Finite Group Databasebeta

← #35458448

Subgroups — #35458448

Order 21248.

1002 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128051C211yesmaximal, characteristic
2C2×#19127996C211yesmaximal, characteristic
3#24912668C211yesmaximal, characteristic
4#24908706C214yesmaximal
5#24908400C211yesmaximal, characteristic
6#24908387C211yesmaximal, characteristic
7#24907717C214yesmaximal
8C4×#14581716C211yesmaximal, characteristic
9C2×#557×C83C211yesmaximal, characteristic
10C2×#14581792C2211yescharacteristic
11#19128051—24no—
12#19127996—24no—
13C2×#14581725C2212yes—
14C2×C4⋊(C83⋊C8)C2211yescharacteristic
15C2×#14581716C2214yes—
16C2×#14581716C2212yes—
17C2×#14581716C2214yes—
18C2×#14581716C2212yes—
19C2×C4×C83⋊C8C2211yescharacteristic
20#19127621—22no—
21#19127608—21no—
22#19127599C2214yes—
23#19127570—21no—
24C4×C4⋊Dic83C2211yescharacteristic
25#19127463C2212yes—
26#19127456C2214yes—
27#557×C83—24no—
28#557×C83—24no—
29C2×C4⋊Q8×C83C2212yes—
30C2×C4⋊Q8×C83C2211yescharacteristic
31C2×#125×C83—21no—
32C2×#125×C83—22no—
33C2×C4×Q8×C83—21no—
34C2×C4×Q8×C83C2212yes—
35C2×C4×Q8×C83C2211yescharacteristic
36C2×C4×Q8×C83C2211yescharacteristic
37C22×Dic166—21no—
38C22×C83⋊C8C2312yes—
39C22×C83⋊C8C2312yes—
40#14581792—22no—
41#14581792—22no—
42C2×C4⋊Dic83—21no—
43C2×C4⋊Dic83C2312yes—
44C2×C4⋊Dic83C2312yes—
45C2×Dic83⋊C4—24no—
46C2×C4×Dic83—21no—
47#14581725—24no—
48C4⋊(C83⋊C8)—22no—
49#14581716—24no—
50#14581716—28no—
51#14581716—24no—
52#14581716—28no—
53C4×C83⋊C8—22no—
54C83⋊#95—21no—
55C83⋊#95—24no—
56C22×Q8×C83D411yescharacteristic
57C22×Q8×C83C2312yes—
58C22×Q8×C83C2311yescharacteristic
59C4⋊Q8×C83—24no—
60C4⋊Q8×C83—24no—
61C4⋊Q8×C83—22no—
62C4⋊Q8×C83D414yes—
63#125×C83—216no—
64#125×C83—28no—
65C4×Q8×C83—28no—
66C4×Q8×C83—24no—
67C4×Q8×C83—24no—
68C4×Q8×C83—22no—
69C4×Q8×C83—22no—
70C4×Q8×C83—22no—
71C4×Q8×C83D414yes—
72C2×C4⋊C4×C83—22no—
73C2×C4⋊C4×C83—21no—
74C2×C4⋊C4×C83—22no—
75C2×C4⋊C4×C83—21no—
76C2×C4⋊C4×C83—22no—
77C2×C4⋊C4×C83—21no—
78C2×C4⋊C4×C83D411yescharacteristic
79C2×C4⋊C4×C83D412yes—
80C2×C4⋊C4×C83C2311yescharacteristic
81C2×C42×C83—22no—
82C2×C42×C83—21no—
83C2×C42×C83C2312yes—
84C2×C42×C83C2311yescharacteristic
85C2×Dic166—42no—
86C2×Dic166—82no—
87C22×Dic83—22no—
88C22×Dic83—22no—
89C2×C83⋊C8—22no—
90C2×C83⋊C8—22no—
91C2×C83⋊C8—24no—
92C2×C83⋊C8—22no—
93C2×C83⋊C8—22no—
94C4⋊Dic83—22no—
95C4⋊Dic83—22no—
96C4⋊Dic83—22no—
97C4⋊Dic83—42no—
98C4⋊Dic83#4114yes—
99Dic83⋊C4—84no—
100C4×Dic83—42no—
101C2×Q8×C83—22no—
102C2×Q8×C83—22no—
103C2×Q8×C83—22no—
104C2×Q8×C83—21no—
105C2×Q8×C83—22no—
106C2×Q8×C83—24no—
107C2×Q8×C83—22no—
108C2×Q8×C83—24no—
109C2×Q8×C83Q1614yes—
110C2×Q8×C83C2×D414yes—
111C2×Q8×C83C2×D412yes—
112C22×C4×C83—21no—
113C22×C4×C83—22no—
114C22×C4×C83—21no—
115C22×C4×C83C2×D412yes—
116C22×C4×C83C2×D411yescharacteristic
117C22×C4×C83C2×D412yes—
118C22×C4×C83C2×D411yescharacteristic
119C22×C4×C83C2411yescharacteristic
120C4⋊C4×C83—28no—
121C4⋊C4×C83—28no—
122C4⋊C4×C83—28no—
123C4⋊C4×C83—24no—
124C4⋊C4×C83—24no—
125C4⋊C4×C83—24no—
126C4⋊C4×C83—24no—
127C4⋊C4×C83—22no—
128C4⋊C4×C83—21no—
129C4⋊C4×C83C2×D412yes—
130C42×C83—28no—
131C42×C83—24no—
132C42×C83—22no—
133C42×C83—22no—
134C42×C83C2×D412yes—
135C42×C83C2×D412yes—
136C23×C83C22×D411yescharacteristic
137Q8×C83—28no—
138Q8×C83—22no—
139Q8×C83—22no—
140Q8×C83—28no—
141Q8×C83—42no—
142Q8×C83—42no—
143Q8×C83—42no—
144C2×C4×C83—22no—
145C2×C4×C83—22no—
146C2×C4×C83—22no—
147C2×C4×C83—22no—
148C2×C4×C83—22no—
149C2×C4×C83—24no—
150C2×C4×C83—22no—
151C2×C4×C83—24no—
152C2×C4×C83—42no—
153C2×C4×C83—44no—
154C2×C4×C83—42no—
155C2×C4×C83#12012yes—
156C2×C4×C83#12012yes—
157C2×C4×C83C2×Q1614yes—
158C2×C4×C83C2×Q1612yes—
159C2×C4×C83#13712yes—
160C2×C4×C83C22×D411yescharacteristic
161C2×C4×C83C22×D411yescharacteristic
162C2×C4×C83C2×#4112yes—
163C2×C4×C832−1+412yes—
164C2×Dic83—24no—
165C2×Dic83—42no—
166C2×Dic83—42no—
167C2×Dic83—42no—
168C2×Dic83—44no—
169Dic166—84no—
170C83⋊C8—42no—
171C83⋊C8—42no—
172C22×C83C2×#12011yescharacteristic
173C22×C83#54812yes—
174C22×C83C22×Q1611yescharacteristic
175C22×C83C2×#13711yescharacteristic
176C22×C83#57712yes—
177C4×C83—24no—
178C4×C83—22no—
179C4×C83—42no—
180C4×C83—44no—
181C4×C83—42no—
182C4×C83—42no—
183C4×C83#45114yes—
184C4×C83#53211yescharacteristic
185C4×C83#53611yescharacteristic
186C4×C83#54712yes—
187Dic83—44no—
188Dic83—82no—
189#25651—831nomaximal, Sylow
190C2×C83#243211yescharacteristic
191C2×C83C2×#45111yescharacteristic
192C2×C83#301011yescharacteristic
193C2×C83#336312yes—
194C2×C83#337212yes—
195C2×#557D8311yescharacteristic
196C2×#487—831no—
197C2×#477—831no—
198#2567—831no—
199#2043—834no—
200#1984—831no—
201#1981—831no—
202#1863—834no—
203C4×#103—831no—
204C83#2565111yescharacteristic, complemented, Sylow
205#557—24no—
206#557—24no—
207C2×C4⋊Q8—831no—
208C2×C4⋊Q8C2×D8312yes—
209C2×C4⋊Q8C2×D8311yescharacteristic
210C2×#125—21no—
211C2×#125—22no—
212C2×C4×Q8—21no—
213C2×C4×Q8C2×D8312yes—
214C2×C4×Q8C2×D8311yescharacteristic
215C2×C4×Q8C2×D8311yescharacteristic
216#487—1664no—
217#477—1664no—
218C2×#107—832no—
219C2×C4⋊C8—831no—
220C2×#103—832no—
221C2×#103—834no—
222C2×#103—834no—
223C2×#103—832no—
224C2×C4×C8—831no—
225#395—1662no—
226#391—1661no—
227#389—834no—
228#384—1661no—
229C4×C4⋊C4—831no—
230#340—832no—
231#336—834no—
232C22×Q8—1661no—
233C22×Q8C22⋊D8311yescharacteristic
234C22×Q8C22×D8312yes—
235C22×Q8C22×D8311yescharacteristic
236C22×C8—832no—
237C22×C8—832no—
238C4⋊Q8—24no—
239C4⋊Q8—24no—
240C4⋊Q8—22no—
241C4⋊Q8—1662no—
242C4⋊Q8—1662no—
243C4⋊Q8C22⋊D8314yes—
244#125—216no—
245#125—28no—
246C4×Q8—28no—
247C4×Q8—24no—
248C4×Q8—24no—
249C4×Q8—22no—
250C4×Q8—22no—
251C4×Q8—22no—
252C4×Q8C22⋊D8314yes—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—832no—
260C2×C4⋊C4—832no—
261C2×C4⋊C4—1661no—
262C2×C4⋊C4—1664no—
263C2×C4⋊C4C22⋊D8311yescharacteristic
264C2×C4⋊C4C22⋊D8312yes—
265C2×C4⋊C4C22×D8311yescharacteristic
266C2×C42—22no—
267C2×C42—21no—
268C2×C42—1661no—
269C2×C42C22×D8312yes—
270C2×C42C22×D8311yescharacteristic
271#107—1664no—
272C4⋊C8—1662no—
273#103—1668no—
274#103—1664no—
275#103—1664no—
276#103—1668no—
277C4×C8—1662no—
278#95—1661no—
279#95—1664no—
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8—21no—
283C2×Q8—22no—
284C2×Q8—24no—
285C2×Q8—22no—
286C2×Q8—24no—
287C2×Q8—22no—
288C2×Q8—3322no—
289C2×Q8—6642no—
290C2×Q8#1305058014yes—
291C2×Q8C2×C22⋊D8314yes—
292C2×Q8C2×C22⋊D8312yes—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—1662no—
297C22×C4—1662no—
298C22×C4C2×C22⋊D8312yes—
299C22×C4C2×C22⋊D8311yescharacteristic
300C22×C4C2×C22⋊D8312yes—
301C22×C4C2×C22⋊D8311yescharacteristic
302C22×C4C23×D8311yescharacteristic
303C2×C8—1662no—
304C2×C8—1662no—
305C2×C8—1662no—
306C2×C8—1664no—
307C2×C8—1662no—
308C4⋊C4—28no—
309C4⋊C4—28no—
310C4⋊C4—28no—
311C4⋊C4—24no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—24no—
315C4⋊C4—22no—
316C4⋊C4—21no—
317C4⋊C4—3322no—
318C4⋊C4—834no—
319C4⋊C4—6644no—
320C4⋊C4—1662no—
321C4⋊C4—1662no—
322C4⋊C4—1662no—
323C4⋊C4C2×C22⋊D8312yes—
324C42—28no—
325C42—24no—
326C42—22no—
327C42—22no—
328C42—3322no—
329C42C2×C22⋊D8312yes—
330C42C2×C22⋊D8312yes—
331C23C22×C22⋊D8311yescharacteristic
332Q8—28no—
333Q8—28no—
334Q8—22no—
335Q8—22no—
336Q8—42no—
337Q8—42no—
338Q8—42no—
339Q8—6644no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—24no—
346C2×C4—22no—
347C2×C4—24no—
348C2×C4—44no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—3322no—
352C2×C4—3322no—
353C2×C4—3322no—
354C2×C4—3324no—
355C2×C4—1664no—
356C2×C4#1458176112yes—
357C2×C4#1458176112yes—
358C2×C4C2×#1305058014yes—
359C2×C4C2×#1305058012yes—
360C2×C4#1458182312yes—
361C2×C4C22×C22⋊D8311yescharacteristic
362C2×C4C22×C22⋊D8311yescharacteristic
363C2×C4#41×D8312yes—
364C2×C4#1458183912yes—
365C8—3322no—
366C8—3322no—
367C22C2×#1458176111yescharacteristic
368C22#1912835812yes—
369C22C22×#1305058011yescharacteristic
370C22C2×#1458182311yescharacteristic
371C22#1912850912yes—
372C4—22no—
373C4—24no—
374C4—42no—
375C4—42no—
376C4—44no—
377C4—42no—
378C4—3324no—
379C4—6642no—
380C4#1912785714yes—
381C4#1912824711yescharacteristic
382C4#1912827211yescharacteristic
383C4#1912835012yes—
384C2#2491108711yescharacteristic
385C2C2×#1912785711yescharacteristic
386C2#2491674911yescharacteristic
387C2#2492013712yes—
388C2#2492024012yes—
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC2×#557×C83C2
socle, prime radicalC23×C83C22×D4
hypercenterC2×#557D83
hypocenterC83#25651
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
204. C83#25651semidirect

Maximal subgroups

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

subgroupindex
C2×#191280512
C2×#191279962
#249126682
#249087062
#249084002
#249083872
#249077172
C4×#145817162
C2×#557×C832
#2565183