Finite Group Databasebeta

← #11778051

Subgroups — #11778051

Order 1152.

816 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 816 conjugacy classes of subgroups — too many to draw. Only its 137 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#539C211yesmaximal, characteristic, complemented
2#10592745C211yesmaximal, characteristic, complemented
3#10592735C211yesmaximal, characteristic
4#10592690C211yesmaximal, characteristic, complemented
5#10589404C211yesmaximal, characteristic, complemented
6#10589403C211yesmaximal, characteristic, complemented
7#10589394C211yesmaximal, characteristic, complemented
8#71888—34nomaximal
9C22×C4⋊(C3⋊D3)C2211yescharacteristic
10C32×#126—21no—
11C32×#125—21no—
12C32×#123—21no—
13C32×#123—21no—
14C32×C22⋊Q8—21no—
15C32×C22⋊Q8C2211yescharacteristic
16C32×#121C2211yescharacteristic
17C32×C4×D4—21no—
18C2×C32×C4⋊C4C2211yescharacteristic
19C2×#3755—21no—
20C2×#3753—21no—
21C2×#3752—21no—
22C2×#3752—21no—
23#63658C2211yescharacteristic
24#63657C2211yescharacteristic
25#63657C2211yescharacteristic
26#63636—21no—
27#63636—21no—
28C3×#539D314yescomplemented
29#5046—34no—
30#5036—34no—
31#4991—34no—
32#4538—34no—
33#4537—34no—
34#4528—34no—
35C23×C3⋊D3—21no—
36C2×C32×Q8D411yescharacteristic
37C2×C32×D4—21no—
38C2×C32×D4D411yescharacteristic
39C22×C32×C4—21no—
40C22×C32×C4D411yescharacteristic
41C22×C32×C4D411yescharacteristic
42C22×C32×C4C2311yescharacteristic
43C22×C3⋊Dic3—21no—
44C2×C4⋊(C3⋊D3)—21no—
45C2×C4⋊(C3⋊D3)—21no—
46C2×C4⋊(C3⋊D3)—42no—
47C32×C4⋊C4—21no—
48C32×C4⋊C4—21no—
49C32×C4⋊C4—21no—
50C32×C4⋊C4—21no—
51C32×C4⋊C4—21no—
52C32×C4⋊C4—21no—
53C32×C4⋊C4D411yescharacteristic
54C32×C4⋊C4D411yescharacteristic
55C32×C22⋊C4—21no—
56C32×C22⋊C4—21no—
57C32×C22⋊C4—21no—
58C32×C22⋊C4—21no—
59C32×C22⋊C4—21no—
60C32×C42—21no—
61#3755—42no—
62#3753—42no—
63#3752—42no—
64#3752—42no—
65C2×C3⋊(C3⋊C8)—21no—
66C2×C3⋊(C3⋊C8)—21no—
67#1597—91noSylow
68C22×D12—34no—
69C3×#126—24no—
70C3×#125—24no—
71C3×#123—24no—
72C3×#123—24no—
73C3×C22⋊Q8—24no—
74C3×C22⋊Q8C2×D314yes—
75C3×#121C2×D314yes—
76C3×C4×D4—24no—
77C2×C3×C4⋊C4C2×D314yes—
78C2×#215—64no—
79C2×#213—64no—
80C2×(C2×D3)⋊C4—64no—
81C2×(C2×D3)⋊C4—64no—
82C3⋊#95—34no—
83#830—34no—
84#830—34no—
85#809—64no—
86#809—64no—
87C23×C32—21no—
88C23×C32C2×D411yescharacteristic
89C22×C3⋊D3—22no—
90C22×C3⋊D3—41no—
91C22×C3⋊D3—41no—
92C22×C3⋊D3—41no—
93C22×C3⋊D3—41no—
94C22×C3⋊D3—41no—
95C22×C3⋊D3—41no—
96C32×Q8—41no—
97C32×D4—41no—
98C32×D4—41no—
99C32×D4—41no—
100C2×C32×C4—21no—
101C2×C32×C4—21no—
102C2×C32×C4—21no—
103C2×C32×C4—21no—
104C2×C32×C4—21no—
105C2×C32×C4—21no—
106C2×C32×C4—21no—
107C2×C32×C4—41no—
108C2×C32×C4—41no—
109C2×C32×C4—81no—
110C2×C32×C4C2×D411yescharacteristic
111C2×C32×C4C2×D411yescharacteristic
112C2×C3⋊Dic3—41no—
113C2×C3⋊Dic3—41no—
114C2×C3⋊Dic3—41no—
115C4⋊(C3⋊D3)—22no—
116C4⋊(C3⋊D3)—22no—
117C4⋊(C3⋊D3)—41no—
118C4⋊(C3⋊D3)—41no—
119C3⋊(C3⋊C8)—41no—
120C3⋊(C3⋊C8)—41no—
121#539C3⋊D311yescharacteristic, complemented
122#450—91no—
123#447—91no—
124#394—91no—
125#328—91no—
126#327—91no—
127#324—91no—
128C23×D3—64no—
129C2×C3×Q8#6714yes—
130C2×C3×D4—24no—
131C2×C3×D4#6714yes—
132C22×C3×C4—24no—
133C22×C3×C4#6714yes—
134C22×C3×C4#6714yes—
135C22×C3×C4C22×D314yes—
136C22×Dic3—64no—
137C2×D12—64no—
138C2×D12—64no—
139C2×D12—128no—
140C3×C4⋊C4—24no—
141C3×C4⋊C4—24no—
142C3×C4⋊C4—24no—
143C3×C4⋊C4—24no—
144C3×C4⋊C4—24no—
145C3×C4⋊C4—24no—
146C3×C4⋊C4#6714yes—
147C3×C4⋊C4#6714yes—
148C3×C22⋊C4—24no—
149C3×C22⋊C4—24no—
150C3×C22⋊C4—24no—
151C3×C22⋊C4—24no—
152C3×C22⋊C4—24no—
153C3×C42—24no—
154#215—128no—
155#213—128no—
156(C2×D3)⋊C4—128no—
157(C2×D3)⋊C4—128no—
158C2×C3⋊C8—64no—
159C2×C3⋊C8—64no—
160C22×C32—21no—
161C22×C32—21no—
162C22×C32—41no—
163C22×C32—41no—
164C22×C32—41no—
165C22×C32#12011yescharacteristic
166C22×C32#13511yescharacteristic
167C22×C32AGL(1,ℤ/8)11yescharacteristic
168C2×C3⋊D3—42no—
169C2×C3⋊D3—42no—
170C2×C3⋊D3—42no—
171C2×C3⋊D3—81no—
172C2×C3⋊D3—81no—
173C2×C3⋊D3—81no—
174C2×C3⋊D3—81no—
175C32×C4—21no—
176C32×C4—21no—
177C32×C4—41no—
178C32×C4—41no—
179C32×C4—41no—
180C32×C4—41no—
181C32×C4—41no—
182C3⋊Dic3—81no—
183C22×D4—91no—
184C2×SD16—181no—
185C2×D8—181no—
186#126—21no—
187#125—21no—
188#123—21no—
189#123—21no—
190C22⋊Q8—21no—
191C22⋊Q8C2×C3⋊D311yescharacteristic
192#121C2×C3⋊D311yescharacteristic
193C4×D4—21no—
194C2×C4⋊C4C2×C3⋊D311yescharacteristic
195C2×C22⋊C4—181no—
196C2×C22⋊C4—181no—
197#102—181no—
198#102—181no—
199C22⋊C8—91no—
200C22⋊C8—91no—
201#95—91no—
202C23×C3—24no—
203C23×C3C2×#6714yes—
204C22×D3—68no—
205C22×D3—124no—
206C22×D3—124no—
207C22×D3—124no—
208C22×D3—124no—
209C22×D3—124no—
210C22×D3—124no—
211C3×Q8—44no—
212C3×D4—44no—
213C3×D4—44no—
214C3×D4—44no—
215C2×C3×C4—24no—
216C2×C3×C4—24no—
217C2×C3×C4—24no—
218C2×C3×C4—24no—
219C2×C3×C4—24no—
220C2×C3×C4—24no—
221C2×C3×C4—24no—
222C2×C3×C4—44no—
223C2×C3×C4—44no—
224C2×C3×C4—84no—
225C2×C3×C4C2×#6714yes—
226C2×C3×C4C2×#6714yes—
227C2×Dic3—124no—
228C2×Dic3—124no—
229C2×Dic3—124no—
230D12—68no—
231D12—68no—
232D12—124no—
233D12—124no—
234C3⋊C8—124no—
235C3⋊C8—124no—
236C2×C32—21no—
237C2×C32—21no—
238C2×C32—81no—
239C2×C32#44911yescharacteristic
240C2×C32#45411yescharacteristic
241C2×C32B(4,𝔽2)11yescharacteristic
242C3⋊D3—82no—
243C24—181no—
244C2×Q8#63811yescharacteristic
245C2×D4—21no—
246C2×D4—181no—
247C2×D4—181no—
248C2×D4—362no—
249C2×D4#63811yescharacteristic
250C22×C4—21no—
251C22×C4—181no—
252C22×C4#63811yescharacteristic
253C22×C4#63811yescharacteristic
254C22×C4C22×C3⋊D311yescharacteristic
255SD16—362no—
256D8—362no—
257C2×C8—181no—
258C2×C8—181no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4#63811yescharacteristic
266C4⋊C4#63811yescharacteristic
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—21no—
272C22⋊C4—362no—
273C22⋊C4—362no—
274C42—21no—
275C22×C3—24no—
276C22×C3—24no—
277C22×C3—44no—
278C22×C3—44no—
279C22×C3—44no—
280C22×C3#94914yes—
281C22×C3#95014yes—
282C22×C3#95314yes—
283C2×D3—128no—
284C2×D3—128no—
285C2×D3—128no—
286C2×D3—244no—
287C2×D3—244no—
288C2×D3—244no—
289C2×D3—244no—
290C3×C4—24no—
291C3×C4—24no—
292C3×C4—44no—
293C3×C4—44no—
294C3×C4—44no—
295C3×C4—44no—
296C3×C4—44no—
297Dic3—244no—
298C32#159711yescharacteristic, complemented, Sylow
299C23—21no—
300C23—182no—
301C23—361no—
302C23—361no—
303C23—361no—
304C23—361no—
305C23—361no—
306C23—361no—
307C23C2×#63811yescharacteristic
308Q8—41no—
309D4—41no—
310D4—41no—
311D4—41no—
312D4—182no—
313D4—182no—
314D4—361no—
315D4—361no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—81no—
326C2×C4—361no—
327C2×C4—361no—
328C2×C4—361no—
329C2×C4C2×#63811yescharacteristic
330C2×C4C2×#63811yescharacteristic
331C8—361no—
332C8—361no—
333C2×C3—24no—
334C2×C3—24no—
335C2×C3—84no—
336C2×C3#523814yes—
337C2×C3#524414yes—
338C2×C3#524514yes—
339D3—248no—
340C22—21no—
341C22—21no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—362no—
346C22—362no—
347C22—362no—
348C22—721no—
349C22—721no—
350C22—721no—
351C22—721no—
352C22#6415811yescharacteristic
353C22#6415911yescharacteristic
354C22#6416211yescharacteristic
355C4—21no—
356C4—21no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—41no—
361C4—41no—
362C4—721no—
363C3#7188814yescomplemented
364C2—21no—
365C2—21no—
366C2—81no—
367C2—722no—
368C2#1059293711yescharacteristic
369C2#1059294311yescharacteristic
370C2#1059294411yescharacteristic
subgroupG/H
centerC22#64162
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C22×C3⋊D3
FittingC32×#539C2
socleC22×C32#120
prime radical#10592735C2
hypercenter#539C3⋊D3
hypocenterC32#1597
LCSG ▷ C22×C32×C4 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ #539

Complements

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

NH ≅ G/Ntype
1. C32×#539C2semidirect
2. #10592745C2semidirect
4. #10592690C2semidirect
5. #10589404C2semidirect
6. #10589403C2semidirect
7. #10589394C2semidirect
28. C3×#539D3semidirect
121. #539C3⋊D3semidirect
298. C32#1597semidirect
363. C3#71888semidirect

Maximal subgroups

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

subgroupindex
C32×#5392
#105927452
#105927352
#105926902
#105894042
#105894032
#105893942
#718883