Finite Group Databasebeta

← #35433529

Subgroups — #35433529

Order 21248.

766 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24916053C211yesmaximal, characteristic
2#24916052C211yesmaximal, characteristic
3C2×#19127737C211yesmaximal, characteristic
4C2×#19127729C211yesmaximal, characteristic
5#24908977C214yesmaximal, complemented
6#24908972C214yesmaximal, complemented
7#24907201C211yesmaximal, characteristic
8C2×#19127465C211yesmaximal, characteristic
9C83×#1742C211yesmaximal, characteristic
10C22×#13050574C2211yescharacteristic
11C23×C83⋊C8C2211yescharacteristic
12C2×#14581751C2211yescharacteristic
13C2×#14581751C2211yescharacteristic
14#19127744—22no—
15#19127743—22no—
16#19127743C2214yes—
17#19127741C2214yes—
18#19127739—22no—
19#19127737—22no—
20#19127737—22no—
21#19127737—22no—
22C2×#14581725C2212yes—
23C2×#14581723C2212yes—
24#19127729—22no—
25#19127729C2212yes—
26#19127729C2212yes—
27C2×Dic83⋊C8—22no—
28#19127469C2214yes—
29#19127465C2212yes—
30#19127465C2212yes—
31C22×M16×C83C2211yescharacteristic
32C22×M16×C83C2211yescharacteristic
33C2×#117×C83C2211yescharacteristic
34#343×C83C2214yes—
35C2×#13050574—21no—
36C2×#13050574—22no—
37C2×#13050574Q814yes—
38C2×#13050574C2312yes—
39C22×C83⋊C8—21no—
40C22×C83⋊C8—22no—
41C22×C83⋊C8—22no—
42C22×C83⋊C8D412yes—
43C22×C83⋊C8D412yes—
44C22×C83⋊C8C2311yescharacteristic
45C22×C83⋊C8C2311yescharacteristic
46#14581751—21no—
47#14581751—21no—
48#14581751—21no—
49#14581751—21no—
50#14581751—21no—
51#14581751—21no—
52#14581751D412yes—
53#14581751C2312yes—
54C2×C4⋊Dic83D412yes—
55C2×C4⋊Dic83C2312yes—
56C2×C22⋊Dic83—21no—
57C2×C22⋊Dic83—21no—
58C2×C4×Dic83—21no—
59C2×C4×Dic83—21no—
60#14581726—24no—
61#14581725—24no—
62#14581725—24no—
63#14581724—24no—
64#14581723—24no—
65#14581723—24no—
66Dic83⋊C8—24no—
67Dic83⋊C8—24no—
68C23×C4×C83C2311yescharacteristic
69C2×M16×C83—21no—
70C2×M16×C83—22no—
71C2×M16×C83—21no—
72C2×M16×C83—22no—
73C2×M16×C83—22no—
74C2×M16×C83Q814yes—
75C2×M16×C83C2312yes—
76C2×M16×C83C2312yes—
77C22×C8×C83—21no—
78C22×C8×C83—21no—
79#117×C83—21no—
80#117×C83—21no—
81#117×C83—21no—
82#117×C83—21no—
83C2×C4⋊C4×C83C2312yes—
84C2×C22⋊C4×C83—21no—
85C2×C42×C83—21no—
86C22×Dic83—22no—
87C22×Dic83—22no—
88#13050574—24no—
89#13050574—22no—
90#13050574C2×Q814yes—
91C2×C83⋊C8—24no—
92C2×C83⋊C8—24no—
93C2×C83⋊C8—24no—
94C2×C83⋊C8—22no—
95C2×C83⋊C8—21no—
96C2×C83⋊C8—21no—
97C2×C83⋊C8—21no—
98C2×C83⋊C8—21no—
99C2×C83⋊C8C2×D412yes—
100C2×C83⋊C8C2×D412yes—
101C4⋊Dic83—22no—
102C4⋊Dic83—22no—
103C4⋊Dic83—22no—
104C4⋊Dic83C2×D414yes—
105C22⋊Dic83—22no—
106C22⋊Dic83—22no—
107C22⋊Dic83—41no—
108C22⋊Dic83—41no—
109C4×Dic83—22no—
110C4×Dic83—22no—
111C4×Dic83—41no—
112C4×Dic83—41no—
113C24×C83#4111yescharacteristic
114C22×C4×C83—22no—
115C22×C4×C83C2×D411yescharacteristic
116C22×C4×C83C2×D411yescharacteristic
117C22×C4×C83C2×D411yescharacteristic
118C22×C4×C83C2×D411yescharacteristic
119C22×C4×C83C2×Q812yes—
120C22×C4×C83C2×Q812yes—
121C22×C4×C83#4112yes—
122C22×C4×C83#4111yescharacteristic
123C22×C4×C83#4112yes—
124C22×C4×C83C2411yescharacteristic
125M16×C83—24no—
126M16×C83—22no—
127M16×C83—22no—
128M16×C83—22no—
129M16×C83—42no—
130M16×C83C2×Q814yes—
131C2×C8×C83—24no—
132C2×C8×C83—22no—
133C2×C8×C83—22no—
134C2×C8×C83—42no—
135C4⋊C4×C83—22no—
136C4⋊C4×C83—22no—
137C22⋊C4×C83—41no—
138C22⋊C4×C83—41no—
139C42×C83—41no—
140C42×C83—41no—
141C23×C83—42no—
142C23×C83#12111yescharacteristic
143C23×C83#12111yescharacteristic
144C23×C83#12512yes—
145C23×C83#12512yes—
146C23×C83C2×#4111yescharacteristic
147C2×C4×C83—42no—
148C2×C4×C83—82no—
149C2×C4×C83#12111yescharacteristic
150C2×C4×C83#12111yescharacteristic
151C2×C4×C83C22⋊Q812yes—
152C2×C4×C83C22⋊Q812yes—
153C2×C4×C83C22⋊Q812yes—
154C2×C4×C83C22⋊Q812yes—
155C2×C4×C83#12312yes—
156C2×C4×C83#12312yes—
157C2×C4×C83#12512yes—
158C2×C4×C83#12512yes—
159C2×C4×C83C4⋊Q812yes—
160C2×C4×C83C4⋊Q812yes—
161C2×C4×C83C22×D411yescharacteristic
162C2×C4×C83C22×D411yescharacteristic
163C2×C4×C83C22×Q811yescharacteristic
164C2×C4×C83C22×Q811yescharacteristic
165C2×C4×C83C2×#4111yescharacteristic
166C2×C4×C83C2×#4111yescharacteristic
167C2×Dic83—24no—
168C2×Dic83—44no—
169C2×Dic83—42no—
170C2×Dic83—82no—
171C8×C83—24no—
172C8×C83—42no—
173C83⋊C8—24no—
174C83⋊C8—22no—
175C83⋊C8—22no—
176C22×C83—22no—
177C22×C83—22no—
178C22×C83—22no—
179C22×C83—42no—
180C22×C83—42no—
181C22×C83#39812yes—
182C22×C83#39812yes—
183C22×C83C2×#12111yescharacteristic
184C22×C83C2×#12511yescharacteristic
185C22×C83C2×#12511yescharacteristic
186C4×C83—82no—
187C4×C83#39812yes—
188C4×C83#39812yes—
189C4×C83C2×C22⋊Q811yescharacteristic
190C4×C83C2×C22⋊Q811yescharacteristic
191C4×C83C2×#12311yescharacteristic
192C4×C83C2×C4⋊Q811yescharacteristic
193Dic83—44no—
194Dic83—82no—
195#23674—831nomaximal, Sylow
196C2×C83—21no—
197C2×C83—21no—
198C2×C83—22no—
199C2×C83—22no—
200C2×C83C2×#39811yescharacteristic
201C2×C83#306712yes—
202#2911—831no—
203#2911—831no—
204C2×#427—831no—
205C2×#424—831no—
206#2090—834no—
207#2089—834no—
208#1742—831no—
209#1742D8311yescharacteristic
210C2×#341—831no—
211C83#2367411yescharacteristic, complemented, Sylow
212C22×M16—831no—
213C22×M16C2×D8311yescharacteristic
214C22×M16C2×D8311yescharacteristic
215C23×C8—831no—
216C2×#117—831no—
217C2×#117—831no—
218C2×#117C2×D8311yescharacteristic
219#428—834no—
220#428—834no—
221#428—1662no—
222#428—1662no—
223#427—1662no—
224#427—1662no—
225#427—1662no—
226#427—1662no—
227C2×#107—832no—
228C2×#106—832no—
229#424—832no—
230#424—832no—
231#424—1662no—
232C2×C4⋊C8—1662no—
233#343—834no—
234#343C2×D8314yes—
235#341—832no—
236#341—832no—
237C23×C4C22×D8311yescharacteristic
238C2×M16—21no—
239C2×M16—22no—
240C2×M16—21no—
241C2×M16—22no—
242C2×M16—22no—
243C2×M16—834no—
244C2×M16—832no—
245C2×M16—1661no—
246C2×M16—1662no—
247C2×M16Dic16614yes—
248C2×M16C22×D8312yes—
249C2×M16C22×D8312yes—
250C22×C8—21no—
251C22×C8—21no—
252C22×C8—832no—
253C22×C8—832no—
254C22×C8—831no—
255C22×C8—831no—
256C22×C8—1661no—
257C22×C8—1662no—
258C22×C8—1662no—
259#117—21no—
260#117—21no—
261#117—21no—
262#117—21no—
263#117—832no—
264#117—832no—
265#117—1661no—
266#117—1661no—
267#117—1661no—
268#117—1661no—
269#117—1661no—
270#117—1661no—
271C2×C4⋊C4—832no—
272C2×C4⋊C4—832no—
273C2×C4⋊C4C22×D8312yes—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—1661no—
276C2×C22⋊C4—1661no—
277C2×C42—21no—
278C2×C42—1661no—
279C2×C42—1661no—
280#107—1664no—
281#107—1664no—
282#107—1664no—
283#106—1664no—
284#106—1664no—
285#106—1664no—
286C4⋊C8—1664no—
287C4⋊C8—1664no—
288C24#1305059011yescharacteristic
289C22×C4—22no—
290C22×C4—1662no—
291C22×C4—1662no—
292C22×C4D4×D8311yescharacteristic
293C22×C4D4×D8311yescharacteristic
294C22×C4D4×D8311yescharacteristic
295C22×C4D4×D8311yescharacteristic
296C22×C4Q8×D8312yes—
297C22×C4C2×Dic16612yes—
298C22×C4#1305058912yes—
299C22×C4#1305059011yescharacteristic
300C22×C4#1305059012yes—
301C22×C4C23×D8311yescharacteristic
302M16—24no—
303M16—22no—
304M16—22no—
305M16—22no—
306M16—42no—
307M16—834no—
308M16—1664no—
309M16—1662no—
310M16C2×Dic16614yes—
311C2×C8—24no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—42no—
315C2×C8—832no—
316C2×C8—832no—
317C2×C8—1664no—
318C2×C8—1664no—
319C2×C8—1664no—
320C2×C8—1662no—
321C2×C8—1661no—
322C2×C8—1661no—
323C2×C8—1661no—
324C2×C8—1661no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—834no—
328C4⋊C4—1662no—
329C4⋊C4—1662no—
330C4⋊C4—1662no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—3321no—
334C22⋊C4—3321no—
335C22⋊C4—1662no—
336C22⋊C4—1662no—
337C42—41no—
338C42—41no—
339C42—3321no—
340C42—3321no—
341C42—1662no—
342C42—1662no—
343C23—42no—
344C23#1458176511yescharacteristic
345C23#1458176511yescharacteristic
346C23#1458178212yes—
347C23#1458178412yes—
348C23C2×#1305059011yescharacteristic
349C2×C4—42no—
350C2×C4—82no—
351C2×C4—3324no—
352C2×C4—3322no—
353C2×C4—6642no—
354C2×C4—1664no—
355C2×C4#1458176511yescharacteristic
356C2×C4#1458176511yescharacteristic
357C2×C4#1458176912yes—
358C2×C4#1458176912yes—
359C2×C4#1458177012yes—
360C2×C4#1458177012yes—
361C2×C4#1458177412yes—
362C2×C4#1458177412yes—
363C2×C4#1458178212yes—
364C2×C4#1458178412yes—
365C2×C4#1458179112yes—
366C2×C4#1458179112yes—
367C2×C4C2×D4×D8311yescharacteristic
368C2×C4C2×D4×D8311yescharacteristic
369C2×C4C2×Q8×D8311yescharacteristic
370C2×C4C22×Dic16611yescharacteristic
371C2×C4C2×#1305058911yescharacteristic
372C2×C4C2×#1305059011yescharacteristic
373C8—24no—
374C8—42no—
375C8—1664no—
376C8—1662no—
377C8—1662no—
378C22—22no—
379C22—22no—
380C22—22no—
381C22—42no—
382C22—42no—
383C22#1912763512yes—
384C22#1912763512yes—
385C22C2×#1458176511yescharacteristic
386C22C2×#1458178211yescharacteristic
387C22C2×#1458178411yescharacteristic
388C4—82no—
389C4—3324no—
390C4—6642no—
391C4#1912763512yes—
392C4#1912763512yes—
393C4C2×#1458176911yescharacteristic
394C4C2×#1458177011yescharacteristic
395C4C2×#1458177411yescharacteristic
396C4C2×#1458179111yescharacteristic
397C2—22no—
398C2—22no—
399C2—21no—
400C2—21no—
401C2C2×#1912763511yescharacteristic
402C2#2491718212yes—
subgroupG/H
centerC2×C4#14581765
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#1742C2
socleC22×C83C2×#121
prime radicalC24×C83#41
hypercenter#1742D83
hypocenterC83#23674
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1742

Complements

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

NH ≅ G/Ntype
5. #24908977C2semidirect
6. #24908972C2semidirect
211. C83#23674semidirect

Maximal subgroups

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

subgroupindex
#249160532
#249160522
C2×#191277372
C2×#191277292
#249089772
#249089722
#249072012
C2×#191274652
C83×#17422
#2367483