yescharacteristic
131C42×C83C2411yescharacteristic
132C23×C83#12111yescharacteristic
133C22×D83—41no—
134Q8×C83—22no—
135Q8×C83—24no—
136Q8×C83—44no—
137Q8×C83—82no—
138Q8×C83C2×SD1614yes—
139C2×C4×C83—22no—
140C2×C4×C83—24no—
141C2×C4×C83—22no—
142C2×C4×C83—22no—
143C2×C4×C83—41no—
144C2×C4×C83—42no—
145C2×C4×C83—81no—
146C2×C4×C83—82no—
147C2×C4×C83#12111yescharacteristic
148C2×C4×C83#12111yescharacteristic
149C2×C4×C83#12111yescharacteristic
150C2×C4×C83C2×SD1612yes—
151C2×C4×C83#13512yes—
152C2×C4×C83C22×D411yescharacteristic
153C2×C4×C83C22×D411yescharacteristic
154C2×C4×C83C2×#4111yescharacteristic
155C22⋊D83—82no—
156C2×Dic83—41no—
157D332—41no—
158D332—41no—
159D332—42no—
160D332—81no—
161C4×D83—82no—
162Dic166—41no—
163Dic166—41no—
164C8×C83—42no—
165C83⋊C8—24no—
166C83⋊C8—42no—
167C83⋊C8—42no—
168C22×C83—21no—
169C22×C83—21no—
170C22×C83#46512yes—
171C22×C83C2×#12111yescharacteristic
172C2×D83—81no—
173C2×D83—81no—
174C2×D83—81no—
175C4×C83—21no—
176C4×C83—24no—
177C4×C83—21no—
178C4×C83—21no—
179C4×C83—42no—
180C4×C83—82no—
181C4×C83#46512yes—
182C4×C83C22×SD1611yescharacteristic
183C4×C83C2×#13511yescharacteristic
184C4×C83#57712yes—
185Dic83—81no—
186#19397—831nomaximal, Sylow
187C2×C83—22no—
188C2×C83#304311yescharacteristic
189C2×C83C2×#46511yescharacteristic
190C2×C83#306911yescharacteristic
191D83—161no—
192#3206—831no—
193C2×#475D8311yescharacteristic, complemented
194#3037—831no—
195#2982—831no—
196#1680—834no—
197#1666—834no—
198#1570—831no—
199#1549—831no—
200#1540—831no—
201C83#1939711yescharacteristic, complemented, Sylow
202C2×#137—1661no—
203#532—831no—
204C2×C4⋊Q8C2×D8311yescharacteristic
205C2×C4×Q8C2×D8311yescharacteristic
206#492—832no—
207#487—832no—
208#475—22no—
209#475C2×D8314yes—
210#464—1662no—
211#462—832no—
212#461—1662no—
213#460—832no—
214#448—1662no—
215C8×Q8—834no—
216C8×Q8—1662no—
217#431—831no—
218#431—831no—
219#431—1662no—
220C2×#106—21no—
221#423—831no—
222C2×C4⋊C8C2×D8311yescharacteristic
223#417—1661no—
224C2×#103—21no—
225#334—834no—
226#332—834no—
227#332—832no—
228#331—832no—
229C2×#41—1661no—
230C22×Q8—21no—
231C22×Q8C22⋊D8311yescharacteristic
232#137—3324no—
233C2×Q16—1662no—
234C2×SD16—1664no—
235C2×SD16—1662no—
236C2×M16—1661no—
237C22×C8—21no—
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8—21no—
241C4⋊Q8—831no—
242C4⋊Q8C22×D8312yes—
243#127—831no—
244#124—1661no—
245#121—1661no—
246#121—1661no—
247C4×Q8—21no—
248C4×Q8—21no—
249C4×Q8C22⋊D8312yes—
250C4×Q8C22×D8312yes—
251C4×D4—1661no—
252C4×D4—1661no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4C22⋊D8311yescharacteristic
257C2×C42—21no—
258C2×C42C22×D8311yescharacteristic
259#106—24no—
260C4⋊C8—21no—
261C4⋊C8—832no—
262C4⋊C8—832no—
263C4⋊C8—832no—
264C4⋊C8—1664no—
265C4⋊C8—1661no—
266C4⋊C8C22×D8312yes—
267#103—24no—
268#103—1664no—
269#103—1662no—
270#102—1662no—
271C22⋊C8—832no—
272C22⋊C8—1661no—
273C4×C8—832no—
274C4×C8—1664no—
275C4×C8—1661no—
276#41—3322no—
277#41—3322no—
278C2×Q8—21no—
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8—41no—
282C2×Q8—82no—
283C2×Q8—1661no—
284C2×Q8#1305057714yescomplemented
285C2×Q8C2×C22⋊D8312yes—
286C2×D4—3321no—
287C2×D4—1661no—
288C2×D4—1661no—
289C2×D4—1661no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—41no—
293C22×C4—3321no—
294C22×C4D4×D8311yescharacteristic
295C22×C4C2×C22⋊D8311yescharacteristic
296C22×C4#1305058911yescharacteristic
297Q16—3324no—
298SD16—3324no—
299SD16—6644no—
300M16—3322no—
301C2×C8—22no—
302C2×C8—42no—
303C2×C8—834no—
304C2×C8—1662no—
305C2×C8—1662no—
306C4⋊C4—22no—
307C4⋊C4—21no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—1661no—
314C4⋊C4—1661no—
315C4⋊C4C2×C22⋊D8312yes—
316C22⋊C4—3321no—
317C22⋊C4—3321no—
318C42—22no—
319C42—22no—
320C42D4×D8311yescharacteristic
321C42C2×C22⋊D8311yescharacteristic
322C42#1305058911yescharacteristic
323C42C23×D8311yescharacteristic
324C23—3321no—
325C23#1458176411yescharacteristic
326Q8—24no—
327Q8—22no—
328Q8—44no—
329Q8—82no—
330Q8—3321no—
331Q8—3321no—
332Q8C2×#1305057714yes—
333D4—3321no—
334D4—3321no—
335D4—3322no—
336D4—6641no—
337D4—6642no—
338C2×C4—24no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—81no—
345C2×C4—82no—
346C2×C4—3321no—
347C2×C4—6642no—
348C2×C4#1458176411yescharacteristic
349C2×C4#1458176411yescharacteristic
350C2×C4#1458176411yescharacteristic
351C2×C4C2×#1305057712yes—
352C2×C4#1458181212yes—
353C2×C4C2×D4×D8311yescharacteristic
354C2×C4C22×C22⋊D8311yescharacteristic
355C2×C4C2×#1305058911yescharacteristic
356C8—42no—
357C8—3322no—
358C8—3322no—
359C8—1664no—
360C22—21no—
361C22—21no—
362C22—6641no—
363C22—6641no—
364C22—6641no—
365C22#1912793212yes—
366C22C2×#1458176411yescharacteristic
367C4—24no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—42no—
372C4—82no—
373C4—6641no—
374C4#1912793212yes—
375C4C22×#1305057711yescharacteristic
376C4C2×#1458181211yescharacteristic
377C4#1912850712yes—
378C2—22no—
379C2—13281no—
380C2#2491698111yescharacteristic
381C2C2×#1912793211yescharacteristic
382C2#2491719711yescharacteristic
subgroupG/H
centerC22C2×#14581764
commutator, cosocleC42×C83C24
FrattiniC42C23×D83
FittingC2×#475×C83C2
socleC22×C83C2×#121
prime radical#19128251C22
hypercenterC2×#475D83
hypocenterC83#19397
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#475

Complements

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

NH ≅ G/Ntype
3. #24916544C2semidirect
4. #24906891C2semidirect
5. #24906813C2semidirect
6. #24906233C2semidirect
7. #24906148C2semidirect
9. C2×#475×C83C2semidirect
108. C2×Q8×C83SD16semidirect
193. C2×#475D83semidirect
201. C83#19397semidirect
284. C2×Q8#13050577semidirect

Maximal subgroups

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

subgroupindex
#249183882
#249169352
#249165442
#249068912
#249068132
#249062332
#249061482
#249061132
C2×#475×C832
#1939783