Finite Group Databasebeta

← #11788823

Subgroups — #11788823

Order 1152.

761 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 761 conjugacy classes of subgroups — too many to draw. Only its 152 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#10596283C211yesmaximal, characteristic, complemented
2C3×#5816C211yesmaximal, characteristic
3C2×C3×#836C211yesmaximal, characteristic, complemented
4#10591110C212yesmaximal
5#10589158C212yesmaximal, complemented
6#71864—31nomaximal
7#70105—31nomaximal
8C22×C32×Q8C2211yescharacteristic
9C2×C3×#235C2211yescharacteristic
10Q8×C3⋊Dic3—24no—
11#64135—22no—
12#64135C2212yes—
13C3×#947C412yescomplemented
14C3×#946—22no—
15C3×#944C412yescomplemented
16C3×#929—22no—
17C3×#926C2211yescharacteristic
18C2×C3×#208C2212yes—
19C3×#836—22no—
20#63568—24no—
21C3×#536—31no—
22#5816D311yescharacteristic
23#5812—31no—
24#5812—31no—
25#5812—61no—
26C2×C3×#101—31no—
27#5236—32no—
28C2×#836D311yescharacteristic, complemented
29#5139—32no—
30#4557—32no—
31#4531—32no—
32C2×C32×Q8—22no—
33C2×C32×Q8—22no—
34C2×C32×Q8C2×C412yes—
35C2×C32×Q8D412yes—
36C22×C32×C4D412yes—
37C22×C32×C4C2311yescharacteristic
38C2×C3×#67—21no—
39C3×#235—22no—
40C3×#235—22no—
41C2×C3×D12C2×C411yescharacteristic
42C2×C3×C4×D3—21no—
43C2×C3×Dic6C2×C411yescharacteristic
44#3757—22no—
45C4⋊(C3⋊Dic3)—24no—
46C4⋊(C3⋊Dic3)—22no—
47C4×C3⋊Dic3—24no—
48C4×C3⋊Dic3D414yes—
49C3×#217—21no—
50C3×(C2×D3)⋊C4—24no—
51C3×C4⋊Dic3—21no—
52C3×Dic3⋊C4—24no—
53C3×C4×Dic3C2×C412yes—
54C3×#208—22no—
55C3×#208—22no—
56C2×C3×C3⋊C8—22no—
57#1912—91noSylow
58C2×C3×#41—31no—
59C22×C3×Q8—21no—
60C22×C3×Q8C2×D311yescharacteristic
61C22×C3×Q8C2×D311yescharacteristic
62C2×#235C2×D311yescharacteristic
63C2×C3×M16—32no—
64C3×C4⋊Q8—32no—
65C3×#124—32no—
66C3×#123—62no—
67C3×C22⋊Q8—62no—
68C3×#117—31no—
69#947Dic312yescomplemented
70#946—22no—
71Q8×Dic3—64no—
72Q8×Dic3—64no—
73Q8×Dic3—68no—
74#944Dic312yescomplemented
75#929—22no—
76#926—32no—
77#926—32no—
78#926—62no—
79#926—64no—
80#926—62no—
81#926—62no—
82#926C2×D311yescharacteristic
83C2×#208C2×D312yes—
84C3×#101—62no—
85#837—64no—
86#836—22no—
87#825—64no—
88C23×C32C2×D411yescharacteristic
89C22×C3×D3—21no—
90C32×Q8—24no—
91C32×Q8—24no—
92C2×C32×C4—22no—
93C2×C32×C4—22no—
94C2×C32×C4C22⋊C414yes—
95C2×C32×C4C22×C411yescharacteristic
96C2×C32×C4C2×D412yes—
97C2×C32×C4C2×D411yescharacteristic
98C2×C32×C4#4112yes—
99C2×C3⋊Dic3—24no—
100C3×#67—41no—
101C3×#67—41no—
102C2×C3×Dic3—22no—
103C2×C3×Dic3—21no—
104C3×D12—22no—
105C3×C4×D3—42no—
106C3×Dic6—22no—
107C3×C3⋊C8—42no—
108#536—31no—
109#519—91no—
110#421—92no—
111C2×#101—31no—
112H(3,ℤ/4)—92no—
113C3×#41—62no—
114C3×#41—62no—
115C2×C3×Q8—24no—
116C2×C3×Q8—24no—
117C2×C3×Q8—22no—
118C2×C3×Q8—22no—
119C2×C3×Q8—22no—
120C2×C3×Q8—22no—
121C2×C3×Q8—22no—
122C2×C3×Q8—22no—
123C2×C3×Q8—31no—
124C2×C3×Q8C4×D312yes—
125C2×C3×Q8C2×Dic312yes—
126C2×C3×Q8#6712yes—
127C2×C3×Q8#6712yes—
128C2×C3×D4—31no—
129C2×C3×D4—61no—
130C22×C3×C4—22no—
131C22×C3×C4—21no—
132C22×C3×C4—61no—
133C22×C3×C4D1212yes—
134C22×C3×C4#6712yes—
135C22×C3×C4C22×D311yescharacteristic
136C22×C3×C4C22×D311yescharacteristic
137C2×#67—21no—
138#235—22no—
139#235—22no—
140C2×D12C2×Dic311yescharacteristic
141C2×C4×D3—21no—
142C2×Dic6C2×Dic311yescharacteristic
143C3×M16—62no—
144C3×M16—62no—
145C2×C3×C8—62no—
146C3×C4⋊C4—64no—
147C3×C4⋊C4—61no—
148C3×C22⋊C4—61no—
149C3×C22⋊C4—64no—
150C3×C42—32no—
151#217—21no—
152#217—64no—
153#217—62no—
154#217—62no—
155(C2×D3)⋊C4—24no—
156C4⋊Dic3—21no—
157C4⋊Dic3—68no—
158C4⋊Dic3—64no—
159C4⋊Dic3—64no—
160C4⋊Dic3—64no—
161C4⋊Dic3—62no—
162C4⋊Dic3—62no—
163Dic3⋊C4—24no—
164C4×Dic3—34no—
165C4×Dic3—34no—
166C4×Dic3—64no—
167C4×Dic3—68no—
168C4×Dic3—64no—
169C4×Dic3—64no—
170C4×Dic3C2×Dic312yes—
171#208—22no—
172#208—22no—
173C2×C3⋊C8—22no—
174C22×C32—41no—
175C22×C32C2×C22⋊C411yescharacteristic
176C22×C32#12011yescharacteristic
177C22×C32#12311yescharacteristic
178C2×C3×D3—42no—
179C2×C3×D3—41no—
180C32×C4—24no—
181C32×C4C4×D412yes—
182C32×C4#12112yes—
183C3⋊Dic3—44no—
184C3×Dic3—42no—
185C3×Dic3—41no—
186C2×#41—31no—
187C22×Q8D3211yescharacteristic
188C2×M16—32no—
189C4⋊Q8—32no—
190#124—32no—
191#123—62no—
192C22⋊Q8—62no—
193C4×Q8—184no—
194#117—31no—
195#117—92no—
196#117—182no—
197#104—184no—
198#101—62no—
199C23×C3—21no—
200C23×C3—61no—
201C23×C3C2×D1211yescharacteristic
202C23×C3C2×#6711yescharacteristic
203C22×D3—21no—
204C3×Q8—28no—
205C3×Q8—28no—
206C3×Q8—24no—
207C3×Q8—24no—
208C3×Q8—24no—
209C3×Q8—24no—
210C3×Q8—62no—
211C3×D4—62no—
212C3×D4—121no—
213C3×D4—121no—
214C2×C3×C4—24no—
215C2×C3×C4—24no—
216C2×C3×C4—22no—
217C2×C3×C4—22no—
218C2×C3×C4—22no—
219C2×C3×C4—22no—
220C2×C3×C4—24no—
221C2×C3×C4—21no—
222C2×C3×C4—22no—
223C2×C3×C4—21no—
224C2×C3×C4—22no—
225C2×C3×C4—62no—
226C2×C3×C4—61no—
227C2×C3×C4—122no—
228C2×C3×C4(C2×D3)⋊C414yes—
229C2×C3×C4#21714yes—
230C2×C3×C4C2×C4×D311yescharacteristic
231C2×C3×C4D3×D412yes—
232C2×C3×C4D3×D412yes—
233C2×C3×C4#23712yes—
234C2×C3×C4#23712yes—
235C2×C3×C4C22×Dic311yescharacteristic
236C2×C3×C4C2×#6711yescharacteristic
237C2×C3×C4C2×#6711yescharacteristic
238#67—41no—
239#67—41no—
240C2×Dic3—22no—
241C2×Dic3—21no—
242C2×Dic3—68no—
243C2×Dic3—64no—
244C2×Dic3—64no—
245D12—22no—
246C4×D3—42no—
247Dic6—22no—
248C3×C8—122no—
249C3⋊C8—42no—
250C2×C32—21no—
251C2×C32—21no—
252C2×C32—21no—
253C2×C32#38611yescharacteristic
254C3×D3—81no—
255#41—62no—
256#41—62no—
257C2×Q8—22no—
258C2×Q8—22no—
259C2×Q8—31no—
260C2×Q8D3×Dic312yes—
261C2×Q8#62512yes—
262C2×D4—31no—
263C2×D4—61no—
264C22×C4—61no—
265C22×C4#62612yes—
266C22×C4C2×D3211yescharacteristic
267M16—62no—
268M16—62no—
269C2×C8—62no—
270C4⋊C4—64no—
271C4⋊C4—61no—
272C4⋊C4—184no—
273C4⋊C4—182no—
274C22⋊C4—61no—
275C22⋊C4—64no—
276C22⋊C4—182no—
277C42—32no—
278C42—94no—
279C42—184no—
280C22×C3—21no—
281C22×C3—21no—
282C22×C3—21no—
283C22×C3—41no—
284C22×C3—41no—
285C22×C3—42no—
286C22×C3—122no—
287C22×C3—121no—
288C22×C3#88211yescharacteristic
289C22×C3#88611yescharacteristic
290C22×C3C2×(C2×D3)⋊C411yescharacteristic
291C22×C3#93511yescharacteristic
292C22×C3#93711yescharacteristic
293C22×C3C2×#21711yescharacteristic
294C2×D3—42no—
295C2×D3—41no—
296C3×C4—28no—
297C3×C4—24no—
298C3×C4—24no—
299C3×C4—22no—
300C3×C4—22no—
301C3×C4—122no—
302C3×C4—121no—
303C3×C4#67⋊C412yes—
304C3×C4D4×Dic312yes—
305C3×C4#93912yes—
306C3×C4#93912yes—
307Dic3—42no—
308Dic3—41no—
309Dic3—128no—
310Dic3—124no—
311Dic3—124no—
312C32#191211yescharacteristic, complemented, Sylow
313C23—61no—
314C23C2×#62611yescharacteristic
315Q8—24no—
316Q8—24no—
317Q8—62no—
318D4—62no—
319D4—121no—
320D4—121no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—62no—
324C2×C4—61no—
325C2×C4—122no—
326C2×C4—184no—
327C2×C4(C2×D3)⋊Dic314yes—
328C2×C4C2×D3×Dic311yescharacteristic
329C2×C4#380512yes—
330C2×C4C2×#62511yescharacteristic
331C2×C4#381112yes—
332C8—122no—
333C2×C3—22no—
334C2×C3—21no—
335C2×C3—21no—
336C2×C3—21no—
337C2×C3—21no—
338C2×C3—21no—
339C2×C3—21no—
340C2×C3—21no—
341C2×C3—41no—
342C2×C3—41no—
343C2×C3—241no—
344C2×C3#495811yescharacteristic
345C2×C3#522011yescharacteristic
346D3—81no—
347C22—41no—
348C22—122no—
349C22—121no—
350C22C2×(C2×D3)⋊Dic311yescharacteristic
351C22#6396211yescharacteristic
352C22#6398111yescharacteristic
353C4—24no—
354C4—122no—
355C4—121no—
356C4—364no—
357C4#67⋊Dic312yes—
358C4#6398012yes—
359C3—21no—
360C3#7010511yescharacteristic, complemented
361C3#7186411yescharacteristic, complemented
362C2—21no—
363C2—21no—
364C2—21no—
365C2—241no—
366C2#1059171311yescharacteristic
subgroupG/H
centerC2#10591713
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C2×D32
FittingC22×C32×Q8C22
socleC2×C32#386
prime radicalC2×C3×#235C22
cosocleC22×C32×C4C23
hypercenterC22×Q8D32
hypocenterC32#1912
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
1. #10596283C2semidirect
3. C2×C3×#836C2semidirect
5. #10589158C2semidirect
13. C3×#947C4semidirect
15. C3×#944C4semidirect
28. C2×#836D3semidirect
69. #947Dic3semidirect
74. #944Dic3semidirect
312. C32#1912semidirect
360. C3#70105semidirect
361. C3#71864semidirect

Maximal subgroups

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

subgroupindex
#105962832
C3×#58162
C2×C3×#8362
#105911102
#105891582
#718643
#701053