Finite Group Databasebeta

← #13159599

Subgroups — #13159599

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#11723449C211yesmaximal, characteristic
2#11723413C211yesmaximal, characteristic
3#11722190C211yesmaximal, characteristic
4C3×#63643C211yesmaximal, characteristic, complemented
5C32×#818C211yesmaximal, characteristic, complemented
6#11722053C211yesmaximal, characteristic, complemented
7#11722039C211yesmaximal, characteristic, complemented
8#10592587—31nomaximal
9#10590715—34nomaximal
10C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
12C2×C33×C8C2211yescharacteristic
13C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C32×C4⋊Dic3C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17Dic3⋊(C3⋊Dic3)—21no—
18#88804—21no—
19#88804—21no—
20Dic3×C3⋊Dic3—21no—
21#64106—31no—
22#64101—31no—
23#63871—34no—
24#63849—34no—
25C32×#107—31no—
26#63647—31no—
27#63647—34no—
28#63647—64no—
29#63643D311yescharacteristic, complemented
30#63637—31no—
31#63635—31no—
32C3×#818D314yescomplemented
33C3×#816—34no—
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—
43C3×#634—21no—
44C3×#634—21no—
45C2×C32×Dic3—21no—
46C2×C32×Dic3—21no—
47C33×C8—21no—
48C3⋊(C3⋊(C3⋊C8))—21no—
49#4888—94no—
50#4814—91no—
51C2×C4×C3⋊D3—31no—
52C2×C4×C3⋊D3—34no—
53C2×C4×C3⋊D3—64no—
54C2×#634C2×D311yescharacteristic
55C2×C3×Dic6—34no—
56C2×C32×C8—24no—
57C2×C32×C8C2×D314yes—
58C2×C32×C8C2×D311yescharacteristic
59C32×C4⋊C4—31no—
60C32×C4⋊C4—31no—
61#3752—61no—
62#3752—61no—
63C4⋊(C3⋊Dic3)—61no—
64C4⋊(C3⋊Dic3)C2×D311yescharacteristic
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⋊Dic3C2×D314yes—
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—91no—
88#897—94no—
89#892—94no—
90#892—91no—
91C3×#107—34no—
92C3×#103—91no—
93#835—91no—
94(C2×D3)⋊C8—94no—
95(C2×D3)⋊C8—91no—
96(C2×D3)⋊C8—184no—
97#818C3⋊D311yescharacteristic, complemented
98#816—34no—
99#810—94no—
100#808—94no—
101#808—91no—
102C22×C3⋊D3—31no—
103C22×C3⋊D3—34no—
104C22×C3⋊D3—64no—
105C2×C32×C4—24no—
106C2×C32×C4—61no—
107C2×C32×C4—61no—
108C2×C32×C4C22×D314yes—
109C2×C32×C4C22×D311yescharacteristic
110C2×C3⋊Dic3—21no—
111C2×C3⋊Dic3—21no—
112C2×C3⋊Dic3—31no—
113C2×C3⋊Dic3—34no—
114C2×C3⋊Dic3—64no—
115C4×C3⋊D3—61no—
116C4×C3⋊D3—61no—
117C4×C3⋊D3—64no—
118C4×C3⋊D3—64no—
119C4×C3⋊D3—124no—
120C4×C3⋊D3—124no—
121#634—21no—
122#634—21no—
123C2×C3×Dic3—24no—
124C2×C3×Dic3—24no—
125C2×C3×Dic3—64no—
126C2×C3×Dic3—64no—
127C3×Dic6—64no—
128C3×Dic6—64no—
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—91no—
141C2×C4×D3—94no—
142C2×C4×D3—91no—
143C2×C4×D3—184no—
144C2×Dic6—34no—
145C2×C3×C8—24no—
146C2×C3×C8D3214yes—
147C2×C3×C8C2×C3⋊D311yescharacteristic
148C3×C4⋊C4—34no—
149C3×C4⋊C4—34no—
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—184no—
157C4⋊Dic3C2×C3⋊D311yescharacteristic
158C4⋊Dic3C2×C3⋊D311yescharacteristic
159Dic3⋊C4—181no—
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—64no—
171C2×C3⋊D3—61no—
172C2×C3⋊D3—61no—
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#23711yescharacteristic
184C32×C4#23914yes—
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—31no—
197#106—271no—
198#103—91no—
199#103—271no—
200C22⋊C8—271no—
201C33#48511yescharacteristic, complemented, Sylow
202C22×D3—94no—
203C22×D3—91no—
204C22×D3—184no—
205C3×Q8—181no—
206C3×Q8—181no—
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—64no—
228Dic6—64no—
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#88311yescharacteristic
239C2×C32#89414yes—
240C2×C32#91214yes—
241C2×C32#91511yescharacteristic
242C2×C32#91814yes—
243C2×C32#91911yescharacteristic
244C3⋊D3—128no—
245C3⋊D3—124no—
246C3⋊D3—121no—
247C2×Q8—91no—
248C22×C4—271no—
249C2×C8—271no—
250C2×C8D3×C3⋊D311yescharacteristic
251C4⋊C4—31no—
252C4⋊C4—31no—
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#383211yescharacteristic
280Dic3—41no—
281Dic3—41no—
282Dic3—124no—
283Dic3—124no—
284Dic3—181no—
285Dic3—184no—
286Dic3—364no—
287C32—24no—
288C32#481411yescharacteristic, complemented
289C32#488814yescomplemented
290C23—271no—
291Q8—181no—
292Q8—181no—
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#6410311yescharacteristic
310C2×C3#6412111yescharacteristic
311C2×C3#6412711yescharacteristic
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#8901811yescharacteristic
325C4#8903111yescharacteristic
326C3—24no—
327C3#1059071514yescomplemented
328C3#1059258711yescharacteristic, complemented
329C2—1081no—
330C2#1172334311yescharacteristic
331C2#1172334611yescharacteristic
332C2#1172350411yescharacteristic
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×#63643C2semidirect
5. C32×#818C2semidirect
6. #11722053C2semidirect
7. #11722039C2semidirect
29. #63643D3semidirect
32. C3×#818D3semidirect
97. #818C3⋊D3semidirect
201. C33#485semidirect
288. C32#4814semidirect
289. C32#4888semidirect
327. C3#10590715semidirect
328. C3#10592587semidirect

Maximal subgroups

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

subgroupindex
#117234492
#117234132
#117221902
C3×#636432
C32×#8182
#117220532
#117220392
#105925873
#105907153