Finite Group Databasebeta

← #11860789

Subgroups — #11860789

Order 1152.

1368 conjugacy classes of subgroups.

The subgroup lattice has 351 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q16×C3⋊D3C211yesmaximal, characteristic
2#10596043C212yesmaximal
3C32×C4×Q16C211yesmaximal, characteristic, complemented
4Q16×C3⋊Dic3C211yesmaximal, characteristic, complemented
5#10592729C212yesmaximal, complemented
6#10592579C211yesmaximal, characteristic
7#10592572C211yesmaximal, characteristic, complemented
8#10592501C212yesmaximal
9#10592488C212yesmaximal, complemented
10C4×#3746C211yesmaximal, characteristic, complemented
11#10592388C211yesmaximal, characteristic
12#79720—34nomaximal
13C2×Q8×C3⋊D3C2212yes—
14C2×C32×Q16C2211yescharacteristic
15C32×C4×Q8C2212yes—
16Q8×C3⋊Dic3C2212yes—
17C2×#3756C2212yes—
18Q16×C3⋊D3C418yescomplemented
19C2×#3746C2211yescharacteristic
20C2×#3743C2211yescharacteristic
21C2×C8×C3⋊D3C2211yescharacteristic
22#64101—22no—
23#64101C2212yes—
24#64096—22no—
25#64096C2212yes—
26#64081—22no—
27#64081C2211yescharacteristic
28C4×#634—22no—
29C4×#634C2212yes—
30C32×#107C2211yescharacteristic
31C32×#103C2212yes—
32C32×C4×C8C2211yescharacteristic
33#63662C2212yes—
34#63645C2211yescharacteristic
35#63643C2212yes—
36#63642C2211yescharacteristic
37C8×C3⋊Dic3C2211yescharacteristic
38#63637C2212yes—
39#63635C2212yes—
40#63630C2211yescharacteristic
41C2×D3×Q16—34no—
42#5572—38no—
43C3×C4×Q16D314yescomplemented
44Dic3×Q16—34no—
45#5030—38no—
46#4880—34no—
47#4873—34no—
48#4802—38no—
49#4789—38no—
50C4×Dic12—34no—
51#4689—34no—
52C2×C32×Q8C2312yes—
53Q8×C3⋊D3—24no—
54Q8×C3⋊D3C2×C418yes—
55C2×C4×C3⋊D3—22no—
56C2×C4×C3⋊D3C2311yescharacteristic
57C2×#634—22no—
58C2×#634C2312yes—
59C32×Q16C2×C414yes—
60C2×C32×C8C2311yescharacteristic
61C2×C32×C8C2311yescharacteristic
62C32×C4⋊C4—22no—
63C32×C4⋊C4C2312yes—
64C32×C42—22no—
65C32×C42C2311yescharacteristic
66#3756C2×C418yes—
67#3752—22no—
68#3752D411yescharacteristic
69#3752D411yescharacteristic
70C4⋊(C3⋊Dic3)—22no—
71C4⋊(C3⋊Dic3)C2312yes—
72#3750—22no—
73#3750—22no—
74#3750D411yescharacteristic
75#3750D411yescharacteristic
76C4×C3⋊Dic3—22no—
77C4×C3⋊Dic3—22no—
78C4×C3⋊Dic3—22no—
79C4×C3⋊Dic3—22no—
80C4×C3⋊Dic3C2311yescharacteristic
81C2×C3⋊(C3⋊C8)C2311yescharacteristic
82C2×C3⋊(C3⋊C8)C2311yescharacteristic
83#3746C2×C414yes—
84#3743—22no—
85C8×C3⋊D3C2×C414yes—
86#2950—91noSylow
87C2×D3×Q8—38no—
88C2×C3×Q16C2×D314yes—
89C3×C4×Q8C2×D318yes—
90Q8×Dic3—38no—
91C2×#216—38no—
92D3×Q16—332no—
93C2×Dic12—34no—
94C2×ΓL(1,𝔽25)—34no—
95C2×D3×C8—34no—
96#892—38no—
97#892—68no—
98#887—38no—
99#887—68no—
100#872—34no—
101#872—68no—
102C4×Dic6—38no—
103C4×Dic6—68no—
104C3×#107C2×D314yes—
105C3×#103C2×D318yes—
106C3×C4×C8C2×D314yes—
107#835—38no—
108#818—34no—
109#816—38no—
110#815—34no—
111C8×Dic3—34no—
112#810—38no—
113#808—38no—
114#803—34no—
115C22×C3⋊D3C2×D411yescharacteristic
116C32×Q8—22no—
117C32×Q8C22×C414yes—
118C2×C32×C4—22no—
119C2×C32×C4—22no—
120C2×C32×C4C2×D411yescharacteristic
121C2×C32×C4C2×D411yescharacteristic
122C2×C32×C4C2411yescharacteristic
123C2×C3⋊Dic3—22no—
124C2×C3⋊Dic3—22no—
125C2×C3⋊Dic3C2×D411yescharacteristic
126C2×C3⋊Dic3C2×D411yescharacteristic
127C2×C3⋊Dic3C2×D411yescharacteristic
128C4×C3⋊D3—28no—
129C4×C3⋊D3C22×C412yes—
130C4×C3⋊D3#4112yes—
131#634—22no—
132#634—28no—
133#634C22×C414yes—
134C32×C8—21no—
135C32×C8C22×C412yes—
136C3⋊(C3⋊C8)—21no—
137C3⋊(C3⋊C8)C22×C412yes—
138C22×Q16—91no—
139#519—92no—
140#441—92no—
141#441—92no—
142C4×Q16—91no—
143C4×Q16—91no—
144C4×Q16—91no—
145C4×Q16C3⋊D311yescharacteristic, complemented
146#427—91no—
147#419—92no—
148#405—91no—
149C2×C3×Q8C22×D318yes—
150D3×Q8—332no—
151D3×Q8—616no—
152C2×C4×D3—34no—
153C2×C4×D3—68no—
154C2×Dic6—38no—
155C2×Dic6—68no—
156C3×Q16C4×D3116yes—
157C2×C3×C8C22×D314yes—
158C2×C3×C8C22×D314yes—
159C3×C4⋊C4—28no—
160C3×C4⋊C4C22×D318yes—
161C3×C42—28no—
162C3×C42C22×D314yes—
163#216—332no—
164(C2×D3)⋊C4—34no—
165(C2×D3)⋊C4—34no—
166(C2×D3)⋊C4—68no—
167C4⋊Dic3—38no—
168C4⋊Dic3—68no—
169Dic3⋊C4—34no—
170Dic3⋊C4—34no—
171Dic3⋊C4—68no—
172Dic3⋊C4—68no—
173C4×Dic3—34no—
174C4×Dic3—68no—
175C4×Dic3—68no—
176C4×Dic3—68no—
177C4×Dic3—68no—
178C2×C3⋊C8—34no—
179C2×C3⋊C8—34no—
180Dic12—316no—
181ΓL(1,𝔽25)—68no—
182D3×C8—316no—
183C22×C32C22×D411yescharacteristic
184C2×C3⋊D3—21no—
185C2×C3⋊D3—21no—
186C2×C3⋊D3C4×D412yes—
187C32×C4—24no—
188C32×C4—21no—
189C32×C4—21no—
190C32×C4—42no—
191C32×C4C23×C411yescharacteristic
192C32×C4C2×#4111yescharacteristic
193C3⋊Dic3—24no—
194C3⋊Dic3—21no—
195C3⋊Dic3—21no—
196C3⋊Dic3—42no—
197C3⋊Dic3C4×D412yes—
198C22×Q8—92no—
199C2×Q16—92no—
200C2×Q16—98no—
201C2×Q16—91no—
202C2×Q16C2×C3⋊D311yescharacteristic
203C2×M16—91no—
204C22×C8—91no—
205C4×Q8—92no—
206C4×Q8—92no—
207C4×Q8—92no—
208C4×Q8—182no—
209C4×Q8—182no—
210C4×Q8C2×C3⋊D312yes—
211#117—91no—
212#117—92no—
213#117—182no—
214#117—182no—
215#107—91no—
216#107C2×C3⋊D311yescharacteristic
217#106—92no—
218#103—92no—
219#103—92no—
220#103—92no—
221#103C2×C3⋊D312yes—
222#97—91no—
223#97—91no—
224C4×C8—91no—
225C4×C8C2×C3⋊D311yescharacteristic
226C22×D3—34no—
227C3×Q8—28no—
228C3×Q8C2×C4×D3116yes—
229C2×C3×C4—28no—
230C2×C3×C4—28no—
231C2×C3×C4D3×D414yes—
232C2×C3×C4D3×D414yes—
233C2×C3×C4C23×D314yes—
234C2×Dic3—34no—
235C2×Dic3—34no—
236C2×Dic3—34no—
237C2×Dic3—68no—
238C2×Dic3—68no—
239C4×D3—38no—
240C4×D3—38no—
241C4×D3—632no—
242Dic6—316no—
243Dic6—632no—
244Dic6—68no—
245C3×C8—24no—
246C3×C8C2×C4×D318yes—
247C3⋊C8—38no—
248C3⋊C8—64no—
249C2×C32C2×C4×D411yescharacteristic
250C2×C32#57811yescharacteristic
251C2×C32#57811yescharacteristic
252C3⋊D3—22no—
253C2×Q8—98no—
254C2×Q8—92no—
255C2×Q8—182no—
256C2×Q8—184no—
257C2×Q8C22×C3⋊D312yes—
258C22×C4—91no—
259C22×C4—182no—
260Q16—98no—
261Q16—94no—
262Q16C4×C3⋊D314yes—
263M16—182no—
264C2×C8—91no—
265C2×C8—94no—
266C2×C8—91no—
267C2×C8C22×C3⋊D311yescharacteristic
268C2×C8C22×C3⋊D311yescharacteristic
269C4⋊C4—22no—
270C4⋊C4—91no—
271C4⋊C4—91no—
272C4⋊C4—92no—
273C4⋊C4—182no—
274C4⋊C4—182no—
275C4⋊C4—182no—
276C4⋊C4C22×C3⋊D312yes—
277C22⋊C4—91no—
278C22⋊C4—91no—
279C22⋊C4—182no—
280C42—22no—
281C42—91no—
282C42—182no—
283C42—182no—
284C42—182no—
285C42—182no—
286C42C22×C3⋊D311yescharacteristic
287C22×C3C2×D3×D414yes—
288C2×D3—38no—
289C2×D3—64no—
290C2×D3—64no—
291C3×C4—24no—
292C3×C4—24no—
293C3×C4—216no—
294C3×C4—48no—
295C3×C4C22×C4×D314yes—
296C3×C4D3×#4114yes—
297Dic3—38no—
298Dic3—64no—
299Dic3—64no—
300Dic3—616no—
301Dic3—128no—
302C32#295011yescharacteristic, complemented, Sylow
303C23—91no—
304Q8—22no—
305Q8—94no—
306Q8—188no—
307Q8—182no—
308Q8C2×C4×C3⋊D314yes—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—91no—
312C2×C4—92no—
313C2×C4—92no—
314C2×C4—91no—
315C2×C4—91no—
316C2×C4—188no—
317C2×C4—182no—
318C2×C4—182no—
319C2×C4D4×C3⋊D311yescharacteristic
320C2×C4D4×C3⋊D311yescharacteristic
321C2×C4C23×C3⋊D311yescharacteristic
322C8—21no—
323C8—92no—
324C8—181no—
325C8C2×C4×C3⋊D312yes—
326C2×C3C4×D3×D414yes—
327C2×C3#576614yes—
328C2×C3#577114yes—
329D3—68no—
330C22—92no—
331C22—181no—
332C22—181no—
333C22C2×D4×C3⋊D311yescharacteristic
334C4—21no—
335C4—21no—
336C4—24no—
337C4—42no—
338C4—92no—
339C4—181no—
340C4—181no—
341C4—184no—
342C4—362no—
343C4C22×C4×C3⋊D311yescharacteristic
344C4#41×C3⋊D311yescharacteristic
345C3#7972014yescomplemented
346C2—182no—
347C2C4×D4×C3⋊D311yescharacteristic
348C2#1059623711yescharacteristic
349C2#1059624211yescharacteristic
subgroupG/H
centerC22C2×D4×C3⋊D3
commutatorC32×C4C23×C4
FrattiniC2×C4C23×C3⋊D3
FittingC32×C4×Q16C2
socleC22×C32C22×D4
prime radicalC22×C3⋊D3C2×D4
cosocleC2×C32×C4C24
hypercenterC4×Q16C3⋊D3
hypocenterC32#2950
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q16

Complements

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

NH ≅ G/Ntype
3. C32×C4×Q16C2semidirect
4. Q16×C3⋊Dic3C2semidirect
5. #10592729C2semidirect
7. #10592572C2semidirect
9. #10592488C2semidirect
10. C4×#3746C2semidirect
18. Q16×C3⋊D3C4semidirect
43. C3×C4×Q16D3semidirect
145. C4×Q16C3⋊D3semidirect
302. C32#2950semidirect
345. C3#79720semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×C3⋊D32
#105960432
C32×C4×Q162
Q16×C3⋊Dic32
#105927292
#105925792
#105925722
#105925012
#105924882
C4×#37462
#105923882
#797203