Finite Group Databasebeta

← #36285151

Subgroups — #36285151

Order 21248.

1674 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24921273C212yesmaximal
2#24920431C218yesmaximal, complemented
3#24920396C218yesmaximal, complemented
4#24917991C212yesmaximal
5#24917987C211yesmaximal, characteristic
6#24917301C214yesmaximal
7#24917268C212yesmaximal
8#24916403C211yesmaximal, characteristic
9#24916331C212yesmaximal
10C2×#441×C83C211yesmaximal, characteristic
11Q8×Dic166—22no—
12Q8×Dic166C2214yes—
13#19128430—22no—
14#19128430C2214yes—
15#19128427—24no—
16#19128427C2218yes—
17#19128332—24no—
18#19128332C2214yes—
19#19128258—24no—
20#19128258C2212yes—
21#19128258C2211yescharacteristic
22#19128179C2214yes—
23#19128110C2214yes—
24#19128110C2214yes—
25#19128108C2218yes—
26#19128101C2214yes—
27#19128098C2214yes—
28#19128010C2218yes—
29#19128008C2218yes—
30#19128006C2218yes—
31#19127996C2214yes—
32#19127994C2218yes—
33#19127986C2214yes—
34#19127984C2218yes—
35#19127829C2218yes—
36#19127808C2214yes—
37#19127808C2214yes—
38Q16×Dic83C2218yes—
39#19127741C2214yes—
40#19127739C2212yes—
41#19127739C2212yes—
42#19127727C2212yes—
43#19127725C2211yescharacteristic
44#19127710C2214yes—
45#19127656C2212yes—
46C22×Q16×C83C2211yescharacteristic
47C2×C4×Q8×C83C2212yes—
48#441×C83C2218yes—
49C2×#106×C83C2211yescharacteristic
50C2×#103×C83C2212yes—
51C2×#97×C83C2211yescharacteristic
52C2×#13050574C2312yes—
53#14581792—24no—
54#14581792—24no—
55#14581792D412yes—
56#14581792C2312yes—
57#14581792C2312yes—
58#14581791—28no—
59#14581791—24no—
60#14581791—28no—
61#14581791C2318yes—
62#14581784—28no—
63#14581784D412yes—
64#14581784C2314yes—
65#14581783—28no—
66#14581783—24no—
67#14581783—28no—
68#14581783C2318yes—
69#14581771—28no—
70#14581771—22no—
71#14581771—22no—
72#14581771D414yes—
73Q8×Dic83—24no—
74Q8×Dic83—24no—
75Q8×Dic83C2318yes—
76Q8×Dic83C2318yes—
77C4×Dic166—24no—
78C4×Dic166—24no—
79C4×Dic166—28no—
80C4×Dic166—22no—
81C4×Dic166—22no—
82C4×Dic166C2314yes—
83#14581751—28no—
84#14581751C2314yes—
85#14581751C2312yes—
86#14581726C2314yes—
87#14581726C2314yes—
88#14581725C2318yes—
89#14581724C2312yes—
90#14581724C2312yes—
91#14581724C2312yes—
92#14581724C2312yes—
93#14581723C2318yes—
94C4⋊(C83⋊C8)C2314yes—
95Dic83⋊C8C2318yes—
96#14581716C2318yes—
97#14581716C2318yes—
98#14581702C2314yes—
99C8×Dic83C2314yes—
100C22×Q8×C83C2312yes—
101C2×Q16×C83Q818yescomplemented
102C2×Q16×C83C2314yes—
103C22×C8×C83C2311yescharacteristic
104C22×C8×C83C2311yescharacteristic
105C4×Q8×C83—22no—
106C4×Q8×C83—22no—
107C4×Q8×C83C2314yes—
108C4×Q8×C83C2314yes—
109C2×C4⋊C4×C83—22no—
110C2×C4⋊C4×C83C2312yes—
111C2×C42×C83—22no—
112C2×C42×C83C2311yescharacteristic
113#106×C83C2314yes—
114#103×C83C2318yes—
115#97×C83C2314yes—
116C2×Dic166—22no—
117C2×Dic166—22no—
118C2×Dic166—28no—
119C2×Dic166C2×D414yes—
120#13050574—24no—
121C2×C83⋊C8C2414yes—
122C4⋊Dic83—22no—
123C4⋊Dic83—22no—
124C4⋊Dic83—28no—
125C4⋊Dic83—28no—
126C4⋊Dic83—28no—
127C4⋊Dic83C2×D414yes—
128C4⋊Dic83C2×D414yes—
129C4⋊Dic83C2414yes—
130C4⋊Dic83C2414yes—
131Dic83⋊C4—24no—
132Dic83⋊C4—24no—
133Dic83⋊C4—28no—
134Dic83⋊C4—28no—
135Dic83⋊C4C2×D418yes—
136C22⋊Dic83—24no—
137C22⋊Dic83C2×D414yes—
138C4×Dic83—24no—
139C4×Dic83—28no—
140C4×Dic83—28no—
141C4×Dic83C2414yes—
142C2×Q8×C83—22no—
143C2×Q8×C83—24no—
144C2×Q8×C83C2×Q818yes—
145C2×Q8×C83C2414yes—
146C22×C4×C83—22no—
147C22×C4×C83—22no—
148C22×C4×C83C2×D412yes—
149C22×C4×C83C2411yescharacteristic
150Q16×C83C2×Q8116yes—
151C2×C8×C83—22no—
152C2×C8×C83C2×Q814yes—
153C2×C8×C83C2412yes—
154C2×C8×C83C2412yes—
155C4⋊C4×C83—24no—
156C4⋊C4×C83—24no—
157C4⋊C4×C83—22no—
158C4⋊C4×C83C2414yes—
159C42×C83—24no—
160C42×C83—24no—
161C42×C83C2×D411yescharacteristic
162C42×C83C2×D411yescharacteristic
163C42×C83C2412yes—
164C23×C83C22×D411yescharacteristic
165Q8×C83—28no—
166Q8×C83—24no—
167Q8×C83C22×Q818yes—
168C2×C4×C83—28no—
169C2×C4×C83—24no—
170C2×C4×C83—24no—
171C2×C4×C83—42no—
172C2×C4×C83—82no—
173C2×C4×C83C22×D412yes—
174C2×C4×C83C22×D412yes—
175C2×C4×C83C22×D411yescharacteristic
176C2×C4×C83C22×Q812yes—
177C2×C4×C832−1+412yes—
178C2×C4×C83C2511yescharacteristic
179C2×Dic83—24no—
180C2×Dic83—24no—
181C2×Dic83C22×D418yes—
182Dic166—44no—
183Dic166—44no—
184Dic166—88no—
185Dic166—82no—
186C8×C83—22no—
187C8×C83C22×Q814yes—
188C83⋊C8—24no—
189C22×C83—22no—
190C22×C83D4×Q812yes—
191C22×C83C23×D411yescharacteristic
192C4×C83—22no—
193C4×C83—22no—
194C4×C83—28no—
195C4×C83—44no—
196C4×C83D4×Q812yes—
197C4×C83C23×Q811yescharacteristic
198C4×C83C2×2−1+411yescharacteristic
199Dic83—28no—
200Dic83—44no—
201Dic83—44no—
202#61379—831nomaximal, Sylow
203C2×C83—22no—
204C2×C83C2×D4×Q811yescharacteristic
205C2×C83#358612yes—
206#3502—832no—
207#3390—838no—
208#3387—838no—
209#3163—832no—
210#3162—831no—
211#3082—834no—
212#3078—832no—
213#2961—831no—
214#2950—832no—
215C2×#441D8311yescharacteristic
216C83#6137911yescharacteristic, complemented, Sylow
217C22×Q16C2×D8311yescharacteristic
218Q82—834no—
219Q82—1662no—
220#557—838no—
221#557—834no—
222#557—1664no—
223#557—1662no—
224#544—834no—
225#544—1664no—
226#533—831no—
227#533—832no—
228#533—1664no—
229#519—834no—
230C2×C4×Q8C2×D8312yes—
231#501—838no—
232#501—834no—
233#501—834no—
234#499—834no—
235#498—834no—
236#479—838no—
237#479—838no—
238#479—838no—
239#477—834no—
240#477—838no—
241#475—834no—
242#475—838no—
243#446—838no—
244#441—834no—
245#441—834no—
246#441C2×D8318yes—
247C4×Q16—838no—
248#428—834no—
249#427—832no—
250#427—832no—
251C2×#106C2×D8311yescharacteristic
252#424—832no—
253#423—831no—
254#419—834no—
255C2×#103C2×D8312yes—
256#405—832no—
257C2×#97C2×D8311yescharacteristic
258C22×Q8C22×D8312yes—
259C2×Q16Dic16618yescomplemented
260C2×Q16C22×D8314yes—
261C2×M16—832no—
262C22×C8C22×D8311yescharacteristic
263C22×C8C22×D8311yescharacteristic
264C4⋊Q8—838no—
265C4⋊Q8—832no—
266C4⋊Q8—832no—
267C4⋊Q8—832no—
268C4⋊Q8—1664no—
269C4⋊Q8—1664no—
270C4⋊Q8—1664no—
271C4⋊Q8—1668no—
272C4⋊Q8—1668no—
273#125—838no—
274#125—832no—
275#125—834no—
276#125—1668no—
277#125—1664no—
278#125—1668no—
279#125—1668no—
280C22⋊Q8—834no—
281C22⋊Q8—1668no—
282C22⋊Q8—1662no—
283C22⋊Q8—1662no—
284C4×Q8—22no—
285C4×Q8—22no—
286C4×Q8—834no—
287C4×Q8—838no—
288C4×Q8—838no—
289C4×Q8—1668no—
290C4×Q8—1664no—
291C4×Q8—1664no—
292C4×Q8—1664no—
293C4×Q8—1664no—
294C4×Q8—1662no—
295C4×Q8—1662no—
296C4×Q8C22×D8314yes—
297C4×Q8C22×D8314yes—
298#117—832no—
299#117—834no—
300#117—1668no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4C22×D8312yes—
303C2×C42—22no—
304C2×C42C22×D8311yescharacteristic
305#107—834no—
306#107—834no—
307#107—838no—
308#106—832no—
309#106—832no—
310#106—832no—
311#106—832no—
312#106—838no—
313#106C22×D8314yes—
314C4⋊C8—838no—
315C4⋊C8—834no—
316#103—838no—
317#103—838no—
318#103C22×D8318yes—
319#97—834no—
320#97C22×D8314yes—
321C4×C8—834no—
322C2×Q8—22no—
323C2×Q8—24no—
324C2×Q8—834no—
325C2×Q8—1668no—
326C2×Q8—1662no—
327C2×Q8—1662no—
328C2×Q8C2×Dic16618yes—
329C2×Q8C23×D8314yes—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4D4×D8312yes—
333C22×C4C23×D8311yescharacteristic
334Q16C2×Dic166116yes—
335M16—1664no—
336C2×C8—22no—
337C2×C8—834no—
338C2×C8C2×Dic16614yes—
339C2×C8C23×D8312yes—
340C2×C8C23×D8312yes—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—22no—
344C4⋊C4—834no—
345C4⋊C4—838no—
346C4⋊C4—834no—
347C4⋊C4—834no—
348C4⋊C4—834no—
349C4⋊C4—1668no—
350C4⋊C4—1668no—
351C4⋊C4—1668no—
352C4⋊C4—1668no—
353C4⋊C4—1668no—
354C4⋊C4—1664no—
355C4⋊C4—1662no—
356C4⋊C4—1664no—
357C4⋊C4—1662no—
358C4⋊C4C23×D8314yes—
359C22⋊C4—834no—
360C22⋊C4—1664no—
361C42—24no—
362C42—24no—
363C42—834no—
364C42—1668no—
365C42—1668no—
366C42—1664no—
367C42D4×D8311yescharacteristic
368C42D4×D8311yescharacteristic
369C42C23×D8312yes—
370C23C2×D4×D8311yescharacteristic
371Q8—28no—
372Q8—24no—
373Q8—3324no—
374Q8—3324no—
375Q8—6642no—
376Q8—6648no—
377Q8C22×Dic16618yes—
378C2×C4—28no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—42no—
382C2×C4—82no—
383C2×C4—838no—
384C2×C4—1664no—
385C2×C4—1664no—
386C2×C4C2×D4×D8312yes—
387C2×C4C2×D4×D8312yes—
388C2×C4C2×D4×D8311yescharacteristic
389C2×C4C22×Dic16612yes—
390C2×C4#1458183912yes—
391C2×C4C24×D8311yescharacteristic
392C8—22no—
393C8—1664no—
394C8C22×Dic16614yes—
395C22—22no—
396C22D4×Dic16612yes—
397C22C22×D4×D8311yescharacteristic
398C4—28no—
399C4—22no—
400C4—22no—
401C4—44no—
402C4—3324no—
403C4—3324no—
404C4—1668no—
405C4D4×Dic16612yes—
406C4C23×Dic16611yescharacteristic
407C4C2×#1458183911yescharacteristic
408C2—22no—
409C2C2×D4×Dic16611yescharacteristic
410C2#2492224612yes—
subgroupG/H
centerC22C22×D4×D83
commutator, cosocleC2×C4×C83C25
FrattiniC2×C4C24×D83
FittingC2×#441×C83C2
socleC22×C83C23×D4
prime radicalC23×C83C22×D4
hypercenterC2×#441D83
hypocenterC83#61379
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#441

Complements

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

NH ≅ G/Ntype
2. #24920431C2semidirect
3. #24920396C2semidirect
101. C2×Q16×C83Q8semidirect
216. C83#61379semidirect
259. C2×Q16Dic166semidirect

Maximal subgroups

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

subgroupindex
#249212732
#249204312
#249203962
#249179912
#249179872
#249173012
#249172682
#249164032
#249163312
C2×#441×C832
#6137983