Finite Group Databasebeta

← #35560744

Subgroups — #35560744

Order 21248.

534 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24918042C211yesmaximal, characteristic, complemented
2#24909617C212yesmaximal, complemented
3#24909605C211yesmaximal, characteristic, complemented
4#24909600C211yesmaximal, characteristic, complemented
5#24909578C212yesmaximal, complemented
6#24909564C212yesmaximal, complemented
7#24909515C212yesmaximal, complemented
8#24909503C211yesmaximal, characteristic, complemented
9#24909498C211yesmaximal, characteristic, complemented
10#24909219C211yesmaximal, characteristic, complemented
11C83×#3168C211yesmaximal, characteristic, complemented
12#19128251—21no—
13C2×#14581735C2212yes—
14C2×#14581733C2212yes—
15C2×#14581732C2212yes—
16#19128106C2211yescharacteristic
17#19128078C2211yescharacteristic
18#19128048—21no—
19#19127937—21no—
20#19127937C2212yes—
21C4×D664—21no—
22C4×D664C2212yes—
23#19127528C2212yes—
24#19127526C2212yes—
25C4⋊(C83⋊C16)C2211yescharacteristic
26C4⋊(C83⋊C16)C2211yescharacteristic
27C4⋊(C83⋊C16)C2212yes—
28#19127505C2212yes—
29#19127502C2212yes—
30#19127501C2212yes—
31#19127501C2212yes—
32#529×C83—21no—
33#499×C83C2211yescharacteristic
34#495×C83C2211yescharacteristic
35#488×C83—21no—
36#467×C83C2211yescharacteristic
37#466×C83C2211yescharacteristic
38#465×C83—21no—
39C4×Q16×C83C2211yescharacteristic
40C4×SD16×C83—21no—
41C4×D8×C83C2211yescharacteristic
42C2×C4×C8×C83C2211yescharacteristic
43C2×#13050590—41no—
44C2×D664—21no—
45C2×D664C2312yes—
46#14581792—21no—
47#14581789—21no—
48#14581780—41no—
49#14581766—22no—
50#14581766—41no—
51C4×D332—22no—
52C4×D332—41no—
53#14581735—42no—
54#14581733—42no—
55#14581732—42no—
56C2×C83⋊C16C2312yes—
57C2×C83⋊C16C2312yes—
58#14581726—21no—
59#14581726C2312yes—
60#14581714—22no—
61#14581714—22no—
62C2×Q16×C83C2311yescharacteristic
63C2×SD16×C83—21no—
64C2×D8×C83C2311yescharacteristic
65C22×C8×C83D411yescharacteristic
66C22×C8×C83C2311yescharacteristic
67#126×C83—41no—
68#125×C83—21no—
69#124×C83—21no—
70#123×C83—41no—
71C22⋊Q8×C83—21no—
72#121×C83—21no—
73C4×Q8×C83—21no—
74C4×D4×C83—21no—
75C4×D4×C83—41no—
76#117×C83—41no—
77C2×C42×C83D411yescharacteristic
78#107×C83C2311yescharacteristic
79#107×C83C2311yescharacteristic
80#106×C83—21no—
81#103×C83—21no—
82#103×C83—21no—
83#102×C83—21no—
84#102×C83—21no—
85C4×C8×C83D411yescharacteristic
86C4×C8×C83D411yescharacteristic
87C4×C8×C83C2311yescharacteristic
88C4×C8×C83C2311yescharacteristic
89#13050590—84no—
90C2×Dic166—41no—
91C2×D332—22no—
92C2×D332—41no—
93C2×D332—41no—
94C2×C22⋊D83—42no—
95C2×C4×D83—42no—
96D664—22no—
97D664—22no—
98D664—42no—
99C4⋊Dic83—22no—
100C4⋊Dic83—41no—
101(C2×D83)⋊C4—42no—
102(C2×D83)⋊C4—42no—
103C83⋊C16—22no—
104C83⋊C16—22no—
105C2×Q8×C83—21no—
106C2×D4×C83—21no—
107C2×D4×C83—41no—
108C22×C4×C83—41no—
109C22×C4×C83D811yescharacteristic
110C22×C4×C83D811yescharacteristic
111C22×C4×C83C2×D411yescharacteristic
112Q16×C83—21no—
113Q16×C83—21no—
114SD16×C83—42no—
115D8×C83—21no—
116D8×C83—21no—
117C2×C8×C83—21no—
118C2×C8×C83—21no—
119C2×C8×C83—41no—
120C2×C8×C83C2×D411yescharacteristic
121C2×C8×C83C2×D411yescharacteristic
122C2×C8×C83C2×D411yescharacteristic
123C2×C8×C83C2411yescharacteristic
124C4⋊C4×C83—21no—
125C4⋊C4×C83—21no—
126C4⋊C4×C83—41no—
127C4⋊C4×C83—41no—
128C4⋊C4×C83—41no—
129C4⋊C4×C83—41no—
130C22⋊C4×C83—41no—
131C22⋊C4×C83—41no—
132C22⋊C4×C83—41no—
133C22⋊C4×C83—41no—
134C22⋊C4×C83—41no—
135C42×C83—41no—
136C42×C83D811yescharacteristic
137C42×C83D811yescharacteristic
138C42×C83C2×D411yescharacteristic
139C42×C83C2×D411yescharacteristic
140C23×C83—41no—
141C23×C83C2×D811yescharacteristic
142C22×D83—42no—
143Q8×C83—41no—
144Q8×C83—41no—
145D4×C83—41no—
146D4×C83—41no—
147D4×C83—82no—
148C2×C4×C83—21no—
149C2×C4×C83—21no—
150C2×C4×C83—22no—
151C2×C4×C83—41no—
152C2×C4×C83—41no—
153C2×C4×C83—41no—
154C2×C4×C83—41no—
155C2×C4×C83—82no—
156C2×C4×C83C2×D811yescharacteristic
157C2×C4×C83C2×D811yescharacteristic
158C2×C4×C83C2×D811yescharacteristic
159C2×C4×C83C2×D811yescharacteristic
160C2×C4×C83C2×D811yescharacteristic
161C2×C4×C83C22×D411yescharacteristic
162C22⋊D83—84no—
163C2×Dic83—42no—
164D332—42no—
165D332—42no—
166D332—82no—
167D332—82no—
168C4×D83—84no—
169Dic166—82no—
170C8×C83—21no—
171C8×C83—21no—
172C8×C83—21no—
173C8×C83H(5,𝔽2)11yescharacteristic
174C8×C832−1+411yescharacteristic
175C22×C83—21no—
176C22×C83—21no—
177C22×C83—21no—
178C22×C83—81no—
179C22×C83—81no—
180C22×C83—81no—
181C22×C83C22×D811yescharacteristic
182C2×D83—82no—
183C2×D83—82no—
184C2×D83—82no—
185C4×C83—21no—
186C4×C83—21no—
187C4×C83—21no—
188C4×C83—21no—
189C4×C83—81no—
190C4×C83—81no—
191C4×C83—81no—
192C4×C83#50912yes—
193C4×C83#53711yescharacteristic
194C4×C83#57511yescharacteristic
195Dic83—82no—
196#33817—831nomaximal, Sylow
197C2×C83—41no—
198C2×C83—161no—
199C2×C83#324511yescharacteristic
200C2×C83C2×#50911yescharacteristic
201C2×C83#341411yescharacteristic
202D83—162no—
203#3169—831no—
204#3168D8311yescharacteristic, complemented
205#2228—832no—
206#2225—831no—
207#2224—831no—
208#2220—832no—
209#2218—832no—
210#2210—832no—
211#2207—831no—
212#2206—831no—
213#2150—831no—
214C83#3381711yescharacteristic, complemented, Sylow
215#532—1661no—
216#529—21no—
217C2×SD32—832no—
218C2×D16—832no—
219C2×M32—832no—
220#500—831no—
221#499C2×D8311yescharacteristic
222#495C2×D8311yescharacteristic
223#493—831no—
224#488—21no—
225#486—1661no—
226#467C2×D8311yescharacteristic
227#466—832no—
228#466—1661no—
229#466C2×D8311yescharacteristic
230#465—21no—
231C4×Q16C2×D8311yescharacteristic
232C4×SD16—21no—
233C4×D8—832no—
234C4×D8—1661no—
235C4×D8C2×D8311yescharacteristic
236C2×C4×C8C2×D8311yescharacteristic
237#367—832no—
238#366—832no—
239C4⋊C16—831no—
240C4⋊C16—831no—
241C4⋊C16—832no—
242#358—832no—
243#357—832no—
244#357—832no—
245#357—832no—
246C2×#41—3321no—
247C2×Q16C22×D8311yescharacteristic
248C2×SD16—21no—
249C2×D8—832no—
250C2×D8—1661no—
251C2×D8C22×D8311yescharacteristic
252C22×C8C22⋊D8311yescharacteristic
253C22×C8C22×D8311yescharacteristic
254C4⋊Q8—1661no—
255#127—1661no—
256#126—41no—
257#125—21no—
258#124—21no—
259#124—3321no—
260#123—41no—
261C22⋊Q8—21no—
262#121—21no—
263#121—3321no—
264#121—1662no—
265C4×Q8—21no—
266C4×D4—21no—
267C4×D4—41no—
268C4×D4—3321no—
269C4×D4—1662no—
270#117—41no—
271C2×C42C22⋊D8311yescharacteristic
272SD32—3322no—
273D16—3322no—
274M32—3322no—
275C2×C16—832no—
276C2×C16—832no—
277#107—832no—
278#107—1661no—
279#107C22×D8311yescharacteristic
280#107C22×D8311yescharacteristic
281#106—21no—
282#103—21no—
283#103—21no—
284#102—21no—
285#102—21no—
286#102—1662no—
287#102—1662no—
288C4×C8C22⋊D8311yescharacteristic
289C4×C8C22⋊D8311yescharacteristic
290C4×C8C22×D8311yescharacteristic
291C4×C8C22×D8311yescharacteristic
292#41—6644no—
293C2×Q8—21no—
294C2×Q8—3321no—
295C2×D4—21no—
296C2×D4—41no—
297C2×D4—3322no—
298C2×D4—3321no—
299C2×D4—3321no—
300C2×D4—1662no—
301C22×C4—41no—
302C22×C4—3322no—
303C22×C4#1305057511yescharacteristic
304C22×C4#1305057511yescharacteristic
305C22×C4C2×C22⋊D8311yescharacteristic
306Q16—21no—
307Q16—21no—
308SD16—42no—
309D8—21no—
310D8—21no—
311D8—3322no—
312D8—1662no—
313D8—1662no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—41no—
317C2×C8C2×C22⋊D8311yescharacteristic
318C2×C8C2×C22⋊D8311yescharacteristic
319C2×C8C2×C22⋊D8311yescharacteristic
320C2×C8C23×D8311yescharacteristic
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—3321no—
328C4⋊C4—1662no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—3322no—
335C22⋊C4—3322no—
336C42—41no—
337C42#1305057511yescharacteristic
338C42#1305057511yescharacteristic
339C42C2×C22⋊D8311yescharacteristic
340C42C2×C22⋊D8311yescharacteristic
341C16—1662no—
342C16—1662no—
343C23—41no—
344C23—3322no—
345C23C2×#1305057511yescharacteristic
346Q8—41no—
347Q8—41no—
348Q8—6642no—
349D4—41no—
350D4—41no—
351D4—82no—
352D4—3322no—
353D4—3322no—
354D4—6642no—
355D4—6644no—
356D4—6642no—
357C2×C4—22no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—82no—
365C2×C4—3322no—
366C2×C4—6644no—
367C2×C4C2×#1305057511yescharacteristic
368C2×C4C2×#1305057511yescharacteristic
369C2×C4C2×#1305057511yescharacteristic
370C2×C4C2×#1305057511yescharacteristic
371C2×C4C2×#1305057511yescharacteristic
372C2×C4C22×C22⋊D8311yescharacteristic
373C8—21no—
374C8—21no—
375C8—21no—
376C8#1458183811yescharacteristic
377C8#1458183911yescharacteristic
378C22—21no—
379C22—21no—
380C22—21no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—6642no—
385C22—6642no—
386C22—6642no—
387C22C22×#1305057511yescharacteristic
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—81no—
393C4—81no—
394C4—81no—
395C4—6642no—
396C4#1912813612yes—
397C4#1912827911yescharacteristic
398C4#1912849911yescharacteristic
399C2—41no—
400C2—161no—
401C2—13282no—
402C2#2491877011yescharacteristic
403C2C2×#1912813611yescharacteristic
404C2#2492059711yescharacteristic
subgroupG/H
centerC22C22×#13050575
commutator, cosocleC2×C8×C83C24
FrattiniC2×C8C23×D83
FittingC83×#3168C2
socleC22×C83C22×D8
hypercenter#3168D83
hypocenterC83#33817
LCSG ▷ C2×C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ #3168

Complements

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

NH ≅ G/Ntype
1. #24918042C2semidirect
2. #24909617C2semidirect
3. #24909605C2semidirect
4. #24909600C2semidirect
5. #24909578C2semidirect
6. #24909564C2semidirect
7. #24909515C2semidirect
8. #24909503C2semidirect
9. #24909498C2semidirect
10. #24909219C2semidirect
11. C83×#3168C2semidirect
204. #3168D83semidirect
214. C83#33817semidirect

Maximal subgroups

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

subgroupindex
#249180422
#249096172
#249096052
#249096002
#249095782
#249095642
#249095152
#249095032
#249094982
#249092192
C83×#31682
#3381783