Finite Group Databasebeta

← #11778697

Subgroups — #11778697

Order 1152.

708 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 119 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
1#10596105C211yesmaximal, characteristic
2#10592743C211yesmaximal, characteristic, complemented
3#10592421C211yesmaximal, characteristic, complemented
4#10592376C211yesmaximal, characteristic
5C32×#328C211yesmaximal, characteristic, complemented
6#10589334C211yesmaximal, characteristic
7#10589318C211yesmaximal, characteristic, complemented
8#68322—34nomaximal
9C32×C22⋊Q8C2211yescharacteristic
10#64155—21no—
11C2×#3752—21no—
12C2×#3752C2211yescharacteristic
13C2×C4⋊(C3⋊Dic3)—21no—
14C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
15#64133C2211yescharacteristic
16#64106—21no—
17#64105—21no—
18#64105—21no—
19C22⋊C4×C3⋊D3—21no—
20#64087—21no—
21C32×C22⋊C8C2211yescharacteristic
22C32×#95C2211yescharacteristic
23#63662—21no—
24#63657C2211yescharacteristic
25#63645—21no—
26#63643—21no—
27#63634—21no—
28#5634—34no—
29#5044—34no—
30#4722—34no—
31#4677—34no—
32C3×#328D314yescomplemented
33#4468—34no—
34#4452—34no—
35C23×C3⋊D3D411yescharacteristic
36C2×C32×Q8D411yescharacteristic
37C22×C32×C4—21no—
38C22×C32×C4C2311yescharacteristic
39C22×C3⋊Dic3—21no—
40C22×C3⋊Dic3D411yescharacteristic
41C2×C4×C3⋊D3—21no—
42C2×C4×C3⋊D3—21no—
43C2×#634—21no—
44C2×#634D411yescharacteristic
45C2×C32×C8—21no—
46C32×C4⋊C4—21no—
47C32×C4⋊C4D411yescharacteristic
48C32×C22⋊C4—21no—
49#3757—21no—
50#3752—21no—
51#3752—21no—
52#3752—21no—
53#3752—21no—
54#3752—42no—
55C4⋊(C3⋊Dic3)—21no—
56C4⋊(C3⋊Dic3)—21no—
57C4⋊(C3⋊Dic3)—21no—
58C4⋊(C3⋊Dic3)—42no—
59C4⋊(C3⋊Dic3)D411yescharacteristic
60#3750—21no—
61#3750—21no—
62#3750—21no—
63C2×C3⋊(C3⋊C8)—21no—
64#1614—91noSylow
65C3×C22⋊Q8C2×D314yes—
66#946—64no—
67C2×(C2×D3)⋊C4—34no—
68C2×(C2×D3)⋊C4—64no—
69C2×C4⋊Dic3—34no—
70C2×C4⋊Dic3—64no—
71#924—34no—
72#897—64no—
73#896—64no—
74#896—64no—
75D3×C22⋊C4—64no—
76#878—64no—
77C3×C22⋊C8C2×D314yes—
78C3×#95C2×D314yes—
79#835—64no—
80#830—34no—
81#818—64no—
82#816—64no—
83#807—64no—
84C23×C32C2×D411yescharacteristic
85C22×C3⋊D3—21no—
86C22×C3⋊D3—21no—
87C22×C3⋊D3—41no—
88C22×C3⋊D3—41no—
89C22×C3⋊D3Q1612yes—
90C32×Q8—41no—
91C2×C32×C4—22no—
92C2×C32×C4—21no—
93C2×C32×C4—21no—
94C2×C32×C4—41no—
95C2×C32×C4—41no—
96C2×C32×C4—41no—
97C2×C32×C4C2×D411yescharacteristic
98C2×C32×C4C2×D411yescharacteristic
99C2×C3⋊Dic3—21no—
100C2×C3⋊Dic3—21no—
101C2×C3⋊Dic3—21no—
102C2×C3⋊Dic3—41no—
103C2×C3⋊Dic3—41no—
104C2×C3⋊Dic3—41no—
105C2×C3⋊Dic3—41no—
106C4×C3⋊D3—81no—
107C4×C3⋊D3—81no—
108#634—41no—
109#634—81no—
110C32×C8—41no—
111C3⋊(C3⋊C8)—41no—
112#543—91no—
113#484—91no—
114#484—91no—
115#399—91no—
116#328—91no—
117#328C3⋊D311yescharacteristic, complemented
118#323—91no—
119C23×D3—34no—
120C2×C3×Q8D1214yes—
121C22×C3×C4—24no—
122C22×C3×C4C22×D314yes—
123C22×Dic3—34no—
124C22×Dic3—64no—
125C2×C4×D3—64no—
126C2×C4×D3—64no—
127C2×Dic6—34no—
128C2×Dic6—64no—
129C2×C3×C8—24no—
130C3×C4⋊C4—24no—
131C3×C4⋊C4D1214yes—
132C3×C22⋊C4—24no—
133#217—64no—
134(C2×D3)⋊C4—64no—
135(C2×D3)⋊C4—64no—
136(C2×D3)⋊C4—64no—
137(C2×D3)⋊C4—64no—
138(C2×D3)⋊C4—128no—
139C4⋊Dic3—34no—
140C4⋊Dic3—64no—
141C4⋊Dic3—64no—
142C4⋊Dic3—64no—
143C4⋊Dic3—128no—
144Dic3⋊C4—64no—
145Dic3⋊C4—64no—
146Dic3⋊C4—64no—
147C2×C3⋊C8—64no—
148C22×C32—21no—
149C22×C32—21no—
150C22×C32#12011yescharacteristic
151C22×C32C2×Q1611yescharacteristic
152C22×C32#13711yescharacteristic
153C2×C3⋊D3—22no—
154C2×C3⋊D3—41no—
155C2×C3⋊D3—41no—
156C2×C3⋊D3—42no—
157C2×C3⋊D3—81no—
158C32×C4—21no—
159C32×C4—21no—
160C32×C4—42no—
161C32×C4—41no—
162C32×C4—41no—
163C3⋊Dic3—41no—
164C3⋊Dic3—41no—
165C3⋊Dic3—41no—
166C3⋊Dic3—81no—
167C22⋊Q8—91no—
168C22⋊Q8—181no—
169C22⋊Q8—181no—
170C22⋊Q8—181no—
171C22⋊Q8—181no—
172C22⋊Q8—181no—
173C22⋊Q8C2×C3⋊D311yescharacteristic
174C2×C4⋊C4—91no—
175C2×C4⋊C4—181no—
176C2×C22⋊C4—91no—
177C2×C22⋊C4—181no—
178C2×C22⋊C4—181no—
179#107—181no—
180#107—181no—
181#103—181no—
182#103—181no—
183C22⋊C8—91no—
184C22⋊C8C2×C3⋊D311yescharacteristic
185#95C2×C3⋊D311yescharacteristic
186C23×C3D3×D414yes—
187C22×D3—38no—
188C22×D3—64no—
189C22×D3—64no—
190C22×D3—124no—
191C22×D3—124no—
192C3×Q8—44no—
193C2×C3×C4—28no—
194C2×C3×C4—24no—
195C2×C3×C4—24no—
196C2×C3×C4—44no—
197C2×C3×C4—44no—
198C2×C3×C4—44no—
199C2×C3×C4C2×D1214yes—
200C2×C3×C4D3×D414yes—
201C2×Dic3—64no—
202C2×Dic3—64no—
203C2×Dic3—64no—
204C2×Dic3—124no—
205C2×Dic3—124no—
206C2×Dic3—124no—
207C2×Dic3—124no—
208C4×D3—244no—
209C4×D3—244no—
210Dic6—124no—
211Dic6—244no—
212C3×C8—44no—
213C3⋊C8—124no—
214C2×C32—21no—
215C2×C32—21no—
216C2×C32#45111yescharacteristic
217C2×C32#45311yescharacteristic
218C2×C32#45811yescharacteristic
219C3⋊D3—42no—
220C24—91no—
221C2×Q8—91no—
222C2×Q8—181no—
223C2×Q8C4⋊(C3⋊D3)11yescharacteristic
224C22×C4—21no—
225C22×C4—91no—
226C22×C4—181no—
227C22×C4—181no—
228C22×C4—181no—
229C22×C4C22×C3⋊D311yescharacteristic
230C2×C8—21no—
231C2×C8—181no—
232C4⋊C4—21no—
233C4⋊C4—91no—
234C4⋊C4—181no—
235C4⋊C4—181no—
236C4⋊C4—181no—
237C4⋊C4—181no—
238C4⋊C4—181no—
239C4⋊C4—181no—
240C4⋊C4—362no—
241C4⋊C4C4⋊(C3⋊D3)11yescharacteristic
242C22⋊C4—21no—
243C22⋊C4—181no—
244C22⋊C4—181no—
245C22⋊C4—181no—
246C22⋊C4—181no—
247C22⋊C4—181no—
248C22⋊C4—362no—
249C22×C3—24no—
250C22×C3—24no—
251C22×C3#88214yes—
252C22×C3#91514yes—
253C22×C3D3×Q1614yes—
254C2×D3—68no—
255C2×D3—124no—
256C2×D3—124no—
257C2×D3—128no—
258C2×D3—244no—
259C3×C4—24no—
260C3×C4—24no—
261C3×C4—48no—
262C3×C4—44no—
263C3×C4—44no—
264Dic3—124no—
265Dic3—124no—
266Dic3—124no—
267Dic3—244no—
268C32#161411yescharacteristic, complemented, Sylow
269C23—92no—
270C23—181no—
271C23—181no—
272C23—361no—
273C23—361no—
274C23D4×C3⋊D311yescharacteristic
275Q8—41no—
276Q8—361no—
277Q8—721no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—181no—
285C2×C4—181no—
286C2×C4—181no—
287C2×C4—361no—
288C2×C4—361no—
289C2×C4—361no—
290C2×C4—361no—
291C2×C4—721no—
292C2×C4—721no—
293C2×C4C2×C4⋊(C3⋊D3)11yescharacteristic
294C2×C4D4×C3⋊D311yescharacteristic
295C8—41no—
296C8—361no—
297C2×C3—24no—
298C2×C3—24no—
299C2×C3#474214yes—
300C2×C3#480914yes—
301C2×C3#482214yes—
302D3—128no—
303C22—21no—
304C22—21no—
305C22—182no—
306C22—362no—
307C22—361no—
308C22—361no—
309C22—721no—
310C22#6409111yescharacteristic
311C22#6412411yescharacteristic
312C22Q16×C3⋊D311yescharacteristic
313C4—21no—
314C4—21no—
315C4—42no—
316C4—41no—
317C4—41no—
318C4—361no—
319C4—361no—
320C4—361no—
321C4—721no—
322C3#6832214yescomplemented
323C2—21no—
324C2—21no—
325C2—362no—
326C2#1059244111yescharacteristic
327C2#1059250811yescharacteristic
328C2#1059252111yescharacteristic
subgroupG/H
centerC22#64091
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C22×C3⋊D3
FittingC32×#328C2
socleC22×C32#120
prime radicalC23×C3⋊D3D4
hypercenter#328C3⋊D3
hypocenterC32#1614
LCSG ▷ C22×C32×C4 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. #10592743C2semidirect
3. #10592421C2semidirect
5. C32×#328C2semidirect
7. #10589318C2semidirect
32. C3×#328D3semidirect
117. #328C3⋊D3semidirect
268. C32#1614semidirect
322. C3#68322semidirect

Maximal subgroups

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

subgroupindex
#105961052
#105927432
#105924212
#105923762
C32×#3282
#105893342
#105893182
#683223