Finite Group Databasebeta

← #13159601

Subgroups — #13159601

Order 1728.

776 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 776 conjugacy classes of subgroups — too many to draw. Only its 142 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#11723456C211yesmaximal, characteristic
2#11723413C211yesmaximal, characteristic
3#11722190C211yesmaximal, characteristic
4C3×#63645C211yesmaximal, characteristic, complemented
5C32×#816C211yesmaximal, characteristic, complemented
6#11722053C211yesmaximal, characteristic, complemented
7#11722037C211yesmaximal, characteristic, complemented
8#10592513—31nomaximal
9#10590715—34nomaximal
10C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
11C2×C32×Dic6C2211yescharacteristic
12C2×C33×C8C2211yescharacteristic
13C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17#88807—21no—
18#88804—21no—
19#88804—21no—
20Dic3×C3⋊Dic3—21no—
21#64155—31no—
22#64101—31no—
23#63871—34no—
24#63849—34no—
25C32×#103—31no—
26#63662—31no—
27#63647—31no—
28#63647—34no—
29#63647—64no—
30#63645D311yescharacteristic, complemented
31#63635—31no—
32C3×#818—34no—
33C3×#816D314yescomplemented
34#63574—34no—
35#63565—34no—
36C22×C3⋊(C3⋊D3)D411yescharacteristic
37C2×C33×C4C2311yescharacteristic
38C2×C3⋊(C3⋊Dic3)D411yescharacteristic
39C4×C3⋊(C3⋊D3)—21no—
40C4×C3⋊(C3⋊D3)—21no—
41C2×C3×C3⋊Dic3—21no—
42C2×C3×C3⋊Dic3—21no—
43C2×C32×Dic3—21no—
44C2×C32×Dic3—21no—
45C32×Dic6—21no—
46C32×Dic6—21no—
47C33×C8—21no—
48C3⋊(C3⋊(C3⋊C8))—21no—
49#4888—91no—
50#4814—94no—
51C2×C32×Q8—31no—
52C2×C4×C3⋊D3—31no—
53C2×C4×C3⋊D3—34no—
54C2×C4×C3⋊D3—64no—
55C2×C3×Dic6C2×D314yes—
56C2×C32×C8—24no—
57C2×C32×C8C2×D314yes—
58C2×C32×C8C2×D311yescharacteristic
59C32×C4⋊C4—31no—
60#3752—61no—
61#3752—61no—
62C4⋊(C3⋊Dic3)C2×D311yescharacteristic
63C4⋊(C3⋊Dic3)C2×D311yescharacteristic
64#3750—61no—
65C4×C3⋊Dic3—61no—
66C2×C3⋊(C3⋊C8)—31no—
67C2×C3⋊(C3⋊C8)—34no—
68C2×C3⋊(C3⋊C8)—64no—
69C3×C4⋊Dic3—34no—
70C3×C4⋊Dic3—34no—
71C3×C4⋊Dic3C2×D314yes—
72Dic3⋊Dic3—64no—
73#3722—64no—
74#3722—64no—
75Dic32—64no—
76C22×C33C2×D411yescharacteristic
77C2×C3⋊(C3⋊D3)—21no—
78C2×C3⋊(C3⋊D3)—21no—
79C2×C3⋊(C3⋊D3)—21no—
80C33×C4#4111yescharacteristic
81C33×C4#4111yescharacteristic
82C3⋊(C3⋊Dic3)—21no—
83C3×C3⋊Dic3—41no—
84C3×C3⋊Dic3—41no—
85C32×Dic3—41no—
86C32×Dic3—41no—
87#946—94no—
88#897—91no—
89#892—94no—
90#892—91no—
91C3×#107—91no—
92C3×#103—34no—
93#835—94no—
94(C2×D3)⋊C8—94no—
95(C2×D3)⋊C8—91no—
96(C2×D3)⋊C8—184no—
97#818—34no—
98#816C3⋊D311yescharacteristic, complemented
99#810—91no—
100#808—94no—
101#808—91no—
102C22×C3⋊D3—31no—
103C22×C3⋊D3—34no—
104C22×C3⋊D3—64no—
105C32×Q8—61no—
106C32×Q8—61no—
107C2×C32×C4—24no—
108C2×C32×C4—61no—
109C2×C32×C4—61no—
110C2×C32×C4C22×D314yes—
111C2×C32×C4C22×D311yescharacteristic
112C2×C3⋊Dic3—21no—
113C2×C3⋊Dic3—21no—
114C2×C3⋊Dic3—31no—
115C2×C3⋊Dic3—34no—
116C2×C3⋊Dic3—64no—
117C4×C3⋊D3—61no—
118C4×C3⋊D3—61no—
119C4×C3⋊D3—64no—
120C4×C3⋊D3—64no—
121C4×C3⋊D3—124no—
122C4×C3⋊D3—124no—
123C2×C3×Dic3—24no—
124C2×C3×Dic3—24no—
125C2×C3×Dic3—64no—
126C2×C3×Dic3—64no—
127C3×Dic6—24no—
128C3×Dic6—24no—
129C32×C8—21no—
130C32×C8—24no—
131C32×C8—44no—
132C3⋊(C3⋊C8)—61no—
133C3⋊(C3⋊C8)—64no—
134C3⋊(C3⋊C8)—124no—
135#485—271noSylow
136C2×C33#12311yescharacteristic
137C2×C33#13511yescharacteristic
138C2×C33#13711yescharacteristic
139C3⋊(C3⋊D3)—41no—
140C2×C3×Q8—34no—
141C2×C4×D3—94no—
142C2×C4×D3—91no—
143C2×C4×D3—184no—
144C2×Dic6C2×C3⋊D311yescharacteristic
145C2×C3×C8—24no—
146C2×C3×C8D3214yes—
147C2×C3×C8C2×C3⋊D311yescharacteristic
148C3×C4⋊C4—34no—
149C3×C4⋊C4—91no—
150C3×C4⋊C4—91no—
151(C2×D3)⋊C4—181no—
152(C2×D3)⋊C4—181no—
153(C2×D3)⋊C4—184no—
154(C2×D3)⋊C4—184no—
155C4⋊Dic3—34no—
156C4⋊Dic3—34no—
157C4⋊Dic3—181no—
158C4⋊Dic3C2×C3⋊D311yescharacteristic
159Dic3⋊C4—184no—
160C4×Dic3—181no—
161C4×Dic3—184no—
162C2×C3⋊C8—94no—
163C2×C3⋊C8—91no—
164C2×C3⋊C8—184no—
165C22×C32—24no—
166C22×C32D3×D414yes—
167C22×C32D3×D411yescharacteristic
168C2×C3⋊D3—68no—
169C2×C3⋊D3—61no—
170C2×C3⋊D3—61no—
171C2×C3⋊D3—61no—
172C2×C3⋊D3—64no—
173C2×C3⋊D3—64no—
174C2×C3⋊D3—64no—
175C2×C3⋊D3—124no—
176C2×C3⋊D3—124no—
177C32×C4—24no—
178C32×C4—24no—
179C32×C4—121no—
180C32×C4—121no—
181C32×C4#23514yes—
182C32×C4#23511yescharacteristic
183C32×C4#23714yes—
184C32×C4#23911yescharacteristic
185C3⋊Dic3—41no—
186C3⋊Dic3—41no—
187C3⋊Dic3—61no—
188C3⋊Dic3—64no—
189C3⋊Dic3—124no—
190C3×Dic3—44no—
191C3×Dic3—44no—
192C3×Dic3—124no—
193C3×Dic3—124no—
194C22⋊Q8—271no—
195#117—271no—
196#107—91no—
197#106—271no—
198#103—31no—
199#103—271no—
200C22⋊C8—271no—
201C33#48511yescharacteristic, complemented, Sylow
202C22×D3—94no—
203C22×D3—91no—
204C22×D3—184no—
205C3×Q8—64no—
206C3×Q8—64no—
207C2×C3×C4—24no—
208C2×C3×C4—64no—
209C2×C3×C4—64no—
210C2×C3×C4—181no—
211C2×C3×C4—181no—
212C2×C3×C4C2×D3214yes—
213C2×C3×C4C22×C3⋊D311yescharacteristic
214C2×Dic3—21no—
215C2×Dic3—21no—
216C2×Dic3—64no—
217C2×Dic3—64no—
218C2×Dic3—94no—
219C2×Dic3—91no—
220C2×Dic3—184no—
221C4×D3—184no—
222C4×D3—184no—
223C4×D3—181no—
224C4×D3—181no—
225C4×D3—364no—
226C4×D3—364no—
227Dic6—21no—
228Dic6—21no—
229C3×C8—24no—
230C3×C8—21no—
231C3×C8—44no—
232C3⋊C8—184no—
233C3⋊C8—181no—
234C3⋊C8—364no—
235C2×C32—24no—
236C2×C32—24no—
237C2×C32—24no—
238C2×C32#88314yes—
239C2×C32#89411yescharacteristic
240C2×C32#91211yescharacteristic
241C2×C32#91514yes—
242C2×C32#91811yescharacteristic
243C2×C32#91914yes—
244C3⋊D3—128no—
245C3⋊D3—121no—
246C3⋊D3—124no—
247C2×Q8—31no—
248C22×C4—271no—
249C2×C8—271no—
250C2×C8D3×C3⋊D311yescharacteristic
251C4⋊C4—31no—
252C4⋊C4—91no—
253C4⋊C4—91no—
254C4⋊C4—541no—
255C22⋊C4—541no—
256C22⋊C4—541no—
257C42—541no—
258C22×C3—24no—
259C22×C3#380214yes—
260C22×C3D4×C3⋊D311yescharacteristic
261C2×D3—188no—
262C2×D3—184no—
263C2×D3—184no—
264C2×D3—181no—
265C2×D3—184no—
266C2×D3—181no—
267C2×D3—181no—
268C2×D3—364no—
269C2×D3—364no—
270C3×C4—24no—
271C3×C4—24no—
272C3×C4—124no—
273C3×C4—124no—
274C3×C4—361no—
275C3×C4—361no—
276C3×C4#379914yes—
277C3×C4#380414yes—
278C3×C4#382811yescharacteristic
279C3×C4#383011yescharacteristic
280Dic3—41no—
281Dic3—41no—
282Dic3—124no—
283Dic3—124no—
284Dic3—181no—
285Dic3—184no—
286Dic3—364no—
287C32—24no—
288C32#481414yescomplemented
289C32#488811yescharacteristic, complemented
290C23—271no—
291Q8—61no—
292Q8—61no—
293C2×C4—61no—
294C2×C4—61no—
295C2×C4—181no—
296C2×C4—181no—
297C2×C4—271no—
298C2×C4—541no—
299C2×C4—541no—
300C2×C4C2×D3×C3⋊D311yescharacteristic
301C8—21no—
302C8—541no—
303C2×C3—24no—
304C2×C3—24no—
305C2×C3—24no—
306C2×C3#6381014yes—
307C2×C3#6381114yes—
308C2×C3#6385414yes—
309C2×C3#6409211yescharacteristic
310C2×C3#6412411yescharacteristic
311C2×C3#6412811yescharacteristic
312D3—368no—
313D3—361no—
314D3—364no—
315C22—541no—
316C22—541no—
317C22—541no—
318C22#8902611yescharacteristic
319C4—121no—
320C4—121no—
321C4—361no—
322C4—361no—
323C4—541no—
324C4#8902211yescharacteristic
325C4#8902811yescharacteristic
326C3—24no—
327C3#1059071514yescomplemented
328C3#1059251311yescharacteristic, complemented
329C2—1081no—
330C2#1172334411yescharacteristic
331C2#1172334511yescharacteristic
332C2#1172342211yescharacteristic
subgroupG/H
centerC22#89026
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC22×C3⋊(C3⋊D3)D4
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#485
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
4. C3×#63645C2semidirect
5. C32×#816C2semidirect
6. #11722053C2semidirect
7. #11722037C2semidirect
30. #63645D3semidirect
33. C3×#816D3semidirect
98. #816C3⋊D3semidirect
201. C33#485semidirect
288. C32#4814semidirect
289. C32#4888semidirect
327. C3#10590715semidirect
328. C3#10592513semidirect

Maximal subgroups

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

subgroupindex
#117234562
#117234132
#117221902
C3×#636452
C32×#8162
#117220532
#117220372
#105925133
#105907153