Finite Group Databasebeta

← #11897626

Subgroups — #11897626

Order 1152.

1020 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#10596050C211yesmaximal, characteristic
2Q8×#634C211yesmaximal, characteristic, complemented
3C32×#446C211yesmaximal, characteristic, complemented
4#10592549C211yesmaximal, characteristic, complemented
5#10592545C212yesmaximal
6#10592541C211yesmaximal, characteristic
7#10592528C212yesmaximal, complemented
8#10592409C212yesmaximal
9#10592401C212yesmaximal, complemented
10C4×#3744C211yesmaximal, characteristic
11#10592381C211yesmaximal, characteristic, complemented
12#79976—34nomaximal
13C32×C4×Q8C2211yescharacteristic
14#64155—21no—
15Q8×C3⋊Dic3—21no—
16C2×#3744C2211yescharacteristic
17#64105—22no—
18C4⋊C4×C3⋊D3—21no—
19#64097—21no—
20#64097—22no—
21C4×C4⋊(C3⋊D3)C2211yescharacteristic
22C4×C4⋊(C3⋊D3)C2212yes—
23#64079C2212yes—
24#64078C2211yescharacteristic
25#64078C2212yes—
26#64078C2211yescharacteristic
27C4×#634C2212yes—
28C4×#634C2211yescharacteristic
29C32×C4⋊C8C2212yes—
30C32×C4⋊C8C2211yescharacteristic
31C32×#97C2212yes—
32C32×C4×C8C2211yescharacteristic
33#63648C2212yes—
34#63648C2211yescharacteristic
35#63645C2212yes—
36#63645C2212yes—
37#63645C2212yes—
38#63644C2212yes—
39#63644C2211yescharacteristic
40#63643C2211yescharacteristic
41#63643C2212yes—
42#5579—34no—
43Q8×Dic6—34no—
44C3×#446D314yescomplemented
45#4850—34no—
46#4846—38no—
47#4842—34no—
48#4829—38no—
49#4710—38no—
50#4702—38no—
51C4×#204—34no—
52#4682—34no—
53C2×C32×Q8D411yescharacteristic
54C2×C4⋊(C3⋊D3)C2311yescharacteristic
55C2×C4×C3⋊D3—21no—
56C2×C4×C3⋊D3—22no—
57C2×#634—22no—
58C2×#634—21no—
59C2×#634—21no—
60C2×#634C2311yescharacteristic
61C2×C32×C8C2312yes—
62C2×C32×C8C2311yescharacteristic
63C2×C32×C8C2311yescharacteristic
64C32×C4⋊C4D411yescharacteristic
65C32×C4⋊C4D412yes—
66C32×C42C2311yescharacteristic
67C32×C42C2312yes—
68#3752—21no—
69#3752—22no—
70C4⋊(C3⋊Dic3)—22no—
71C4⋊(C3⋊Dic3)—21no—
72C4⋊(C3⋊Dic3)—21no—
73C4⋊(C3⋊Dic3)C2312yes—
74C4⋊(C3⋊Dic3)C2311yescharacteristic
75C4⋊(C3⋊Dic3)C2312yes—
76C4⋊(C3⋊Dic3)C2311yescharacteristic
77#3750—22no—
78#3750—21no—
79#3750—22no—
80#3750—22no—
81C4×C3⋊Dic3—22no—
82C4×C3⋊Dic3—21no—
83#3744Q814yescomplemented
84#3385—91noSylow
85C3×C4×Q8C2×D314yes—
86#946—64no—
87Q8×Dic3—64no—
88C2×#204—34no—
89#896—68no—
90D3×C4⋊C4—64no—
91#888—64no—
92#888—68no—
93C4×D12—34no—
94C4×D12—38no—
95#870—38no—
96#869—34no—
97#869—34no—
98#869—38no—
99C4×Dic6—34no—
100C4×Dic6—38no—
101C3×C4⋊C8C2×D318yes—
102C3×C4⋊C8C2×D314yes—
103C3×#97C2×D318yes—
104C3×C4×C8C2×D314yes—
105#821—34no—
106#821—38no—
107#818—38no—
108#818—38no—
109#818—38no—
110#817—34no—
111#817—38no—
112#816—34no—
113#816—38no—
114C22×C3⋊D3—21no—
115C32×Q8—22no—
116C2×C32×C4C2×D412yes—
117C2×C32×C4C2×D412yes—
118C2×C32×C4C2×D411yescharacteristic
119C2×C32×C4C2×D411yescharacteristic
120C2×C32×C4C2411yescharacteristic
121C2×C3⋊Dic3—21no—
122C2×C3⋊Dic3—22no—
123C2×C3⋊Dic3—21no—
124C2×C3⋊Dic3—21no—
125C2×C3⋊Dic3—22no—
126C4⋊(C3⋊D3)—21no—
127C4⋊(C3⋊D3)C2×Q812yes—
128C4×C3⋊D3—42no—
129C4×C3⋊D3—82no—
130#634—21no—
131#634—81no—
132#634—82no—
133#634—81no—
134#634C2×Q812yes—
135C32×C8—22no—
136C32×C8—21no—
137C32×C8C2×Q812yes—
138Q82—91no—
139#554—91no—
140#501—92no—
141#500—91no—
142#479—92no—
143#477—91no—
144#476—91no—
145#474—92no—
146#446C3⋊D311yescharacteristic, complemented
147#440—92no—
148C4×SD16—91no—
149C2×C3×Q8D1214yes—
150C2×D12—34no—
151C2×C4×D3—64no—
152C2×C4×D3—68no—
153C2×Dic6—34no—
154C2×Dic6—68no—
155C2×Dic6—64no—
156C2×Dic6—64no—
157C2×C3×C8C22×D318yes—
158C2×C3×C8C22×D314yes—
159C2×C3×C8C22×D314yes—
160C3×C4⋊C4D1214yes—
161C3×C4⋊C4D1218yes—
162C3×C42C22×D314yes—
163C3×C42C22×D318yes—
164(C2×D3)⋊C4—64no—
165(C2×D3)⋊C4—68no—
166C4⋊Dic3—34no—
167C4⋊Dic3—38no—
168C4⋊Dic3—34no—
169C4⋊Dic3—38no—
170C4⋊Dic3—68no—
171C4⋊Dic3—64no—
172C4⋊Dic3—64no—
173Dic3⋊C4—68no—
174Dic3⋊C4—64no—
175Dic3⋊C4—68no—
176Dic3⋊C4—68no—
177C4×Dic3—68no—
178C4×Dic3—64no—
179#204—316no—
180C22×C32C22×D411yescharacteristic
181C2×C3⋊D3—22no—
182C2×C3⋊D3—41no—
183C2×C3⋊D3—41no—
184C32×C4—22no—
185C32×C4—21no—
186C32×C4—22no—
187C32×C4#13712yes—
188C32×C4C22×Q811yescharacteristic
189C32×C42−1+411yescharacteristic
190C3⋊Dic3—22no—
191C3⋊Dic3—41no—
192C3⋊Dic3—42no—
193C3⋊Dic3—41no—
194C3⋊Dic3—42no—
195C2×SD16—91no—
196C4⋊Q8—91no—
197C4⋊Q8—92no—
198C4⋊Q8—91no—
199C4⋊Q8—181no—
200C4⋊Q8—182no—
201#125—92no—
202C22⋊Q8—181no—
203C22⋊Q8—182no—
204C4×Q8—91no—
205C4×Q8—92no—
206C4×Q8—181no—
207C4×Q8C2×C3⋊D311yescharacteristic
208C4×D4—91no—
209C4×D4—92no—
210C2×C4⋊C4—181no—
211#107—92no—
212#107—92no—
213#107—92no—
214#106—91no—
215#106—92no—
216C4⋊C8C2×C3⋊D312yes—
217C4⋊C8C2×C3⋊D311yescharacteristic
218#103—91no—
219#103—92no—
220#102—91no—
221#102—92no—
222#97C2×C3⋊D312yes—
223C4×C8C2×C3⋊D311yescharacteristic
224C22×D3—64no—
225C3×Q8—28no—
226C2×C3×C4C2×D1218yes—
227C2×C3×C4C2×D1218yes—
228C2×C3×C4C2×D1214yes—
229C2×C3×C4C2×D1214yes—
230C2×C3×C4C23×D314yes—
231C2×Dic3—64no—
232C2×Dic3—68no—
233C2×Dic3—64no—
234C2×Dic3—64no—
235C2×Dic3—68no—
236D12—38no—
237D12—64no—
238C4×D3—128no—
239C4×D3—248no—
240Dic6—38no—
241Dic6—64no—
242Dic6—244no—
243Dic6—248no—
244Dic6—244no—
245C3×C8—28no—
246C3×C8—24no—
247C3×C8D3×Q818yes—
248C2×C32D4×Q811yescharacteristic
249C2×C32C2×#13711yescharacteristic
250C2×C32#57611yescharacteristic
251C3⋊D3—42no—
252C2×Q8—91no—
253C2×Q8—181no—
254C2×Q8—181no—
255C2×Q8—182no—
256C2×Q8C4⋊(C3⋊D3)11yescharacteristic
257C2×D4—91no—
258C22×C4—181no—
259C22×C4—182no—
260SD16—94no—
261C2×C8C22×C3⋊D311yescharacteristic
262C2×C8C22×C3⋊D312yes—
263C2×C8C22×C3⋊D311yescharacteristic
264C4⋊C4—91no—
265C4⋊C4—92no—
266C4⋊C4—91no—
267C4⋊C4—92no—
268C4⋊C4—182no—
269C4⋊C4—181no—
270C4⋊C4—181no—
271C4⋊C4—182no—
272C4⋊C4—182no—
273C4⋊C4—181no—
274C4⋊C4—182no—
275C4⋊C4C4⋊(C3⋊D3)11yescharacteristic
276C4⋊C4C4⋊(C3⋊D3)12yes—
277C22⋊C4—181no—
278C22⋊C4—182no—
279C42—182no—
280C42—181no—
281C42C22×C3⋊D311yescharacteristic
282C42C22×C3⋊D312yes—
283C22×C3C22×D1214yes—
284C2×D3—68no—
285C2×D3—124no—
286C2×D3—124no—
287C3×C4—28no—
288C3×C4—24no—
289C3×C4—28no—
290C3×C4#90918yes—
291C3×C4C2×D3×Q814yes—
292C3×C4#101014yes—
293Dic3—68no—
294Dic3—124no—
295Dic3—128no—
296Dic3—124no—
297Dic3—128no—
298C32#338511yescharacteristic, complemented, Sylow
299C23—181no—
300Q8—22no—
301Q8—92no—
302Q8—181no—
303Q8—722no—
304Q8—721no—
305Q8—721no—
306D4—92no—
307D4—181no—
308C2×C4—181no—
309C2×C4—182no—
310C2×C4—181no—
311C2×C4—181no—
312C2×C4—182no—
313C2×C4—362no—
314C2×C4—722no—
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—22no—
322C8Q8×C3⋊D312yes—
323C2×C3Q8×D1214yes—
324C2×C3C2×#90914yes—
325C2×C3#575214yes—
326D3—128no—
327C22—182no—
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#6411812yes—
340C4C2×Q8×C3⋊D311yescharacteristic
341C4#6436711yescharacteristic
342C3#7997614yescomplemented
343C2—362no—
344C2Q8×C4⋊(C3⋊D3)11yescharacteristic
345C2C2×#6411811yescharacteristic
346C2#1059622311yescharacteristic
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#3385
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. #10592549C2semidirect
7. #10592528C2semidirect
9. #10592401C2semidirect
11. #10592381C2semidirect
44. C3×#446D3semidirect
83. #3744Q8semidirect
146. #446C3⋊D3semidirect
298. C32#3385semidirect
342. C3#79976semidirect

Maximal subgroups

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

subgroupindex
#105960502
Q8×#6342
C32×#4462
#105925492
#105925452
#105925412
#105925282
#105924092
#105924012
C4×#37442
#105923812
#799763