Finite Group Databasebeta

← #35559117

Subgroups — #35559117

Order 21248.

534 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24917999C211yesmaximal, characteristic, complemented
2#24916339C211yesmaximal, characteristic
3#24909625C212yesmaximal, complemented
4#24909622C212yesmaximal, complemented
5#24909613C212yesmaximal, complemented
6#24909598C212yesmaximal, complemented
7#24909251C211yesmaximal, characteristic, complemented
8C2×#19127526C211yesmaximal, characteristic, complemented
9#24909223C211yesmaximal, characteristic, complemented
10#24909170C211yesmaximal, characteristic
11C83×#2139C211yesmaximal, characteristic
12#19128332—21no—
13#19128184—21no—
14C22×C83⋊C16C2211yescharacteristic
15#19128109—22no—
16#19128105C2212yes—
17#19128102C2212yes—
18#19127802—22no—
19Q16×Dic83C2212yes—
20D8×Dic83C2212yes—
21#19127741—21no—
22#19127739C2211yescharacteristic
23C2×#14581726C2211yescharacteristic
24#19127702—21no—
25#19127656C2211yescharacteristic
26#19127529C2212yes—
27#19127527C2212yes—
28#19127526C2212yes—
29#19127526C2212yes—
30Dic83⋊C16C2212yes—
31Dic83⋊C16C2212yes—
32#19127505C2211yescharacteristic
33#19127505C2211yescharacteristic
34#19127502C2211yescharacteristic
35#19127502C2211yescharacteristic
36C2×#135×C83C2211yescharacteristic
37C2×M32×C83C2211yescharacteristic
38C2×#107×C83C2211yescharacteristic
39#358×C83C2212yes—
40#357×C83C2212yes—
41C2×#13050574D411yescharacteristic
42#14581791—22no—
43#14581791—41no—
44#14581783—22no—
45#14581783—41no—
46#14581771—42no—
47Q8×Dic83—22no—
48D4×Dic83—22no—
49D4×Dic83—42no—
50#14581751—21no—
51#14581751—41no—
52#14581751D411yescharacteristic
53C2×C4⋊Dic83—21no—
54C2×C4⋊Dic83—41no—
55C2×C83⋊C16—22no—
56C2×C83⋊C16C2311yescharacteristic
57C2×C83⋊C16C2311yescharacteristic
58#14581726—21no—
59#14581726C2311yescharacteristic
60#14581726C2311yescharacteristic
61#14581726C2312yes—
62#14581725—22no—
63#14581724—21no—
64#14581723—22no—
65#14581716—22no—
66#14581713—22no—
67#14581702D412yes—
68C8×Dic83C2312yes—
69C2×#41×C83—21no—
70#135×C83—22no—
71#135×C83Q814yescomplemented
72C2×Q16×C83C2311yescharacteristic
73C2×SD16×C83—21no—
74C2×D8×C83C2311yescharacteristic
75C22×C8×C83C2311yescharacteristic
76C2×C4⋊C4×C83—21no—
77M32×C83—22no—
78C2×C16×C83C2312yes—
79#107×C83—21no—
80#107×C83C2312yes—
81C2×Dic166—42no—
82C22×Dic83—41no—
83C22×Dic83—42no—
84#13050574—22no—
85C2×C83⋊C8C2×D412yes—
86C4⋊Dic83—21no—
87C4⋊Dic83—21no—
88C4⋊Dic83—22no—
89C4⋊Dic83—41no—
90C4⋊Dic83—42no—
91C4⋊Dic83—42no—
92C4⋊Dic83D811yescharacteristic
93C4⋊Dic83D811yescharacteristic
94Dic83⋊C4—42no—
95Dic83⋊C4—42no—
96C22⋊Dic83—41no—
97C22⋊Dic83—42no—
98C22⋊Dic83D812yes—
99C4×Dic83—41no—
100C4×Dic83—42no—
101C4×Dic83C2×D412yes—
102C83⋊C16—21no—
103C83⋊C16—21no—
104#41×C83—24no—
105#41×C83—42no—
106C2×Q8×C83—21no—
107C2×D4×C83—21no—
108C2×D4×C83—41no—
109C22×C4×C83—41no—
110C22×C4×C83—41no—
111C22×C4×C83C2×D411yescharacteristic
112Q16×C83—21no—
113Q16×C83C2×Q812yes—
114SD16×C83—24no—
115D8×C83—21no—
116D8×C83C2×Q812yes—
117C2×C8×C83C2×D411yescharacteristic
118C2×C8×C83C2×Q812yes—
119C2×C8×C83#4112yes—
120C2×C8×C83C2411yescharacteristic
121C4⋊C4×C83—22no—
122C4⋊C4×C83—41no—
123C16×C83—22no—
124C23×C83—41no—
125C23×C83C2×D811yescharacteristic
126Q8×C83—22no—
127Q8×C83—41no—
128D4×C83—22no—
129D4×C83—44no—
130D4×C83—41no—
131C2×C4×C83—41no—
132C2×C4×C83—44no—
133C2×C4×C83—42no—
134C2×C4×C83—161no—
135C2×C4×C83C22⋊Q812yes—
136C2×C4×C83C2×D811yescharacteristic
137C2×C4×C83#13712yes—
138C2×C4×C83C22×D411yescharacteristic
139C2×Dic83—41no—
140C2×Dic83—41no—
141C2×Dic83—42no—
142C2×Dic83—162no—
143C2×Dic83—161no—
144C2×Dic83C2×D812yes—
145C2×Dic83C2×D812yes—
146Dic166—162no—
147C8×C83C22⋊Q812yes—
148C8×C83C22×Q811yescharacteristic
149C8×C83C2×#4111yescharacteristic
150C83⋊C8—22no—
151C22×C83—21no—
152C22×C83—21no—
153C22×C83—42no—
154C22×C83—81no—
155C22×C83—81no—
156C22×C83#47412yes—
157C22×C83C22×D811yescharacteristic
158C4×C83—42no—
159C4×C83—82no—
160C4×C83#47412yes—
161C4×C83C2×C22⋊Q811yescharacteristic
162C4×C83C2×#13711yescharacteristic
163Dic83—22no—
164Dic83—22no—
165Dic83—81no—
166Dic83—82no—
167Dic83—81no—
168#33633—831nomaximal, Sylow
169C2×C83—22no—
170C2×C83—82no—
171C2×C83C2×#47411yescharacteristic
172C2×C83#341511yescharacteristic
173C2×C83#341711yescharacteristic
174#3164—831no—
175#2951—831no—
176#2229—832no—
177#2228—832no—
178#2227—832no—
179#2224—832no—
180#2159—831no—
181C2×#366—831no—
182#2151—831no—
183#2139D8311yescharacteristic
184#2138—831no—
185C83#3363311yescharacteristic, complemented, Sylow
186C2×#135C2×D8311yescharacteristic
187#544—1661no—
188#520—1661no—
189C2×M32C2×D8311yescharacteristic
190C22×C16—831no—
191#501—1662no—
192#500—832no—
193#499—832no—
194#440—1662no—
195C4×Q16—832no—
196C4×D8—832no—
197#428—1661no—
198#427—831no—
199C2×#107—831no—
200C2×#107C2×D8311yescharacteristic
201#417—1661no—
202#405—831no—
203#367—832no—
204#366—832no—
205#366—832no—
206#366—832no—
207C4⋊C16—832no—
208C4⋊C16—832no—
209#358—831no—
210#358—831no—
211#358C2×D8312yes—
212#357—831no—
213#357—831no—
214#357C2×D8312yes—
215C2×#41—21no—
216#135—22no—
217#135Dic16614yescomplemented
218C2×Q16C22×D8311yescharacteristic
219C2×SD16—21no—
220C2×D8C22×D8311yescharacteristic
221C2×M16—831no—
222C22×C8C22×D8311yescharacteristic
223C4⋊Q8—3321no—
224C4⋊Q8—1662no—
225#125—3321no—
226#125—1662no—
227C22⋊Q8—3322no—
228C4×Q8—1662no—
229C4×D4—3322no—
230C4×D4—1662no—
231#117—3321no—
232#117—831no—
233#117—1661no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—3321no—
236C2×C4⋊C4—1661no—
237M32—22no—
238C2×C16—831no—
239C2×C16—831no—
240C2×C16—1662no—
241C2×C16C22×D8312yes—
242#107—21no—
243#107—831no—
244#107—831no—
245#107—832no—
246#107—1662no—
247#107—1661no—
248#107C22×D8312yes—
249#106—1662no—
250#106—1661no—
251#103—1662no—
252#102—1662no—
253#97—832no—
254C4×C8—832no—
255#41—24no—
256#41—42no—
257C2×Q8—21no—
258C2×Q8—3322no—
259C2×D4—21no—
260C2×D4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—3321no—
264C22×C4—3322no—
265C22×C4D4×D8311yescharacteristic
266Q16—21no—
267Q16C2×Dic16612yes—
268SD16—24no—
269D8—21no—
270D8C2×Dic16612yes—
271M16—1662no—
272C2×C8—832no—
273C2×C8D4×D8311yescharacteristic
274C2×C8C2×Dic16612yes—
275C2×C8#1305058912yes—
276C2×C8C23×D8311yescharacteristic
277C4⋊C4—22no—
278C4⋊C4—41no—
279C4⋊C4—3322no—
280C4⋊C4—3322no—
281C4⋊C4—3322no—
282C4⋊C4—3322no—
283C4⋊C4—3321no—
284C4⋊C4—831no—
285C4⋊C4—831no—
286C4⋊C4—1662no—
287C4⋊C4—1661no—
288C4⋊C4—1661no—
289C22⋊C4—3321no—
290C22⋊C4—3322no—
291C22⋊C4—832no—
292C42—3322no—
293C42—3321no—
294C42—832no—
295C16—22no—
296C16—1661no—
297C16—1661no—
298C23—41no—
299C23D8×D8311yescharacteristic
300Q8—22no—
301Q8—41no—
302Q8—13282no—
303D4—22no—
304D4—44no—
305D4—41no—
306C2×C4—41no—
307C2×C4—44no—
308C2×C4—42no—
309C2×C4—161no—
310C2×C4—13281no—
311C2×C4—13282no—
312C2×C4—3321no—
313C2×C4—3322no—
314C2×C4—3321no—
315C2×C4—832no—
316C2×C4—832no—
317C2×C4#1458176912yes—
318C2×C4D8×D8311yescharacteristic
319C2×C4#1458182112yes—
320C2×C4C2×D4×D8311yescharacteristic
321C8—1662no—
322C8#1458176912yes—
323C8C22×Dic16611yescharacteristic
324C8C2×#1305058911yescharacteristic
325C22—21no—
326C22—21no—
327C22—42no—
328C22—81no—
329C22—81no—
330C22#1912797912yes—
331C22C2×D8×D8311yescharacteristic
332C4—42no—
333C4—82no—
334C4—6642no—
335C4—6641no—
336C4—6641no—
337C4—1662no—
338C4—1662no—
339C4#1912797912yes—
340C4C2×#1458176911yescharacteristic
341C4C2×#1458182111yescharacteristic
342C2—22no—
343C2—82no—
344C2C2×#1912797911yescharacteristic
345C2#2492060311yescharacteristic
346C2#2492061511yescharacteristic
subgroupG/H
centerC22C2×D8×D83
commutator, cosocleC2×C8×C83C24
FrattiniC2×C8C23×D83
FittingC83×#2139C2
socleC22×C83C22×D8
prime radicalC2×#135×C83C22
hypercenter#2139D83
hypocenterC83#33633
LCSG ▷ C2×C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2139

Complements

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

NH ≅ G/Ntype
1. #24917999C2semidirect
3. #24909625C2semidirect
4. #24909622C2semidirect
5. #24909613C2semidirect
6. #24909598C2semidirect
7. #24909251C2semidirect
8. C2×#19127526C2semidirect
9. #24909223C2semidirect
71. #135×C83Q8semidirect
185. C83#33633semidirect
217. #135Dic166semidirect

Maximal subgroups

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

subgroupindex
#249179992
#249163392
#249096252
#249096222
#249096132
#249095982
#249092512
C2×#191275262
#249092232
#249091702
C83×#21392
#3363383