Finite Group Databasebeta

← #11780422

Subgroups — #11780422

Order 1152.

720 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 161 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×#547C211yesmaximal, characteristic
2#10592779C211yesmaximal, characteristic, complemented
3#10592718C211yesmaximal, characteristic, complemented
4#10592706C211yesmaximal, characteristic, complemented
5#10589413C211yesmaximal, characteristic, complemented
6D4⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
7D4⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
8#71913—34nomaximal
9C2×C32×#41—21no—
10C32×C4⋊Q8C2211yescharacteristic
11C32×#123—22no—
12C32×C22⋊Q8—21no—
13C32×#121—21no—
14C32×C4×D4C2211yescharacteristic
15C32×C4×D4C2211yescharacteristic
16C2×C32×C4⋊C4—21no—
17C2×#3754—21no—
18#64078C2211yescharacteristic
19#63659—21no—
20#63637—21no—
21#63635—22no—
22#63634—21no—
23C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
24C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
25C4×C3⋊(C3⋊C8)C2211yescharacteristic
26C3×#547D314yes—
27#5080—34no—
28#5019—34no—
29#5007—34no—
30#4547—34no—
31D4⋊(C3⋊C8)—34no—
32D4⋊(C3⋊C8)—34no—
33C32×#41—42no—
34C32×#41—42no—
35C2×C32×Q8—21no—
36C2×C32×D4—22no—
37C2×C32×D4D411yescharacteristic
38C2×C32×D4D411yescharacteristic
39C22×C32×C4—22no—
40C22×C32×C4—21no—
41C22×C32×C4—21no—
42C2×#634—21no—
43C32×C4⋊C4—22no—
44C32×C4⋊C4—21no—
45C32×C4⋊C4D411yescharacteristic
46C32×C4⋊C4D412yes—
47C32×C4⋊C4D411yescharacteristic
48C32×C22⋊C4—22no—
49C32×C22⋊C4—21no—
50C32×C22⋊C4—21no—
51C32×C42C2311yescharacteristic
52#3754—42no—
53C4⋊(C3⋊Dic3)—21no—
54C4⋊(C3⋊Dic3)—21no—
55C2×C3⋊(C3⋊C8)—21no—
56C2×C3⋊(C3⋊C8)—21no—
57C2×C3⋊(C3⋊C8)—21no—
58#1654—91noSylow
59C2×C3×#41—24no—
60C3×C4⋊Q8C2×D314yes—
61C3×#123—28no—
62C3×C22⋊Q8—24no—
63C3×#121—24no—
64C3×C4×D4C2×D314yes—
65C3×C4×D4C2×D314yes—
66C2×C3×C4⋊C4—24no—
67C2×#214—64no—
68#869—34no—
69#832—64no—
70#810—64no—
71#808—68no—
72#807—64no—
73C4⋊(C3⋊C8)—34no—
74C4⋊(C3⋊C8)—34no—
75C4×C3⋊C8—34no—
76C23×C32—21no—
77C23×C32—21no—
78C32×Q8—41no—
79C32×Q8—41no—
80C32×D4—21no—
81C32×D4—21no—
82C32×D4—21no—
83C32×D4—44no—
84C32×D4SD1612yescomplemented
85C2×C32×C4—21no—
86C2×C32×C4—21no—
87C2×C32×C4—22no—
88C2×C32×C4—44no—
89C2×C32×C4—41no—
90C2×C32×C4—41no—
91C2×C32×C4—41no—
92C2×C32×C4—41no—
93C2×C32×C4C2×D411yescharacteristic
94C2×C32×C4C2×D411yescharacteristic
95C2×C32×C4C2×D411yescharacteristic
96C2×C3⋊Dic3—41no—
97#634—41no—
98#634—41no—
99C3⋊(C3⋊C8)—22no—
100C3⋊(C3⋊C8)—41no—
101C3⋊(C3⋊C8)—41no—
102#547C3⋊D311yescharacteristic
103#498—91no—
104#474—91no—
105#461—91no—
106#332—91no—
107D4⋊C8—91no—
108D4⋊C8—91no—
109C3×#41—48no—
110C3×#41—48no—
111C2×C3×Q8—24no—
112C2×C3×D4—28no—
113C2×C3×D4#6714yes—
114C2×C3×D4#6714yes—
115C22×C3×C4—28no—
116C22×C3×C4—24no—
117C22×C3×C4—24no—
118C2×Dic6—64no—
119C3×C4⋊C4—28no—
120C3×C4⋊C4—24no—
121C3×C4⋊C4#6714yes—
122C3×C4⋊C4#6718yes—
123C3×C4⋊C4#6714yes—
124C3×C22⋊C4—28no—
125C3×C22⋊C4—24no—
126C3×C22⋊C4—24no—
127C3×C42C22×D314yes—
128#214—128no—
129C4⋊Dic3—64no—
130C4⋊Dic3—64no—
131C2×C3⋊C8—64no—
132C2×C3⋊C8—64no—
133C2×C3⋊C8—64no—
134C22×C32—22no—
135C22×C32—41no—
136C22×C32—41no—
137C22×C32—41no—
138C22×C32—41no—
139C22×C32—41no—
140C22×C32#12011yescharacteristic
141C32×C4—21no—
142C32×C4—41no—
143C32×C4—41no—
144C32×C4—42no—
145C32×C4C2×SD1611yescharacteristic
146C32×C4#13511yescharacteristic
147C32×C4AGL(1,ℤ/8)11yescharacteristic
148C32×C4AGL(1,ℤ/8)11yescharacteristic
149C3⋊Dic3—81no—
150C2×#41—21no—
151C2×SD16—181no—
152C4⋊Q8—91no—
153C4⋊Q8C2×C3⋊D311yescharacteristic
154#123—22no—
155C22⋊Q8—21no—
156#121—21no—
157C4×D4C2×C3⋊D311yescharacteristic
158C4×D4C2×C3⋊D311yescharacteristic
159C2×C4⋊C4—21no—
160#107—181no—
161#106—182no—
162C4⋊C8—91no—
163C4⋊C8—91no—
164#103—181no—
165#102—181no—
166C4×C8—91no—
167C23×C3—24no—
168C23×C3—24no—
169C3×Q8—44no—
170C3×Q8—44no—
171C3×D4—24no—
172C3×D4—24no—
173C3×D4—24no—
174C3×D4—416no—
175C3×D4#21418yescomplemented
176C2×C3×C4—24no—
177C2×C3×C4—24no—
178C2×C3×C4—28no—
179C2×C3×C4—416no—
180C2×C3×C4—44no—
181C2×C3×C4—44no—
182C2×C3×C4—44no—
183C2×C3×C4—44no—
184C2×C3×C4C2×#6714yes—
185C2×C3×C4C2×#6714yes—
186C2×C3×C4C2×#6714yes—
187C2×Dic3—124no—
188Dic6—124no—
189Dic6—124no—
190C3⋊C8—68no—
191C3⋊C8—124no—
192C3⋊C8—124no—
193C2×C32—42no—
194C2×C32—81no—
195C2×C32#44911yescharacteristic
196C2×C32#45011yescharacteristic
197C2×C32#45611yescharacteristic
198#41—42no—
199#41—42no—
200C2×Q8—21no—
201C2×Q8—181no—
202C2×D4—22no—
203C2×D4#63811yescharacteristic
204C2×D4#63811yescharacteristic
205C22×C4—22no—
206C22×C4—21no—
207C22×C4—21no—
208SD16—362no—
209C2×C8—181no—
210C2×C8—181no—
211C2×C8—181no—
212C4⋊C4—22no—
213C4⋊C4—21no—
214C4⋊C4—181no—
215C4⋊C4—181no—
216C4⋊C4#63811yescharacteristic
217C4⋊C4#63812yes—
218C4⋊C4#63811yescharacteristic
219C22⋊C4—22no—
220C22⋊C4—21no—
221C22⋊C4—21no—
222C42C22×C3⋊D311yescharacteristic
223C22×C3—28no—
224C22×C3—44no—
225C22×C3—44no—
226C22×C3—44no—
227C22×C3—44no—
228C22×C3—44no—
229C22×C3#95314yes—
230C3×C4—24no—
231C3×C4—44no—
232C3×C4—48no—
233C3×C4—44no—
234C3×C4#93214yes—
235C3×C4C2×#21414yes—
236C3×C4#94914yes—
237C3×C4#95014yes—
238Dic3—244no—
239C32#165411yescharacteristic, complemented, Sylow
240C23—21no—
241C23—21no—
242Q8—41no—
243Q8—41no—
244Q8—361no—
245Q8—361no—
246D4—21no—
247D4—21no—
248D4—21no—
249D4—44no—
250D4#375412yescomplemented
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—44no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—361no—
260C2×C4C2×#63811yescharacteristic
261C2×C4C2×#63811yescharacteristic
262C2×C4C2×#63811yescharacteristic
263C8—182no—
264C8—361no—
265C8—361no—
266C2×C3—48no—
267C2×C3—84no—
268C2×C3#521814yes—
269C2×C3#523814yes—
270C2×C3#524614yes—
271C22—22no—
272C22—41no—
273C22—41no—
274C22—41no—
275C22—41no—
276C22—41no—
277C22#6416211yescharacteristic
278C4—21no—
279C4—41no—
280C4—42no—
281C4—41no—
282C4—721no—
283C4#6414111yescharacteristic
284C4C2×#375411yescharacteristic
285C4#6415811yescharacteristic
286C4#6415911yescharacteristic
287C3#7191314yescomplemented
288C2—42no—
289C2—81no—
290C2#1059291711yescharacteristic
291C2#1059293711yescharacteristic
292C2#1059294511yescharacteristic
subgroupG/H
centerC22#64162
commutator, cosocleC32×C42C23
FrattiniC42C22×C3⋊D3
Fitting, prime radicalC32×#547C2
socleC22×C32#120
hypercenter#547C3⋊D3
hypocenterC32#1654
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #547

Complements

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

NH ≅ G/Ntype
2. #10592779C2semidirect
3. #10592718C2semidirect
4. #10592706C2semidirect
5. #10589413C2semidirect
6. D4⋊(C3⋊(C3⋊C8))C2semidirect
7. D4⋊(C3⋊(C3⋊C8))C2semidirect
84. C32×D4SD16semidirect
175. C3×D4#214semidirect
239. C32#1654semidirect
250. D4#3754semidirect
287. C3#71913semidirect

Maximal subgroups

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

subgroupindex
C32×#5472
#105927792
#105927182
#105927062
#105894132
D4⋊(C3⋊(C3⋊C8))2
#719133