Finite Group Databasebeta

← #11897617

Subgroups — #11897617

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#10596201C211yesmaximal, characteristic
2Q8×#634C211yesmaximal, characteristic, complemented
3C32×#477C211yesmaximal, characteristic, complemented
4#10592777C211yesmaximal, characteristic, complemented
5C4×#3755C211yesmaximal, characteristic
6#10592589C212yesmaximal
7#10592575C212yesmaximal, complemented
8#10592541C211yesmaximal, characteristic
9#10592530C211yesmaximal, characteristic, complemented
10#10592516C212yesmaximal
11#10592498C212yesmaximal, complemented
12#81137—34nomaximal
13C32×C4⋊Q8C2211yescharacteristic
14C32×C4×Q8C2211yescharacteristic
15#64155—22no—
16Q8×C3⋊Dic3C2212yes—
17C2×#3755C2211yescharacteristic
18#64106—21no—
19#64102C2212yes—
20C4⋊C4×C3⋊D3—21no—
21#64099C2212yes—
22#64097—22no—
23#64097C2212yes—
24C4×C4⋊(C3⋊D3)C2211yescharacteristic
25#64078—21no—
26#64078C2211yescharacteristic
27C4×#634—21no—
28C4×#634C2211yescharacteristic
29C32×#107C2212yes—
30C32×C4⋊C8C2211yescharacteristic
31C32×#103C2212yes—
32#63662C2211yescharacteristic
33#63648C2212yes—
34#63644C2212yes—
35#63642C2212yes—
36#63641C2212yes—
37#63636C2211yescharacteristic
38#63635C2211yescharacteristic
39#63634C2212yes—
40#63634C2212yes—
41C4×C3⋊(C3⋊C8)C2211yescharacteristic
42#5730—34no—
43Q8×Dic6—34no—
44C3×#477D314yescomplemented
45#5078—34no—
46C4×#215—34no—
47#4890—38no—
48#4876—38no—
49#4842—34no—
50#4831—34no—
51#4817—38no—
52#4799—38no—
53C2×C32×Q8—21no—
54C2×C32×Q8C2311yescharacteristic
55C2×C4⋊(C3⋊D3)C2311yescharacteristic
56C2×C4×C3⋊D3—21no—
57C2×C4×C3⋊D3—22no—
58C2×#634—21no—
59C2×#634—22no—
60C2×#634D411yescharacteristic
61C2×C32×C8C2312yes—
62C32×C4⋊C4—21no—
63C32×C4⋊C4—21no—
64C32×C4⋊C4C2311yescharacteristic
65C32×C4⋊C4C2312yes—
66C32×C42—21no—
67C32×C42C2311yescharacteristic
68#3755Q814yescomplemented
69#3752—21no—
70#3752—22no—
71C4⋊(C3⋊Dic3)—21no—
72C4⋊(C3⋊Dic3)—22no—
73C4⋊(C3⋊Dic3)—22no—
74C4⋊(C3⋊Dic3)D411yescharacteristic
75C4⋊(C3⋊Dic3)C2312yes—
76C4⋊(C3⋊Dic3)C2311yescharacteristic
77#3750—22no—
78#3750—22no—
79#3750D412yes—
80C4×C3⋊Dic3—22no—
81C4×C3⋊Dic3C2312yes—
82C2×C3⋊(C3⋊C8)C2311yescharacteristic
83C2×C3⋊(C3⋊C8)C2311yescharacteristic
84#3385—91noSylow
85C3×C4⋊Q8C2×D314yes—
86C3×C4×Q8C2×D314yes—
87#946—68no—
88Q8×Dic3—38no—
89C2×#215—34no—
90#897—64no—
91#893—38no—
92D3×C4⋊C4—64no—
93#890—38no—
94#888—38no—
95#888—68no—
96C4×D12—34no—
97#869—34no—
98#869—64no—
99C4×Dic6—34no—
100C4×Dic6—64no—
101C3×#107C2×D318yes—
102C3×C4⋊C8C2×D314yes—
103C3×#103C2×D318yes—
104#835—34no—
105#821—38no—
106#817—38no—
107#815—38no—
108Dic3⋊C8—38no—
109#809—34no—
110#808—34no—
111#807—38no—
112#807—38no—
113C4×C3⋊C8—34no—
114C22×C3⋊D3—21no—
115C32×Q8—21no—
116C32×Q8—81no—
117C32×Q8C2×Q812yes—
118C2×C32×C4—21no—
119C2×C32×C4—21no—
120C2×C32×C4—22no—
121C2×C32×C4C2×D411yescharacteristic
122C2×C32×C4C2×D411yescharacteristic
123C2×C32×C4C2411yescharacteristic
124C2×C3⋊Dic3—22no—
125C2×C3⋊Dic3—21no—
126C2×C3⋊Dic3C2×D412yes—
127C2×C3⋊Dic3C2×D412yes—
128C4⋊(C3⋊D3)—21no—
129C4⋊(C3⋊D3)C2×Q812yes—
130C4×C3⋊D3—42no—
131C4×C3⋊D3—82no—
132#634—22no—
133#634—82no—
134#634—81no—
135C32×C8—22no—
136C3⋊(C3⋊C8)—21no—
137C3⋊(C3⋊C8)C2×Q812yes—
138Q82—91no—
139#554—91no—
140#501—92no—
141#500—91no—
142#479—92no—
143#477C3⋊D311yescharacteristic, complemented
144#476—91no—
145#474—92no—
146#446—91no—
147#440—92no—
148C4×SD16—91no—
149C2×C3×Q8—24no—
150C2×C3×Q8C22×D314yes—
151C2×D12—34no—
152C2×C4×D3—64no—
153C2×C4×D3—68no—
154C2×Dic6—34no—
155C2×Dic6—68no—
156C2×Dic6—64no—
157C2×C3×C8C22×D318yes—
158C3×C4⋊C4—24no—
159C3×C4⋊C4—24no—
160C3×C4⋊C4C22×D318yes—
161C3×C4⋊C4C22×D314yes—
162C3×C42—24no—
163C3×C42C22×D314yes—
164#215—316no—
165(C2×D3)⋊C4—68no—
166(C2×D3)⋊C4—64no—
167C4⋊Dic3—34no—
168C4⋊Dic3—38no—
169C4⋊Dic3—34no—
170C4⋊Dic3—68no—
171C4⋊Dic3—68no—
172C4⋊Dic3—64no—
173Dic3⋊C4—38no—
174Dic3⋊C4—68no—
175Dic3⋊C4—68no—
176C4×Dic3—38no—
177C4×Dic3—68no—
178C2×C3⋊C8—34no—
179C2×C3⋊C8—34no—
180C22×C32C22×D411yescharacteristic
181C2×C3⋊D3—22no—
182C2×C3⋊D3—41no—
183C2×C3⋊D3—41no—
184C32×C4—22no—
185C32×C4—21no—
186C32×C4—41no—
187C32×C4—42no—
188C32×C4#13712yes—
189C32×C4C22×Q811yescharacteristic
190C32×C42−1+411yescharacteristic
191C3⋊Dic3—22no—
192C3⋊Dic3—22no—
193C3⋊Dic3—41no—
194C3⋊Dic3—42no—
195C2×SD16—91no—
196C4⋊Q8—92no—
197C4⋊Q8—91no—
198C4⋊Q8—181no—
199C4⋊Q8—182no—
200C4⋊Q8C2×C3⋊D311yescharacteristic
201#125—92no—
202C22⋊Q8—181no—
203C22⋊Q8—182no—
204C4×Q8—91no—
205C4×Q8—92no—
206C4×Q8—181no—
207C4×Q8C2×C3⋊D311yescharacteristic
208C4×D4—92no—
209C4×D4—91no—
210C2×C4⋊C4—181no—
211#107—92no—
212#107—92no—
213#107C2×C3⋊D312yes—
214#106—92no—
215#106—91no—
216C4⋊C8—92no—
217C4⋊C8C2×C3⋊D311yescharacteristic
218#103—91no—
219#103C2×C3⋊D312yes—
220#102—92no—
221#102—91no—
222#97—92no—
223C4×C8—91no—
224C22×D3—64no—
225C3×Q8—24no—
226C3×Q8—84no—
227C3×Q8D3×Q818yes—
228C2×C3×C4—24no—
229C2×C3×C4—24no—
230C2×C3×C4—28no—
231C2×C3×C4D3×D414yes—
232C2×C3×C4D3×D414yes—
233C2×C3×C4C23×D314yes—
234C2×Dic3—38no—
235C2×Dic3—38no—
236C2×Dic3—68no—
237C2×Dic3—64no—
238D12—38no—
239D12—64no—
240C4×D3—128no—
241C4×D3—248no—
242Dic6—68no—
243Dic6—248no—
244Dic6—244no—
245C3×C8—28no—
246C3⋊C8—38no—
247C3⋊C8—64no—
248C2×C32D4×Q811yescharacteristic
249C2×C32C2×#13711yescharacteristic
250C2×C32#57611yescharacteristic
251C3⋊D3—42no—
252C2×Q8—21no—
253C2×Q8—91no—
254C2×Q8—182no—
255C2×Q8—181no—
256C2×Q8C22×C3⋊D311yescharacteristic
257C2×D4—91no—
258C22×C4—182no—
259C22×C4—181no—
260SD16—94no—
261C2×C8—91no—
262C2×C8—91no—
263C2×C8C22×C3⋊D312yes—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—91no—
267C4⋊C4—92no—
268C4⋊C4—92no—
269C4⋊C4—91no—
270C4⋊C4—182no—
271C4⋊C4—182no—
272C4⋊C4—182no—
273C4⋊C4—182no—
274C4⋊C4—181no—
275C4⋊C4C22×C3⋊D311yescharacteristic
276C4⋊C4C22×C3⋊D312yes—
277C22⋊C4—182no—
278C22⋊C4—181no—
279C42—21no—
280C42—92no—
281C42—182no—
282C42C22×C3⋊D311yescharacteristic
283C22×C3C2×D3×D414yes—
284C2×D3—68no—
285C2×D3—124no—
286C2×D3—124no—
287C3×C4—28no—
288C3×C4—24no—
289C3×C4—44no—
290C3×C4—48no—
291C3×C4#91818yes—
292C3×C4C2×D3×Q814yes—
293C3×C4#101014yes—
294Dic3—68no—
295Dic3—68no—
296Dic3—124no—
297Dic3—128no—
298C32#338511yescharacteristic, complemented, Sylow
299C23—181no—
300Q8—21no—
301Q8—81no—
302Q8—182no—
303Q8—721no—
304Q8—722no—
305Q8Q8×C3⋊D312yes—
306D4—92no—
307D4—181no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—92no—
312C2×C4—92no—
313C2×C4—182no—
314C2×C4—181no—
315C2×C4—362no—
316C2×C4—722no—
317C2×C4D4×C3⋊D311yescharacteristic
318C2×C4D4×C3⋊D311yescharacteristic
319C2×C4C23×C3⋊D311yescharacteristic
320C8—22no—
321C8—92no—
322C8—181no—
323C2×C3#563214yes—
324C2×C3C2×#91814yes—
325C2×C3#577414yes—
326D3—128no—
327C22—182no—
328C22—361no—
329C22—361no—
330C22C2×D4×C3⋊D311yescharacteristic
331C4—22no—
332C4—21no—
333C4—42no—
334C4—41no—
335C4—182no—
336C4—182no—
337C4—361no—
338C4—362no—
339C4#6412712yes—
340C4C2×Q8×C3⋊D311yescharacteristic
341C4#6436711yescharacteristic
342C3#8113714yescomplemented
343C2—362no—
344C2#1059610311yescharacteristic
345C2C2×#6412711yescharacteristic
346C2#1059624511yescharacteristic
subgroupG/H
centerC22C2×D4×C3⋊D3
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C23×C3⋊D3
FittingC32×#477C2
socleC22×C32C22×D4
prime radicalC2×C4⋊(C3⋊D3)C23
hypercenter#477C3⋊D3
hypocenterC32#3385
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
2. Q8×#634C2semidirect
3. C32×#477C2semidirect
4. #10592777C2semidirect
7. #10592575C2semidirect
9. #10592530C2semidirect
11. #10592498C2semidirect
44. C3×#477D3semidirect
68. #3755Q8semidirect
143. #477C3⋊D3semidirect
298. C32#3385semidirect
342. C3#81137semidirect

Maximal subgroups

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

subgroupindex
#105962012
Q8×#6342
C32×#4772
#105927772
C4×#37552
#105925892
#105925752
#105925412
#105925302
#105925162
#105924982
#811373