Finite Group Databasebeta

← #35408646

Subgroups — #35408646

Order 21248.

798 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
1C2×C4×#13050580C211yesmaximal, characteristic
2#24909941C211yesmaximal, characteristic
3#24909937C211yesmaximal, characteristic
4#24908331C212yesmaximal
5#24908200C212yesmaximal
6#24908154C212yesmaximal
7#24908150C212yesmaximal
8C4×#14581723C211yesmaximal, characteristic
9C4×#14581717C211yesmaximal, characteristic
10C4×#14581702C211yesmaximal, characteristic
11C4×#103×C83C211yesmaximal, characteristic
12C22×#13050580C2211yescharacteristic
13C2×C4×Dic166C2211yescharacteristic
14C4×#13050580C418yescomplemented
15C2×#14581725C2211yescharacteristic
16C2×#14581723C2212yes—
17C2×#14581717C2212yes—
18C2×#14581716C2211yescharacteristic
19C2×#14581715C2211yescharacteristic
20C2×#14581702C2212yes—
21C2×#14581702C2212yes—
22C2×C4×C83⋊C8C2211yescharacteristic
23#19127608C2212yes—
24#19127606C2212yes—
25#19127600C2212yes—
26#19127597C2212yes—
27#19127587—22no—
28#19127583—22no—
29C4×Dic83⋊C4—21no—
30C4×C4⋊Dic83—21no—
31C4×C4⋊Dic83C2211yescharacteristic
32#19127547—21no—
33#19127547—21no—
34C42×Dic83C2211yescharacteristic
35#19127464C2212yes—
36#19127463C2212yes—
37C2×C4×Q8×C83C2211yescharacteristic
38C2×#103×C83C2212yes—
39C2×C4×C8×C83C2211yescharacteristic
40C4×C4⋊C4×C83C2211yescharacteristic
41#340×C83C2212yes—
42C22×Dic166C2311yescharacteristic
43C2×#13050580—24no—
44C2×#13050580C2×C414yes—
45C22×C83⋊C8C2311yescharacteristic
46C22×C83⋊C8C2311yescharacteristic
47C4×Dic166—22no—
48C4×Dic166C2×C414yes—
49C2×C4⋊Dic83—22no—
50C2×C4⋊Dic83C2312yes—
51C2×C4⋊Dic83C2311yescharacteristic
52C2×Dic83⋊C4—22no—
53C2×Dic83⋊C4—21no—
54C2×C4×Dic83—21no—
55C2×C4×Dic83—21no—
56C2×C4×Dic83D411yescharacteristic
57C2×C4×Dic83D411yescharacteristic
58C2×C4×Dic83D412yes—
59C2×C4×Dic83C2312yes—
60#14581725C2×C414yes—
61#14581723—24no—
62#14581717—24no—
63#14581716C2×C414yes—
64#14581715C2×C414yes—
65#14581702—24no—
66#14581702—24no—
67C4×C83⋊C8C2×C414yes—
68C83⋊#95—22no—
69C83⋊#95—22no—
70C83⋊#95—22no—
71C83⋊#95—22no—
72C83⋊#95—22no—
73C83⋊#95—22no—
74C83⋊#95—21no—
75C22×Q8×C83C2311yescharacteristic
76C22×C8×C83C2312yes—
77C4×Q8×C83—22no—
78C4×Q8×C83C2×C414yes—
79C2×C4⋊C4×C83—21no—
80C2×C4⋊C4×C83C2312yes—
81C2×C4⋊C4×C83C2311yescharacteristic
82C2×C42×C83—21no—
83C2×C42×C83—21no—
84C2×C42×C83C2311yescharacteristic
85#103×C83—24no—
86C4×C8×C83—22no—
87#95×C83—21no—
88C2×Dic166—22no—
89C2×Dic166—22no—
90C2×Dic166—21no—
91C2×Dic166C22×C412yes—
92C22×Dic83—21no—
93C22×Dic83—21no—
94C22×Dic83—22no—
95C22×Dic83C2×D412yes—
96C22×Dic83C2×D412yes—
97#13050580—28no—
98C2×C83⋊C8—22no—
99C2×C83⋊C8C22×C412yes—
100C2×C83⋊C8C22×C412yes—
101C2×C83⋊C8#4114yes—
102C4⋊Dic83—21no—
103C4⋊Dic83—22no—
104C4⋊Dic83—22no—
105C4⋊Dic83—44no—
106C4⋊Dic83C22×C412yes—
107Dic83⋊C4—24no—
108Dic83⋊C4—44no—
109C4×Dic83—24no—
110C4×Dic83—22no—
111C4×Dic83—22no—
112C4×Dic83—22no—
113C4×Dic83—22no—
114C4×Dic83—42no—
115C4×Dic83—42no—
116C2×Q8×C83—21no—
117C2×Q8×C83—22no—
118C2×Q8×C83—22no—
119C2×Q8×C83C22×C412yes—
120C22×C4×C83—21no—
121C22×C4×C83—21no—
122C22×C4×C83—22no—
123C22×C4×C83C2×D411yescharacteristic
124C22×C4×C83C2×D411yescharacteristic
125C22×C4×C83C2411yescharacteristic
126C2×C8×C83—22no—
127C2×C8×C83—22no—
128C2×C8×C83#4114yes—
129C4⋊C4×C83—24no—
130C4⋊C4×C83—21no—
131C4⋊C4×C83—22no—
132C4⋊C4×C83—22no—
133C4⋊C4×C83C22×C412yes—
134C42×C83—24no—
135C42×C83—42no—
136C42×C83C22×C412yes—
137C42×C83#4112yes—
138C23×C83C22×D411yescharacteristic
139Q8×C83—22no—
140Q8×C83—22no—
141Q8×C83—42no—
142C2×C4×C83—22no—
143C2×C4×C83—22no—
144C2×C4×C83—22no—
145C2×C4×C83—42no—
146C2×C4×C83—42no—
147C2×C4×C83—42no—
148C2×C4×C83—42no—
149C2×C4×C83—42no—
150C2×C4×C83C4×D412yes—
151C2×C4×C83C4×D412yes—
152C2×C4×C83#13712yes—
153C2×C4×C83#13712yes—
154C2×C4×C83C23×C411yescharacteristic
155C2×C4×C83C2×#4112yes—
156C2×C4×C83C2×#4111yescharacteristic
157C2×C4×C83C2×#4112yes—
158C2×Dic83—22no—
159C2×Dic83—22no—
160C2×Dic83—22no—
161C2×Dic83—22no—
162C2×Dic83—22no—
163C2×Dic83—22no—
164C2×Dic83—22no—
165C2×Dic83—22no—
166C2×Dic83—42no—
167C2×Dic83—42no—
168C2×Dic83—42no—
169C2×Dic83—42no—
170C2×Dic83—42no—
171Dic166—22no—
172Dic166—22no—
173Dic166—42no—
174C8×C83—24no—
175C83⋊C8—22no—
176C83⋊C8#11714yes—
177C22×C83C2×C4×D411yescharacteristic
178C22×C83#53212yes—
179C22×C83C2×#13711yescharacteristic
180C22×C83C2×#13711yescharacteristic
181C22×C83#57512yes—
182C4×C83—22no—
183C4×C83—42no—
184C4×C83—42no—
185C4×C83—82no—
186C4×C83#44114yes—
187C4×C83C2×#11711yescharacteristic
188C4×C83C4×#4111yescharacteristic
189C4×C83#52912yes—
190Dic83—42no—
191Dic83—42no—
192Dic83—42no—
193Dic83—42no—
194Dic83—82no—
195#21278—831nomaximal, Sylow
196C2×C83#229311yescharacteristic
197C2×C83C2×#44111yescharacteristic
198C2×C83#294311yescharacteristic
199C2×C83#317212yes—
200C2×C83#317212yes—
201C2×C4×Q16—831no—
202#2295—831no—
203#2295—831no—
204#1971—832no—
205#1943—832no—
206#1935—832no—
207#1935—832no—
208C4×#106—831no—
209C4×#103—831no—
210C4×#103D8311yescharacteristic
211C4×#97—831no—
212C83#2127811yescharacteristic, complemented, Sylow
213C22×Q16—831no—
214C2×C4×Q8—831no—
215C2×C4×Q8C2×D8311yescharacteristic
216C4×Q16—838no—
217C2×#107—831no—
218C2×#106—832no—
219C2×#103—831no—
220C2×#103—831no—
221C2×#103—832no—
222C2×#103C2×D8312yes—
223C2×#97—832no—
224C2×#97—832no—
225C2×C4×C8—831no—
226C2×C4×C8C2×D8311yescharacteristic
227#391—832no—
228#391—832no—
229#389—832no—
230#389—832no—
231#387—1662no—
232#387—1662no—
233C4×C4⋊C4—831no—
234C4×C4⋊C4—1661no—
235C4×C4⋊C4—1661no—
236C4×C4⋊C4C2×D8311yescharacteristic
237#376—1661no—
238#376—1661no—
239C43—831no—
240#340—832no—
241#340—832no—
242#340C2×D8312yes—
243C22×Q8—831no—
244C22×Q8C22×D8311yescharacteristic
245C2×Q16—834no—
246C2×Q16—1664no—
247C22×C8—831no—
248C22×C8—831no—
249C22×C8C22×D8312yes—
250C4×Q8—22no—
251C4×Q8—834no—
252C4×Q8—1662no—
253C4×Q8C4×D8314yes—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—831no—
256C2×C4⋊C4—832no—
257C2×C4⋊C4—1662no—
258C2×C4⋊C4—1662no—
259C2×C4⋊C4—1661no—
260C2×C4⋊C4C22×D8312yes—
261C2×C4⋊C4C22×D8311yescharacteristic
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—832no—
265C2×C42—831no—
266C2×C42—831no—
267C2×C42—832no—
268C2×C42—1661no—
269C2×C42—1661no—
270C2×C42C22×D8311yescharacteristic
271#107—834no—
272#106—1664no—
273#103—24no—
274#103—834no—
275#103—834no—
276#103—1664no—
277#97—1664no—
278#97—1664no—
279C4×C8—22no—
280C4×C8—834no—
281#95—21no—
282#95—1662no—
283#95—1662no—
284#95—1662no—
285#95—1662no—
286#95—1662no—
287#95—1662no—
288#95—1661no—
289C2×Q8—21no—
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—832no—
293C2×Q8—1662no—
294C2×Q8—1662no—
295C2×Q8—1661no—
296C2×Q8C2×C4×D8312yes—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—832no—
301C22×C4—832no—
302C22×C4—1662no—
303C22×C4—1661no—
304C22×C4—1661no—
305C22×C4D4×D8311yescharacteristic
306C22×C4D4×D8311yescharacteristic
307C22×C4C23×D8311yescharacteristic
308Q16—1668no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—832no—
312C2×C8—834no—
313C2×C8—832no—
314C2×C8—1662no—
315C2×C8#1305059014yes—
316C4⋊C4—21no—
317C4⋊C4—24no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—3324no—
321C4⋊C4—3324no—
322C4⋊C4—832no—
323C4⋊C4—1662no—
324C4⋊C4—1662no—
325C4⋊C4—1661no—
326C4⋊C4—1664no—
327C4⋊C4C2×C4×D8312yes—
328C42—24no—
329C42—42no—
330C42—3322no—
331C42—3322no—
332C42—1662no—
333C42—1662no—
334C42—1662no—
335C42—1662no—
336C42—1664no—
337C42C2×C4×D8312yes—
338C42#1305058912yes—
339C23C2×D4×D8311yescharacteristic
340Q8—22no—
341Q8—22no—
342Q8—42no—
343Q8—3322no—
344Q8—1662no—
345Q8—1662no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—3322no—
355C2×C4—3322no—
356C2×C4—3322no—
357C2×C4—3322no—
358C2×C4—3322no—
359C2×C4—1662no—
360C2×C4—1662no—
361C2×C4—1662no—
362C2×C4—1662no—
363C2×C4—1662no—
364C2×C4—1662no—
365C2×C4—1662no—
366C2×C4—1662no—
367C2×C4(C22⋊D83)⋊C412yes—
368C2×C4(C22⋊D83)⋊C412yes—
369C2×C4#1458182112yes—
370C2×C4#1458182412yes—
371C2×C4C22×C4×D8311yescharacteristic
372C2×C4#41×D8312yes—
373C2×C4C2×#1305058911yescharacteristic
374C2×C4C2×#1305059012yes—
375C8—24no—
376C8—834no—
377C8—1662no—
378C22C2×(C22⋊D83)⋊C411yescharacteristic
379C22#1912824812yes—
380C22C2×#1458182111yescharacteristic
381C22C2×#1458182411yescharacteristic
382C22#1912849812yes—
383C4—22no—
384C4—42no—
385C4—42no—
386C4—82no—
387C4—3322no—
388C4—3322no—
389C4—3322no—
390C4—3322no—
391C4—6642no—
392C4#1912780514yes—
393C4#117×D8311yescharacteristic
394C4C4×#1305059011yescharacteristic
395C4#1912823112yes—
396C2#2490992611yescharacteristic
397C2C2×#1912780511yescharacteristic
398C2#2491625911yescharacteristic
399C2#2491806512yes—
400C2#2491806712yes—
subgroupG/H
centerC22×C4D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC4×#103×C83C2
socle, prime radicalC23×C83C22×D4
cosocleC22×C4×C83C24
hypercenterC4×#103D83
hypocenterC83#21278
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C4×#103

Complements

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

NH ≅ G/Ntype
14. C4×#13050580C4semidirect
212. C83#21278semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#130505802
#249099412
#249099372
#249083312
#249082002
#249081542
#249081502
C4×#145817232
C4×#145817172
C4×#145817022
C4×#103×C832
#2127883