Finite Group Databasebeta

← #11895526

Subgroups — #11895526

Order 1152.

1008 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10596055C211yesmaximal, characteristic
2#10596038C211yesmaximal, characteristic, complemented
3C32×C8×Q8C211yesmaximal, characteristic, complemented
4#10592548C211yesmaximal, characteristic, complemented
5#10592545C211yesmaximal, characteristic
6#10592544C212yesmaximal
7#10592528C212yesmaximal, complemented
8C4×#3746C212yesmaximal, complemented
9#10592396C212yesmaximal
10C4×#3744C211yesmaximal, characteristic
11#10592380C211yesmaximal, characteristic, complemented
12#79934—34nomaximal
13C32×C4×Q8C2211yescharacteristic
14#64155—21no—
15#64153—21no—
16C2×#3746C2212yes—
17C2×#3744C2211yescharacteristic
18#64107—22no—
19#64101—21no—
20#64098—22no—
21#64096—21no—
22#64084C2212yes—
23C4×C4⋊(C3⋊D3)C2211yescharacteristic
24#64079C2212yes—
25#64078C2211yescharacteristic
26#64078C2211yescharacteristic
27C4×#634C2212yes—
28C4×#634C2212yes—
29C4×#634C2211yescharacteristic
30C32×C4⋊C8C2212yes—
31C32×C4⋊C8C2211yescharacteristic
32C32×C4×C8C2211yescharacteristic
33C32×C4×C8C2212yes—
34#63648C2212yes—
35#63648C2211yescharacteristic
36#63645C2212yes—
37#63644C2211yescharacteristic
38#63643C2212yes—
39#63643C2212yes—
40#63643C2212yes—
41#63643C2211yescharacteristic
42#5584—34no—
43#5567—34no—
44C3×C8×Q8D314yescomplemented
45#4849—34no—
46#4846—34no—
47#4845—38no—
48#4829—38no—
49C4×Dic12—38no—
50#4697—38no—
51C4×#204—34no—
52#4681—34no—
53C2×C32×Q8D411yescharacteristic
54C2×C4⋊(C3⋊D3)C2311yescharacteristic
55C2×C4×C3⋊D3—21no—
56C2×#634—21no—
57C2×#634C2312yes—
58C2×#634C2311yescharacteristic
59C2×C32×C8C2311yescharacteristic
60C2×C32×C8C2312yes—
61C2×C32×C8C2311yescharacteristic
62C32×C4⋊C4D411yescharacteristic
63C32×C4⋊C4D412yes—
64C32×C42C2312yes—
65C32×C42C2311yescharacteristic
66#3752—21no—
67#3752—22no—
68#3752—22no—
69C4⋊(C3⋊Dic3)—21no—
70C4⋊(C3⋊Dic3)—21no—
71C4⋊(C3⋊Dic3)—21no—
72C4⋊(C3⋊Dic3)C2311yescharacteristic
73C4⋊(C3⋊Dic3)C2312yes—
74C4⋊(C3⋊Dic3)C2311yescharacteristic
75#3750—21no—
76#3750—22no—
77#3750—22no—
78#3750—22no—
79#3750—22no—
80C4×C3⋊Dic3—22no—
81C4×C3⋊Dic3—21no—
82C4×C3⋊Dic3—22no—
83#3746—24no—
84#3744—22no—
85#3364—91noSylow
86C3×C4×Q8C2×D314yes—
87#946—64no—
88#944—64no—
89C2×Dic12—38no—
90C2×#204—34no—
91#898—68no—
92#892—64no—
93#889—68no—
94#887—64no—
95#875—38no—
96C4×D12—34no—
97#870—38no—
98#869—34no—
99#869—34no—
100C4×Dic6—34no—
101C4×Dic6—38no—
102C4×Dic6—38no—
103C3×C4⋊C8C2×D318yes—
104C3×C4⋊C8C2×D314yes—
105C3×C4×C8C2×D314yes—
106C3×C4×C8C2×D318yes—
107#821—34no—
108#821—38no—
109#818—38no—
110#817—34no—
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—21no—
125C2×C3⋊Dic3—22no—
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—42no—
134#634#4112yes—
135C32×C8—21no—
136C32×C8—22no—
137C32×C8—21no—
138#557—91no—
139#556—91no—
140#488—92no—
141#487—91no—
142#479—92no—
143#474—91no—
144#464—92no—
145#462—91no—
146C8×Q8C3⋊D311yescharacteristic, complemented
147C4×Q16—92no—
148C4×SD16—91no—
149C2×C3×Q8D1214yes—
150C2×D12—34no—
151C2×C4×D3—64no—
152C2×Dic6—34no—
153C2×Dic6—38no—
154C2×Dic6—64no—
155C2×C3×C8C22×D314yes—
156C2×C3×C8C22×D314yes—
157C2×C3×C8C22×D318yes—
158C3×C4⋊C4D1214yes—
159C3×C4⋊C4D1218yes—
160C3×C42C22×D318yes—
161C3×C42C22×D314yes—
162(C2×D3)⋊C4—64no—
163(C2×D3)⋊C4—68no—
164(C2×D3)⋊C4—68no—
165C4⋊Dic3—34no—
166C4⋊Dic3—38no—
167C4⋊Dic3—34no—
168C4⋊Dic3—64no—
169C4⋊Dic3—64no—
170C4⋊Dic3—64no—
171Dic3⋊C4—64no—
172Dic3⋊C4—68no—
173Dic3⋊C4—68no—
174Dic3⋊C4—68no—
175Dic3⋊C4—68no—
176C4×Dic3—68no—
177C4×Dic3—64no—
178C4×Dic3—68no—
179Dic12—616no—
180#204—68no—
181C22×C32C22×D411yescharacteristic
182C2×C3⋊D3—41no—
183C2×C3⋊D3—41no—
184C2×C3⋊D3—41no—
185C32×C4—22no—
186C32×C4—22no—
187C32×C4—21no—
188C32×C4#13512yes—
189C32×C4C2×#4111yescharacteristic
190C32×C42−1+411yescharacteristic
191C3⋊Dic3—22no—
192C3⋊Dic3—42no—
193C3⋊Dic3—42no—
194C3⋊Dic3—41no—
195C3⋊Dic3—41no—
196C2×Q16—92no—
197C2×SD16—91no—
198C4⋊Q8—91no—
199C4⋊Q8—91no—
200C4⋊Q8—181no—
201#126—182no—
202#125—92no—
203#125—182no—
204#124—92no—
205C22⋊Q8—181no—
206C4×Q8—92no—
207C4×Q8—91no—
208C4×Q8—92no—
209C4×Q8—181no—
210C4×Q8C2×C3⋊D311yescharacteristic
211C4×D4—91no—
212#117—181no—
213#107—92no—
214#106—91no—
215C4⋊C8C2×C3⋊D311yescharacteristic
216C4⋊C8C2×C3⋊D312yes—
217#103—91no—
218#103—92no—
219#103—92no—
220#103—92no—
221#102—91no—
222#102—92no—
223C4×C8C2×C3⋊D311yescharacteristic
224C4×C8C2×C3⋊D312yes—
225C22×D3—64no—
226C3×Q8—28no—
227C2×C3×C4C2×D1218yes—
228C2×C3×C4C2×D1218yes—
229C2×C3×C4C2×D1214yes—
230C2×C3×C4C2×D1214yes—
231C2×C3×C4C23×D314yes—
232C2×Dic3—64no—
233C2×Dic3—64no—
234C2×Dic3—64no—
235C2×Dic3—68no—
236C2×Dic3—68no—
237D12—64no—
238D12—64no—
239C4×D3—128no—
240Dic6—38no—
241Dic6—64no—
242Dic6—68no—
243Dic6—68no—
244Dic6—128no—
245C3×C8—24no—
246C3×C8—24no—
247C3×C8—28no—
248C2×C32#54811yescharacteristic
249C2×C32C2×#13511yescharacteristic
250C2×C32#57811yescharacteristic
251C3⋊D3—81no—
252C2×Q8—91no—
253C2×Q8—92no—
254C2×Q8—181no—
255C2×Q8C4⋊(C3⋊D3)11yescharacteristic
256C2×D4—91no—
257C22×C4—181no—
258Q16—184no—
259SD16—182no—
260C2×C8C22×C3⋊D311yescharacteristic
261C2×C8C22×C3⋊D312yes—
262C2×C8C22×C3⋊D311yescharacteristic
263C4⋊C4—91no—
264C4⋊C4—92no—
265C4⋊C4—91no—
266C4⋊C4—181no—
267C4⋊C4—182no—
268C4⋊C4—181no—
269C4⋊C4—182no—
270C4⋊C4—182no—
271C4⋊C4—182no—
272C4⋊C4—181no—
273C4⋊C4—181no—
274C4⋊C4C4⋊(C3⋊D3)11yescharacteristic
275C4⋊C4C4⋊(C3⋊D3)12yes—
276C22⋊C4—182no—
277C22⋊C4—181no—
278C22⋊C4—182no—
279C42—182no—
280C42—182no—
281C42—181no—
282C42C22×C3⋊D311yescharacteristic
283C42C22×C3⋊D312yes—
284C22×C3C22×D1214yes—
285C2×D3—124no—
286C2×D3—124no—
287C2×D3—124no—
288C3×C4—28no—
289C3×C4—28no—
290C3×C4—24no—
291C3×C4#90418yes—
292C3×C4#100714yes—
293C3×C4D3×#4114yes—
294Dic3—68no—
295Dic3—128no—
296Dic3—124no—
297Dic3—124no—
298Dic3—128no—
299C32#336411yescharacteristic, complemented, Sylow
300C23—181no—
301Q8—22no—
302Q8—92no—
303Q8—181no—
304Q8—182no—
305Q8—182no—
306Q8—362no—
307D4—181no—
308D4—181no—
309C2×C4—181no—
310C2×C4—181no—
311C2×C4—182no—
312C2×C4—181no—
313C2×C4—182no—
314C2×C4—362no—
315C2×C4C2×C4⋊(C3⋊D3)12yes—
316C2×C4C2×C4⋊(C3⋊D3)11yescharacteristic
317C2×C4C2×C4⋊(C3⋊D3)12yes—
318C2×C4C2×C4⋊(C3⋊D3)11yescharacteristic
319C2×C4C23×C3⋊D311yescharacteristic
320C8—21no—
321C8—21no—
322C8—22no—
323C2×C3#557614yes—
324C2×C3C2×#90414yes—
325C2×C3#575314yes—
326D3—244no—
327C22—361no—
328C22—361no—
329C22—361no—
330C22C22×C4⋊(C3⋊D3)11yescharacteristic
331C4—22no—
332C4—22no—
333C4—21no—
334C4—182no—
335C4—361no—
336C4—362no—
337C4—361no—
338C4—362no—
339C4#6411312yes—
340C4#6436411yescharacteristic
341C4#41×C3⋊D311yescharacteristic
342C3#7993414yescomplemented
343C2—721no—
344C2#1059604711yescharacteristic
345C2C2×#6411311yescharacteristic
346C2#1059622411yescharacteristic
subgroupG/H
centerC22C22×C4⋊(C3⋊D3)
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C23×C3⋊D3
FittingC32×C8×Q8C2
socleC22×C32C22×D4
prime radicalC2×C4⋊(C3⋊D3)C23
hypercenterC8×Q8C3⋊D3
hypocenterC32#3364
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4×Q8 ◁ C8×Q8

Complements

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

NH ≅ G/Ntype
2. #10596038C2semidirect
3. C32×C8×Q8C2semidirect
4. #10592548C2semidirect
7. #10592528C2semidirect
8. C4×#3746C2semidirect
11. #10592380C2semidirect
44. C3×C8×Q8D3semidirect
146. C8×Q8C3⋊D3semidirect
299. C32#3364semidirect
342. C3#79934semidirect

Maximal subgroups

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

subgroupindex
#105960552
#105960382
C32×C8×Q82
#105925482
#105925452
#105925442
#105925282
C4×#37462
#105923962
C4×#37442
#105923802
#799343