Finite Group Databasebeta

← #11896031

Subgroups — #11896031

Order 1152.

1014 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
1#10596055C211yesmaximal, characteristic
2Q8×#634C211yesmaximal, characteristic, complemented
3C32×#446C211yesmaximal, characteristic, complemented
4#10592550C212yesmaximal, complemented
5#10592548C211yesmaximal, characteristic, complemented
6#10592544C212yesmaximal
7#10592541C211yesmaximal, characteristic
8#10592415C212yesmaximal, complemented
9#10592413C212yesmaximal
10C4×#3744C211yesmaximal, characteristic
11#10592380C211yesmaximal, characteristic, complemented
12#79981—34nomaximal
13C32×C4×Q8C2211yescharacteristic
14#64155—21no—
15Q8×C3⋊Dic3—21no—
16C2×#3746C2212yes—
17C2×#3744C2211yescharacteristic
18#64107—22no—
19#64101—21no—
20#64097—21no—
21#64097—22no—
22#64084C2212yes—
23C4×C4⋊(C3⋊D3)C2211yescharacteristic
24#64078C2211yescharacteristic
25#64078C2211yescharacteristic
26#64078C2212yes—
27C4×#634C2212yes—
28C4×#634C2211yescharacteristic
29C4×#634C2212yes—
30C32×C4⋊C8C2211yescharacteristic
31C32×C4⋊C8C2212yes—
32C32×#97C2212yes—
33C32×C4×C8C2211yescharacteristic
34#63648C2212yes—
35#63648C2211yescharacteristic
36#63644C2212yes—
37#63644C2211yescharacteristic
38#63643C2212yes—
39#63643C2212yes—
40#63643C2212yes—
41#63643C2211yescharacteristic
42#5584—34no—
43Q8×Dic6—34no—
44C3×#446D314yescomplemented
45#4851—38no—
46#4849—34no—
47#4845—38no—
48#4842—34no—
49#4716—38no—
50#4714—38no—
51C4×#204—34no—
52#4681—34no—
53C2×C32×Q8D411yescharacteristic
54C2×C4⋊(C3⋊D3)C2311yescharacteristic
55C2×C4×C3⋊D3—21no—
56C2×#634—22no—
57C2×#634—21no—
58C2×#634C2312yes—
59C2×#634C2311yescharacteristic
60C2×C32×C8C2311yescharacteristic
61C2×C32×C8C2312yes—
62C2×C32×C8C2311yescharacteristic
63C32×C4⋊C4D411yescharacteristic
64C32×C4⋊C4D412yes—
65C32×C42C2312yes—
66C32×C42C2311yescharacteristic
67#3752—21no—
68#3752—22no—
69#3752—22no—
70C4⋊(C3⋊Dic3)—22no—
71C4⋊(C3⋊Dic3)—21no—
72C4⋊(C3⋊Dic3)—21no—
73C4⋊(C3⋊Dic3)—21no—
74C4⋊(C3⋊Dic3)C2311yescharacteristic
75C4⋊(C3⋊Dic3)C2312yes—
76C4⋊(C3⋊Dic3)C2311yescharacteristic
77#3750—22no—
78#3750—21no—
79#3750—22no—
80C4×C3⋊Dic3—22no—
81C4×C3⋊Dic3—21no—
82C4×C3⋊Dic3—22no—
83#3746—24no—
84#3744—22no—
85#3369—91noSylow
86C3×C4×Q8C2×D314yes—
87#946—64no—
88Q8×Dic3—64no—
89C2×Dic12—38no—
90C2×#204—34no—
91#898—68no—
92#892—64no—
93#888—64no—
94#888—68no—
95#875—38no—
96C4×D12—34no—
97#869—34no—
98#869—38no—
99#869—34no—
100C4×Dic6—34no—
101C4×Dic6—38no—
102C4×Dic6—38no—
103C3×C4⋊C8C2×D314yes—
104C3×C4⋊C8C2×D318yes—
105C3×#97C2×D318yes—
106C3×C4×C8C2×D314yes—
107#821—34no—
108#821—38no—
109#817—34no—
110#817—38no—
111#816—38no—
112#816—38no—
113#816—34no—
114#816—38no—
115C22×C3⋊D3—21no—
116C32×Q8—22no—
117C2×C32×C4C2×D412yes—
118C2×C32×C4C2×D412yes—
119C2×C32×C4C2×D411yescharacteristic
120C2×C32×C4C2×D411yescharacteristic
121C2×C32×C4C2411yescharacteristic
122C2×C3⋊Dic3—21no—
123C2×C3⋊Dic3—21no—
124C2×C3⋊Dic3—22no—
125C2×C3⋊Dic3—21no—
126C2×C3⋊Dic3—22no—
127C4⋊(C3⋊D3)—21no—
128C4⋊(C3⋊D3)—21no—
129C4×C3⋊D3—42no—
130#634—21no—
131#634—22no—
132#634—22no—
133#634—82no—
134#634—81no—
135#634#4112yes—
136C32×C8—21no—
137C32×C8—22no—
138C32×C8—21no—
139Q82—91no—
140#556—91no—
141#489—92no—
142#487—91no—
143#476—91no—
144#475—92no—
145#464—92no—
146#462—91no—
147#446C3⋊D311yescharacteristic, complemented
148#441—92no—
149C4×SD16—91no—
150C2×C3×Q8D1214yes—
151C2×D12—34no—
152C2×C4×D3—64no—
153C2×Dic6—34no—
154C2×Dic6—38no—
155C2×Dic6—68no—
156C2×Dic6—64no—
157C2×C3×C8C22×D314yes—
158C2×C3×C8C22×D314yes—
159C2×C3×C8C22×D318yes—
160C3×C4⋊C4D1214yes—
161C3×C4⋊C4D1218yes—
162C3×C42C22×D314yes—
163C3×C42C22×D318yes—
164(C2×D3)⋊C4—64no—
165(C2×D3)⋊C4—68no—
166(C2×D3)⋊C4—68no—
167C4⋊Dic3—38no—
168C4⋊Dic3—34no—
169C4⋊Dic3—34no—
170C4⋊Dic3—68no—
171C4⋊Dic3—64no—
172C4⋊Dic3—64no—
173C4⋊Dic3—64no—
174Dic3⋊C4—68no—
175Dic3⋊C4—64no—
176Dic3⋊C4—68no—
177C4×Dic3—68no—
178C4×Dic3—64no—
179C4×Dic3—68no—
180Dic12—616no—
181#204—68no—
182C22×C32C22×D411yescharacteristic
183C2×C3⋊D3—41no—
184C2×C3⋊D3—41no—
185C2×C3⋊D3—41no—
186C32×C4—22no—
187C32×C4—22no—
188C32×C4—21no—
189C32×C4#13712yes—
190C32×C4C2×#4111yescharacteristic
191C32×C42−1+411yescharacteristic
192C3⋊Dic3—22no—
193C3⋊Dic3—42no—
194C3⋊Dic3—41no—
195C3⋊Dic3—41no—
196C3⋊Dic3—42no—
197C2×Q16—92no—
198C2×SD16—91no—
199C4⋊Q8—92no—
200C4⋊Q8—91no—
201C4⋊Q8—91no—
202C4⋊Q8—181no—
203C4⋊Q8—182no—
204#126—182no—
205#124—92no—
206C22⋊Q8—181no—
207C4×Q8—92no—
208C4×Q8—91no—
209C4×Q8—92no—
210C4×Q8—181no—
211C4×Q8C2×C3⋊D311yescharacteristic
212C4×D4—91no—
213#117—181no—
214#106—91no—
215#106—92no—
216C4⋊C8C2×C3⋊D311yescharacteristic
217C4⋊C8C2×C3⋊D312yes—
218#103—91no—
219#103—92no—
220#103—92no—
221#103—92no—
222#102—91no—
223#102—92no—
224#97C2×C3⋊D312yes—
225C4×C8C2×C3⋊D311yescharacteristic
226C22×D3—64no—
227C3×Q8—28no—
228C2×C3×C4C2×D1218yes—
229C2×C3×C4C2×D1218yes—
230C2×C3×C4C2×D1214yes—
231C2×C3×C4C2×D1214yes—
232C2×C3×C4C23×D314yes—
233C2×Dic3—64no—
234C2×Dic3—64no—
235C2×Dic3—68no—
236C2×Dic3—64no—
237C2×Dic3—68no—
238D12—64no—
239D12—64no—
240C4×D3—128no—
241Dic6—38no—
242Dic6—64no—
243Dic6—68no—
244Dic6—68no—
245Dic6—248no—
246Dic6—244no—
247C3×C8—24no—
248C3×C8—24no—
249C3×C8—28no—
250C2×C32#54811yescharacteristic
251C2×C32C2×#13711yescharacteristic
252C2×C32#57711yescharacteristic
253C3⋊D3—81no—
254C2×Q8—91no—
255C2×Q8—92no—
256C2×Q8—182no—
257C2×Q8—181no—
258C2×Q8C4⋊(C3⋊D3)11yescharacteristic
259C2×D4—91no—
260C22×C4—181no—
261Q16—184no—
262SD16—182no—
263C2×C8C22×C3⋊D311yescharacteristic
264C2×C8C22×C3⋊D311yescharacteristic
265C2×C8C22×C3⋊D312yes—
266C4⋊C4—91no—
267C4⋊C4—92no—
268C4⋊C4—91no—
269C4⋊C4—181no—
270C4⋊C4—182no—
271C4⋊C4—182no—
272C4⋊C4—181no—
273C4⋊C4—182no—
274C4⋊C4—181no—
275C4⋊C4—181no—
276C4⋊C4C4⋊(C3⋊D3)11yescharacteristic
277C4⋊C4C4⋊(C3⋊D3)12yes—
278C22⋊C4—182no—
279C22⋊C4—181no—
280C22⋊C4—182no—
281C42—182no—
282C42—182no—
283C42—181no—
284C42C22×C3⋊D311yescharacteristic
285C42C22×C3⋊D312yes—
286C22×C3C22×D1214yes—
287C2×D3—124no—
288C2×D3—124no—
289C2×D3—124no—
290C3×C4—28no—
291C3×C4—28no—
292C3×C4—24no—
293C3×C4#90918yes—
294C3×C4#100714yes—
295C3×C4D3×#4114yes—
296Dic3—68no—
297Dic3—124no—
298Dic3—124no—
299Dic3—128no—
300Dic3—128no—
301C32#336911yescharacteristic, complemented, Sylow
302C23—181no—
303Q8—22no—
304Q8—92no—
305Q8—181no—
306Q8—182no—
307Q8—182no—
308Q8—722no—
309Q8—721no—
310D4—181no—
311D4—181no—
312C2×C4—181no—
313C2×C4—181no—
314C2×C4—182no—
315C2×C4—182no—
316C2×C4—181no—
317C2×C4—362no—
318C2×C4C2×C4⋊(C3⋊D3)12yes—
319C2×C4C2×C4⋊(C3⋊D3)12yes—
320C2×C4C2×C4⋊(C3⋊D3)11yescharacteristic
321C2×C4C2×C4⋊(C3⋊D3)11yescharacteristic
322C2×C4C23×C3⋊D311yescharacteristic
323C8—21no—
324C8—21no—
325C8—22no—
326C2×C3#557614yes—
327C2×C3C2×#90914yes—
328C2×C3#575114yes—
329D3—244no—
330C22—361no—
331C22—361no—
332C22—361no—
333C22C22×C4⋊(C3⋊D3)11yescharacteristic
334C4—22no—
335C4—22no—
336C4—21no—
337C4—182no—
338C4—361no—
339C4—362no—
340C4—361no—
341C4—362no—
342C4#6411812yes—
343C4#6436411yescharacteristic
344C4#41×C3⋊D311yescharacteristic
345C3#7998114yescomplemented
346C2—721no—
347C2#1059604711yescharacteristic
348C2C2×#6411811yescharacteristic
349C2#1059622211yescharacteristic
subgroupG/H
centerC22C22×C4⋊(C3⋊D3)
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C23×C3⋊D3
FittingC32×#446C2
socleC22×C32C22×D4
prime radicalC2×C4⋊(C3⋊D3)C23
hypercenter#446C3⋊D3
hypocenterC32#3369
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #446

Complements

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

NH ≅ G/Ntype
2. Q8×#634C2semidirect
3. C32×#446C2semidirect
4. #10592550C2semidirect
5. #10592548C2semidirect
8. #10592415C2semidirect
11. #10592380C2semidirect
44. C3×#446D3semidirect
147. #446C3⋊D3semidirect
301. C32#3369semidirect
345. C3#79981semidirect

Maximal subgroups

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

subgroupindex
#105960552
Q8×#6342
C32×#4462
#105925502
#105925482
#105925442
#105925412
#105924152
#105924132
C4×#37442
#105923802
#799813