Finite Group Databasebeta

← #11896037

Subgroups — #11896037

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#10596203C211yesmaximal, characteristic
2Q8×#634C211yesmaximal, characteristic, complemented
3C32×#462C211yesmaximal, characteristic, complemented
4#10592884C212yesmaximal
5#10592883C212yesmaximal, complemented
6#10592778C211yesmaximal, characteristic, complemented
7C4×#3755C211yesmaximal, characteristic
8#10592541C211yesmaximal, characteristic
9#10592530C211yesmaximal, characteristic, complemented
10#10592493C212yesmaximal
11#10592490C212yesmaximal, complemented
12#80464—34nomaximal
13C2×C32×Q16C2212yes—
14C32×C4⋊Q8C2211yescharacteristic
15C32×C4×Q8C2211yescharacteristic
16#64156C2212yes—
17Q8×C3⋊Dic3C2212yes—
18Q8×C3⋊Dic3C2212yes—
19C2×#3755C2211yescharacteristic
20#64107—22no—
21#64106—21no—
22#64101—21no—
23#64097—22no—
24#64097C2212yes—
25C4×C4⋊(C3⋊D3)C2211yescharacteristic
26#64078—21no—
27#64078C2211yescharacteristic
28C4×#634—21no—
29C4×#634C2211yescharacteristic
30C32×C4⋊C8C2211yescharacteristic
31C32×#103C2212yes—
32#63662C2212yes—
33#63662C2212yes—
34#63662C2211yescharacteristic
35#63648C2212yes—
36#63644C2212yes—
37#63642C2212yes—
38#63641C2212yes—
39#63636C2211yescharacteristic
40#63635C2211yescharacteristic
41C4×C3⋊(C3⋊C8)C2211yescharacteristic
42#5732—34no—
43Q8×Dic6—34no—
44C3×#462D314yescomplemented
45#5185—38no—
46#5184—38no—
47#5079—34no—
48C4×#215—34no—
49#4842—34no—
50#4831—34no—
51#4794—38no—
52#4791—38no—
53C2×C32×Q8C2312yes—
54C2×C32×Q8C2311yescharacteristic
55C2×C4⋊(C3⋊D3)C2311yescharacteristic
56C2×C4×C3⋊D3—21no—
57C2×#634—21no—
58C2×#634—22no—
59C2×#634D411yescharacteristic
60C32×Q16—24no—
61C2×C32×C8C2312yes—
62C32×C4⋊C4—21no—
63C32×C4⋊C4—21no—
64C32×C4⋊C4—21no—
65C32×C4⋊C4C2311yescharacteristic
66C32×C42—21no—
67C32×C42C2311yescharacteristic
68#3755—22no—
69#3752—21no—
70#3752—22no—
71#3752—22no—
72C4⋊(C3⋊Dic3)—21no—
73C4⋊(C3⋊Dic3)—22no—
74C4⋊(C3⋊Dic3)—22no—
75C4⋊(C3⋊Dic3)D411yescharacteristic
76C4⋊(C3⋊Dic3)C2312yes—
77C4⋊(C3⋊Dic3)C2311yescharacteristic
78#3750—22no—
79#3750D412yes—
80C4×C3⋊Dic3—22no—
81C4×C3⋊Dic3—22no—
82C4×C3⋊Dic3C2312yes—
83C2×C3⋊(C3⋊C8)C2311yescharacteristic
84C2×C3⋊(C3⋊C8)C2311yescharacteristic
85#3369—91noSylow
86C2×C3×Q16C2×D318yes—
87C3×C4⋊Q8C2×D314yes—
88C3×C4×Q8C2×D314yes—
89#947—38no—
90Q8×Dic3—38no—
91Q8×Dic3—38no—
92C2×#215—34no—
93#898—68no—
94#897—64no—
95#892—64no—
96#888—38no—
97#888—68no—
98C4×D12—34no—
99#869—34no—
100#869—64no—
101C4×Dic6—34no—
102C4×Dic6—64no—
103C3×C4⋊C8C2×D314yes—
104C3×#103C2×D318yes—
105#835—34no—
106#835—38no—
107#835—38no—
108#821—38no—
109#817—38no—
110#815—38no—
111Dic3⋊C8—38no—
112#809—34no—
113#808—34no—
114C4×C3⋊C8—34no—
115C22×C3⋊D3—21no—
116C32×Q8—21no—
117C32×Q8—22no—
118C32×Q8—22no—
119C32×Q8#4112yes—
120C2×C32×C4—21no—
121C2×C32×C4—22no—
122C2×C32×C4—21no—
123C2×C32×C4C2×D411yescharacteristic
124C2×C32×C4C2×D411yescharacteristic
125C2×C32×C4C2411yescharacteristic
126C2×C3⋊Dic3—21no—
127C2×C3⋊Dic3—22no—
128C2×C3⋊Dic3C2×D412yes—
129C2×C3⋊Dic3C2×D412yes—
130C4⋊(C3⋊D3)—21no—
131C4⋊(C3⋊D3)—21no—
132C4×C3⋊D3—42no—
133#634—22no—
134#634—82no—
135#634—81no—
136C32×C8—22no—
137C3⋊(C3⋊C8)—21no—
138C3⋊(C3⋊C8)—21no—
139Q82—91no—
140#556—91no—
141#489—92no—
142#487—91no—
143#476—91no—
144#475—92no—
145#464—92no—
146#462C3⋊D311yescharacteristic, complemented
147#446—91no—
148#441—92no—
149C4×SD16—91no—
150C2×C3×Q8C22×D318yes—
151C2×C3×Q8C22×D314yes—
152C2×D12—34no—
153C2×C4×D3—64no—
154C2×Dic6—34no—
155C2×Dic6—68no—
156C2×Dic6—64no—
157C3×Q16—216no—
158C2×C3×C8C22×D318yes—
159C3×C4⋊C4—24no—
160C3×C4⋊C4—24no—
161C3×C4⋊C4—24no—
162C3×C4⋊C4C22×D314yes—
163C3×C42—24no—
164C3×C42C22×D314yes—
165#215—68no—
166(C2×D3)⋊C4—68no—
167(C2×D3)⋊C4—68no—
168(C2×D3)⋊C4—64no—
169C4⋊Dic3—34no—
170C4⋊Dic3—34no—
171C4⋊Dic3—38no—
172C4⋊Dic3—68no—
173C4⋊Dic3—68no—
174C4⋊Dic3—64no—
175Dic3⋊C4—38no—
176Dic3⋊C4—68no—
177C4×Dic3—38no—
178C4×Dic3—68no—
179C4×Dic3—68no—
180C2×C3⋊C8—34no—
181C2×C3⋊C8—34no—
182C22×C32C22×D411yescharacteristic
183C2×C3⋊D3—41no—
184C2×C3⋊D3—41no—
185C2×C3⋊D3—41no—
186C32×C4—22no—
187C32×C4—21no—
188C32×C4—42no—
189C32×C4—41no—
190C32×C4#13712yes—
191C32×C4C2×#4111yescharacteristic
192C32×C42−1+411yescharacteristic
193C3⋊Dic3—22no—
194C3⋊Dic3—22no—
195C3⋊Dic3—41no—
196C3⋊Dic3—42no—
197C2×Q16C2×C3⋊D312yes—
198C2×SD16—91no—
199C4⋊Q8—92no—
200C4⋊Q8—91no—
201C4⋊Q8—181no—
202C4⋊Q8—182no—
203C4⋊Q8C2×C3⋊D311yescharacteristic
204#126—182no—
205#124—92no—
206C22⋊Q8—181no—
207C4×Q8—91no—
208C4×Q8—92no—
209C4×Q8—92no—
210C4×Q8—181no—
211C4×Q8C2×C3⋊D311yescharacteristic
212C4×D4—91no—
213#117—181no—
214#106—92no—
215#106—91no—
216C4⋊C8—92no—
217C4⋊C8C2×C3⋊D311yescharacteristic
218#103—91no—
219#103—92no—
220#103—92no—
221#103C2×C3⋊D312yes—
222#102—92no—
223#102—91no—
224#97—92no—
225C4×C8—91no—
226C22×D3—64no—
227C3×Q8—24no—
228C3×Q8—28no—
229C3×Q8—28no—
230C3×Q8#23718yes—
231C2×C3×C4—24no—
232C2×C3×C4—28no—
233C2×C3×C4—24no—
234C2×C3×C4D3×D414yes—
235C2×C3×C4D3×D414yes—
236C2×C3×C4C23×D314yes—
237C2×Dic3—38no—
238C2×Dic3—38no—
239C2×Dic3—64no—
240C2×Dic3—68no—
241D12—64no—
242D12—64no—
243C4×D3—128no—
244Dic6—68no—
245Dic6—248no—
246Dic6—244no—
247C3×C8—28no—
248C3⋊C8—64no—
249C3⋊C8—64no—
250C2×C32#54811yescharacteristic
251C2×C32C2×#13711yescharacteristic
252C2×C32#57711yescharacteristic
253C3⋊D3—81no—
254C2×Q8—91no—
255C2×Q8—182no—
256C2×Q8—181no—
257C2×Q8C22×C3⋊D312yes—
258C2×Q8C22×C3⋊D311yescharacteristic
259C2×D4—91no—
260C22×C4—181no—
261Q16—24no—
262SD16—182no—
263C2×C8—91no—
264C2×C8—91no—
265C2×C8C22×C3⋊D312yes—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—91no—
270C4⋊C4—92no—
271C4⋊C4—92no—
272C4⋊C4—91no—
273C4⋊C4—182no—
274C4⋊C4—182no—
275C4⋊C4—182no—
276C4⋊C4—181no—
277C4⋊C4C22×C3⋊D311yescharacteristic
278C22⋊C4—182no—
279C22⋊C4—182no—
280C22⋊C4—181no—
281C42—21no—
282C42—92no—
283C42—182no—
284C42—182no—
285C42C22×C3⋊D311yescharacteristic
286C22×C3C2×D3×D414yes—
287C2×D3—124no—
288C2×D3—124no—
289C2×D3—124no—
290C3×C4—24no—
291C3×C4—28no—
292C3×C4—44no—
293C3×C4—48no—
294C3×C4#91818yes—
295C3×C4C2×#23714yes—
296C3×C4#101014yes—
297Dic3—68no—
298Dic3—68no—
299Dic3—124no—
300Dic3—128no—
301C32#336911yescharacteristic, complemented, Sylow
302C23—181no—
303Q8—21no—
304Q8—22no—
305Q8—22no—
306Q8—182no—
307Q8—721no—
308Q8—722no—
309Q8#383012yes—
310D4—181no—
311D4—181no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—22no—
315C2×C4—92no—
316C2×C4—92no—
317C2×C4—181no—
318C2×C4—182no—
319C2×C4—362no—
320C2×C4D4×C3⋊D311yescharacteristic
321C2×C4D4×C3⋊D311yescharacteristic
322C2×C4C23×C3⋊D311yescharacteristic
323C8—22no—
324C8—181no—
325C8—181no—
326C2×C3#559814yes—
327C2×C3C2×#91814yes—
328C2×C3#577814yes—
329D3—244no—
330C22—361no—
331C22—361no—
332C22—361no—
333C22C2×D4×C3⋊D311yescharacteristic
334C4—21no—
335C4—22no—
336C4—41no—
337C4—42no—
338C4—182no—
339C4—182no—
340C4—361no—
341C4—362no—
342C4#6412712yes—
343C4C2×#383011yescharacteristic
344C4#6436711yescharacteristic
345C3#8046414yescomplemented
346C2—721no—
347C2#1059606911yescharacteristic
348C2C2×#6412711yescharacteristic
349C2#1059624911yescharacteristic
subgroupG/H
centerC22C2×D4×C3⋊D3
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C23×C3⋊D3
FittingC32×#462C2
socleC22×C32C22×D4
prime radicalC2×C4⋊(C3⋊D3)C23
hypercenter#462C3⋊D3
hypocenterC32#3369
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
2. Q8×#634C2semidirect
3. C32×#462C2semidirect
5. #10592883C2semidirect
6. #10592778C2semidirect
9. #10592530C2semidirect
11. #10592490C2semidirect
44. C3×#462D3semidirect
146. #462C3⋊D3semidirect
301. C32#3369semidirect
345. C3#80464semidirect

Maximal subgroups

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

subgroupindex
#105962032
Q8×#6342
C32×#4622
#105928842
#105928832
#105927782
C4×#37552
#105925412
#105925302
#105924932
#105924902
#804643