Finite Group Databasebeta

← #35375259

Subgroups — #35375259

Order 21248.

584 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24918018C211yesmaximal, characteristic
2#24916560C211yesmaximal, characteristic, complemented
3#24916539C211yesmaximal, characteristic
4#24906344C212yesmaximal, complemented
5#24906317C212yesmaximal, complemented
6#24906309C212yesmaximal, complemented
7#24906296C212yesmaximal, complemented
8#24906224C211yesmaximal, characteristic, complemented
9#24905937C211yesmaximal, characteristic, complemented
10#24905756C211yesmaximal, characteristic, complemented
11C83×#1496C211yesmaximal, characteristic
12#19128258C2211yescharacteristic
13#19128097C411yescharacteristic
14#19128073C411yescharacteristic
15#19128053C411yescharacteristic
16#19128047C411yescharacteristic
17#19127928C412yes—
18#19127928C412yes—
19#19127829—22no—
20#19127828C2212yes—
21#19127825C2212yes—
22C4×#13050578C2212yes—
23C4×#13050578C2212yes—
24#19127779C2212yes—
25#19127773—22no—
26D4×C83⋊C8C2212yes—
27#19127759—22no—
28#19127758C2211yescharacteristic
29#19127754C2211yescharacteristic
30#19127668—22no—
31C2×C4×C83⋊C8C2211yescharacteristic
32#19127453C2212yes—
33#19127451C2212yes—
34#19127433C2212yes—
35#19127433C2212yes—
36#19127423C2211yescharacteristic
37#19127423C2211yescharacteristic
38#19127422C2211yescharacteristic
39#19127422C2211yescharacteristic
40#532×C83C2211yescharacteristic
41#432×C83C2211yescharacteristic
42#431×C83C2211yescharacteristic
43D4⋊C8×C83C2212yes—
44D4⋊C8×C83C2212yes—
45C2×#13050578C2×C412yes—
46C2×#13050578C2×C412yes—
47C2×#13050574—22no—
48C22×C83⋊C8—22no—
49C22×C83⋊C8C2×C411yescharacteristic
50C22×C83⋊C8C2×C411yescharacteristic
51#14581792C2×C411yescharacteristic
52#14581792C2×C411yescharacteristic
53#14581784—21no—
54#14581771—21no—
55#14581771C2×C412yes—
56C4×Dic166—21no—
57C4×Dic166C2312yes—
58#14581751—21no—
59#14581723C2×C412yes—
60#14581723C2×C412yes—
61C4⋊(C83⋊C8)D412yes—
62C4⋊(C83⋊C8)C2312yes—
63Dic83⋊C8—22no—
64Dic83⋊C8—22no—
65#14581715C2×C412yes—
66#14581715C2×C412yes—
67#14581713C2×C412yes—
68#14581713C2×C412yes—
69C22⋊(C83⋊C8)—22no—
70C22⋊(C83⋊C8)—22no—
71C22⋊(C83⋊C8)—21no—
72C22⋊(C83⋊C8)D412yes—
73#14581703—21no—
74#14581702—22no—
75C4×C83⋊C8C2×C411yescharacteristic
76C4×C83⋊C8C2×C411yescharacteristic
77C4×C83⋊C8C2311yescharacteristic
78C4×C83⋊C8C2311yescharacteristic
79C4×C83⋊C8C2311yescharacteristic
80C4×C83⋊C8C2311yescharacteristic
81C8×Dic83—22no—
82C2×#41×C83—21no—
83C4⋊Q8×C83C2×C411yescharacteristic
84#127×C83C2×C411yescharacteristic
85#124×C83—21no—
86#121×C83—21no—
87#121×C83C2×C412yes—
88C4×D4×C83—21no—
89C4×D4×C83C2312yes—
90C2×C42×C83C2311yescharacteristic
91C4⋊C8×C83—21no—
92C4⋊C8×C83C2312yes—
93C22⋊C8×C83—21no—
94C22⋊C8×C83—21no—
95#97×C83—21no—
96C4×C8×C83C2311yescharacteristic
97C4×C8×C83C2311yescharacteristic
98C2×Dic166—21no—
99C2×Dic166C22×C412yes—
100#13050578—42no—
101#13050578—42no—
102#13050574—82no—
103C2×C83⋊C8—22no—
104C2×C83⋊C8—41no—
105C2×C83⋊C8—41no—
106C2×C83⋊C8—82no—
107C2×C83⋊C8C22×C411yescharacteristic
108C2×C83⋊C8C22×C411yescharacteristic
109C2×C83⋊C8C22×C411yescharacteristic
110C2×C83⋊C8C22×C411yescharacteristic
111C2×C83⋊C8C2×D412yes—
112C2×C83⋊C8C2×D412yes—
113C4⋊Dic83—21no—
114C4⋊Dic83—21no—
115C4⋊Dic83—21no—
116C4⋊Dic83C22×C412yes—
117Dic83⋊C4—22no—
118Dic83⋊C4—22no—
119C22⋊Dic83—22no—
120C4×Dic83—22no—
121#41×C83—82no—
122C2×Q8×C83—21no—
123C2×D4×C83—22no—
124C2×D4×C83—21no—
125C2×D4×C83—21no—
126C2×D4×C83C22×C412yes—
127C22×C4×C83—22no—
128C22×C4×C83C22×C411yescharacteristic
129C22×C4×C83C2×D411yescharacteristic
130C22×C4×C83#4111yescharacteristic
131C2×C8×C83—22no—
132C2×C8×C83—21no—
133C2×C8×C83—21no—
134C4⋊C4×C83—21no—
135C4⋊C4×C83C22×C412yes—
136C22⋊C4×C83—22no—
137C22⋊C4×C83—22no—
138C42×C83C22×C411yescharacteristic
139C42×C83C2×D411yescharacteristic
140C42×C83#4111yescharacteristic
141C42×C83C2411yescharacteristic
142C23×C83—22no—
143C23×C83C4×D411yescharacteristic
144Q8×C83—81no—
145D4×C83—22no—
146D4×C83—22no—
147D4×C83—81no—
148D4×C83—81no—
149D4×C83—82no—
150C2×C4×C83—21no—
151C2×C4×C83—21no—
152C2×C4×C83—22no—
153C2×C4×C83—22no—
154C2×C4×C83—41no—
155C2×C4×C83—42no—
156C2×C4×C83—42no—
157C2×C4×C83C4×D411yescharacteristic
158C2×C4×C83C4×D411yescharacteristic
159C2×C4×C83C4×D411yescharacteristic
160C2×C4×C83C23×C411yescharacteristic
161C2×C4×C83C22×D411yescharacteristic
162C2×C4×C83C2×#4111yescharacteristic
163C2×Dic83—22no—
164C2×Dic83—22no—
165Dic166—22no—
166Dic166—22no—
167Dic166—81no—
168C8×C83—41no—
169C8×C83—41no—
170C8×C83—42no—
171C83⋊C8—21no—
172C83⋊C8—21no—
173C83⋊C8—21no—
174C83⋊C8—21no—
175C83⋊C8—22no—
176C83⋊C8—22no—
177C83⋊C8—42no—
178C22×C83—21no—
179C22×C83—21no—
180C22×C83—21no—
181C22×C83—42no—
182C22×C83—42no—
183C22×C83—42no—
184C22×C83C2×C4×D411yescharacteristic
185C4×C83—21no—
186C4×C83—21no—
187C4×C83—21no—
188C4×C83—21no—
189C4×C83—42no—
190C4×C83#56811yescharacteristic
191C4×C83#56811yescharacteristic
192C4×C83#57712yes—
193Dic83—42no—
194Dic83—42no—
195#17990—831nomaximal, Sylow
196C2×C83—41no—
197C2×C83—82no—
198C2×C83#293211yescharacteristic
199C2×C83#294911yescharacteristic
200C2×C83#295711yescharacteristic
201#3167—831no—
202#2985—831no—
203#2981—831no—
204#1588—832no—
205#1585—832no—
206#1583—832no—
207#1582—832no—
208#1568—831no—
209#1497—831no—
210#1496D8311yescharacteristic
211#1452—831no—
212C83#1799011yescharacteristic, complemented, Sylow
213#533—831no—
214#532C2×D8311yescharacteristic
215#498—831no—
216#492—831no—
217#487—831no—
218#486—831no—
219#465—832no—
220#465—832no—
221#446—832no—
222#446—1662no—
223C8×Q8—832no—
224C4×SD16—832no—
225C4×SD16—832no—
226#436—832no—
227#435—1662no—
228C8×D4—832no—
229#433—1662no—
230#432—831no—
231#432C2×D8311yescharacteristic
232#431—831no—
233#431C2×D8311yescharacteristic
234#408—1662no—
235C2×C4×C8—831no—
236#335—832no—
237#334—832no—
238Q8⋊C8—832no—
239Q8⋊C8—832no—
240D4⋊C8C2×D8312yes—
241D4⋊C8C2×D8312yes—
242#322—831no—
243#322—831no—
244#322—831no—
245#322—831no—
246C2×#41—21no—
247C2×SD16—832no—
248C2×SD16—832no—
249C2×M16—1662no—
250C22×C8—831no—
251C22×C8—831no—
252C22×C8—1662no—
253C4⋊Q8—831no—
254C4⋊Q8—831no—
255C4⋊Q8C4×D8311yescharacteristic
256#127C4×D8311yescharacteristic
257#125—1661no—
258#124—21no—
259C22⋊Q8—832no—
260C22⋊Q8—1661no—
261#121—21no—
262#121C4×D8312yes—
263C4×Q8—832no—
264C4×Q8—1661no—
265C4×D4—21no—
266C4×D4C22×D8312yes—
267#117—1661no—
268C2×C42C22×D8311yescharacteristic
269#106—832no—
270#106—832no—
271C4⋊C8—21no—
272C4⋊C8—832no—
273C4⋊C8—832no—
274C4⋊C8—1662no—
275C4⋊C8—1662no—
276C4⋊C8C22×D8312yes—
277#103—832no—
278#103—832no—
279#102—832no—
280#102—832no—
281C22⋊C8—21no—
282C22⋊C8—21no—
283C22⋊C8—832no—
284C22⋊C8—1662no—
285C22⋊C8—1662no—
286C22⋊C8—1661no—
287#97—21no—
288#97—1662no—
289#97—1661no—
290C4×C8—831no—
291C4×C8—831no—
292C4×C8—831no—
293C4×C8—831no—
294C4×C8—831no—
295C4×C8—831no—
296C4×C8—1662no—
297C4×C8C22×D8311yescharacteristic
298C4×C8C22×D8311yescharacteristic
299#41—82no—
300C2×Q8—21no—
301C2×Q8—832no—
302C2×Q8—1661no—
303C2×D4—22no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4C2×C4×D8312yes—
307C22×C4—22no—
308C22×C4C2×C4×D8311yescharacteristic
309C22×C4D4×D8311yescharacteristic
310C22×C4#1305058911yescharacteristic
311SD16—3322no—
312SD16—3322no—
313M16—6642no—
314C2×C8—22no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—3321no—
318C2×C8—3321no—
319C2×C8—831no—
320C2×C8—831no—
321C2×C8—831no—
322C2×C8—831no—
323C2×C8—832no—
324C2×C8—832no—
325C2×C8—6642no—
326C2×C8—1662no—
327C4⋊C4—21no—
328C4⋊C4—832no—
329C4⋊C4—1662no—
330C4⋊C4—1661no—
331C4⋊C4—1661no—
332C4⋊C4—1662no—
333C4⋊C4—1661no—
334C4⋊C4C2×C4×D8312yes—
335C22⋊C4—22no—
336C22⋊C4—22no—
337C22⋊C4—1662no—
338C42—1662no—
339C42C2×C4×D8311yescharacteristic
340C42D4×D8311yescharacteristic
341C42#1305058911yescharacteristic
342C42C23×D8311yescharacteristic
343C23—22no—
344C23(C22⋊D83)⋊C411yescharacteristic
345Q8—81no—
346Q8—6641no—
347Q8—1662no—
348Q8—1662no—
349D4—22no—
350D4—22no—
351D4—81no—
352D4—81no—
353D4—82no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—1662no—
362C2×C4—1662no—
363C2×C4(C22⋊D83)⋊C411yescharacteristic
364C2×C4(C22⋊D83)⋊C411yescharacteristic
365C2×C4(C22⋊D83)⋊C411yescharacteristic
366C2×C4C22×C4×D8311yescharacteristic
367C2×C4C2×D4×D8311yescharacteristic
368C2×C4C2×#1305058911yescharacteristic
369C8—41no—
370C8—41no—
371C8—42no—
372C8—3322no—
373C8—1661no—
374C8—1661no—
375C8—1661no—
376C8—1661no—
377C8—1662no—
378C8—1662no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—42no—
383C22—42no—
384C22—42no—
385C22C2×(C22⋊D83)⋊C411yescharacteristic
386C4—21no—
387C4—21no—
388C4—21no—
389C4—21no—
390C4—42no—
391C4—3322no—
392C4—3322no—
393C4#1912846611yescharacteristic
394C4#1912846611yescharacteristic
395C4#1912851112yes—
396C2—41no—
397C2—82no—
398C2#2491617811yescharacteristic
399C2#2491632511yescharacteristic
400C2#2491637611yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC42C23×D83
FittingC83×#1496C2
socleC22×C83C2×C4×D4
prime radical#532×C83C22
cosocleC42×C83C24
hypercenter#1496D83
hypocenterC83#17990
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1496

Complements

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

NH ≅ G/Ntype
2. #24916560C2semidirect
4. #24906344C2semidirect
5. #24906317C2semidirect
6. #24906309C2semidirect
7. #24906296C2semidirect
8. #24906224C2semidirect
9. #24905937C2semidirect
10. #24905756C2semidirect
212. C83#17990semidirect

Maximal subgroups

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

subgroupindex
#249180182
#249165602
#249165392
#249063442
#249063172
#249063092
#249062962
#249062242
#249059372
#249057562
C83×#14962
#1799083