Finite Group Databasebeta

← #13159706

Subgroups — #13159706

Order 1728.

800 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 148 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#11723536C211yesmaximal, characteristic, complemented
2#11723501C211yesmaximal, characteristic
3#11722184C211yesmaximal, characteristic, complemented
4C3×#63648C211yesmaximal, characteristic
5C32×#818C211yesmaximal, characteristic, complemented
6#11722056C211yesmaximal, characteristic, complemented
7#11722033C211yesmaximal, characteristic
8#10592589—31nomaximal
9#10590778—34nomaximal
10C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
11C2×#6293—21no—
12C2×Dic3×C3⋊D3—21no—
13C2×C33×C8C2211yescharacteristic
14C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19#88806—21no—
20#88803—21no—
21#64102—31no—
22#64097—31no—
23#63919—34no—
24#63898—34no—
25C32×#107—31no—
26#63648D311yescharacteristic
27#63641—31no—
28#63641—34no—
29#63641—64no—
30#63636—31no—
31#63634—31no—
32C3×#821—34no—
33C3×#818D314yescomplemented
34#63576—34no—
35#63562—34no—
36C22×C3×C3⋊D3—21no—
37C2×C33×C4C2311yescharacteristic
38C2×C3⋊(C3⋊Dic3)D411yescharacteristic
39C2×C3⋊(C3⋊Dic3)D411yescharacteristic
40C2×C3×C3⋊Dic3—21no—
41C3×C4⋊(C3⋊D3)—21no—
42C3×C4⋊(C3⋊D3)Q812yescomplemented
43C2×C32×Dic3—21no—
44C2×C32×Dic3—21no—
45#6293—42no—
46Dic3×C3⋊D3—42no—
47C33×C8—21no—
48C3⋊(C3⋊(C3⋊C8))—21no—
49#4890—94no—
50#4759—91no—
51C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
52C2×C4×C3⋊D3—61no—
53C2×#634—61no—
54C2×C3×D12—34no—
55C2×#627—64no—
56C2×D3×Dic3—64no—
57C2×C32×C8—24no—
58C2×C32×C8C2×D314yes—
59C2×C32×C8C2×D311yescharacteristic
60C32×C4⋊C4—31no—
61C32×C4⋊C4—31no—
62#3752—61no—
63C4⋊(C3⋊Dic3)C2×D311yescharacteristic
64#3750—61no—
65C4×C3⋊Dic3—31no—
66C4×C3⋊Dic3—34no—
67C4×C3⋊Dic3—64no—
68C2×C3⋊(C3⋊C8)—31no—
69C2×C3⋊(C3⋊C8)—34no—
70C2×C3⋊(C3⋊C8)—64no—
71C3×C4⋊Dic3—34no—
72C3×C4⋊Dic3C2×D314yes—
73C3×C4⋊Dic3C2×D314yes—
74#3724—64no—
75(C2×D3)⋊Dic3—64no—
76C22×C33C2×D411yescharacteristic
77C2×C3×C3⋊D3—22no—
78C2×C3×C3⋊D3—41no—
79C2×C3×C3⋊D3—41no—
80C33×C4C2×Q811yescharacteristic
81C33×C4#4111yescharacteristic
82C3⋊(C3⋊Dic3)—21no—
83C3⋊(C3⋊Dic3)D812yes—
84C3×C3⋊Dic3—41no—
85C32×Dic3—41no—
86C32×Dic3—41no—
87D4×Dic3—91no—
88#893—94no—
89#888—94no—
90#888—91no—
91C3×#107—34no—
92C3×#102—91no—
93#832—91no—
94#821—34no—
95#818C3⋊D311yescharacteristic, complemented
96Dic3⋊C8—94no—
97Dic3⋊C8—91no—
98Dic3⋊C8—184no—
99#809—94no—
100#807—94no—
101#807—91no—
102C22×C3⋊D3—21no—
103C22×C3×D3—64no—
104C2×C32×C4—24no—
105C2×C32×C4—61no—
106C2×C32×C4—61no—
107C2×C32×C4C22×D314yes—
108C2×C32×C4C22×D311yescharacteristic
109C2×C3⋊Dic3—21no—
110C2×C3⋊Dic3—31no—
111C2×C3⋊Dic3—31no—
112C2×C3⋊Dic3—34no—
113C2×C3⋊Dic3—34no—
114C2×C3⋊Dic3—64no—
115C2×C3⋊Dic3—64no—
116C4⋊(C3⋊D3)—21no—
117C4⋊(C3⋊D3)Dic612yescomplemented
118C4×C3⋊D3—122no—
119#634—122no—
120C2×C3×Dic3—24no—
121C2×C3×Dic3—24no—
122C2×C3×Dic3—64no—
123C3×D12—38no—
124C3×D12—64no—
125#627—128no—
126D3×Dic3—128no—
127C32×C8—21no—
128C32×C8—24no—
129C32×C8—44no—
130C3⋊(C3⋊C8)—61no—
131C3⋊(C3⋊C8)—64no—
132C3⋊(C3⋊C8)—124no—
133#474—271noSylow
134C2×C33C22⋊Q811yescharacteristic
135C2×C33C2×D811yescharacteristic
136C2×C33#13711yescharacteristic
137C3×C3⋊D3—42no—
138C2×C3×D4—91no—
139C22×Dic3—181no—
140C2×D12—34no—
141C2×C4×D3—184no—
142C2×Dic6—181no—
143C2×Dic6—184no—
144C2×C3×C8—24no—
145C2×C3×C8D3214yes—
146C2×C3×C8C2×C3⋊D311yescharacteristic
147C3×C4⋊C4—34no—
148C3×C4⋊C4—34no—
149C3×C4⋊C4—91no—
150#217—181no—
151(C2×D3)⋊C4—184no—
152C4⋊Dic3—34no—
153C4⋊Dic3C2×C3⋊D311yescharacteristic
154C4⋊Dic3C2×C3⋊D311yescharacteristic
155Dic3⋊C4—181no—
156Dic3⋊C4—184no—
157C4×Dic3—94no—
158C4×Dic3—91no—
159C4×Dic3—184no—
160C2×C3⋊C8—94no—
161C2×C3⋊C8—91no—
162C2×C3⋊C8—184no—
163C22×C32—24no—
164C22×C32D3×D414yes—
165C22×C32D3×D411yescharacteristic
166C2×C3⋊D3—22no—
167C2×C3⋊D3—41no—
168C2×C3⋊D3—41no—
169C2×C3×D3—68no—
170C2×C3×D3—124no—
171C2×C3×D3—124no—
172C32×C4—24no—
173C32×C4—24no—
174C32×C4—121no—
175C32×C4—121no—
176C32×C4C2×Dic611yescharacteristic
177C32×C4#23514yes—
178C32×C4#23711yescharacteristic
179C32×C4D3×Q814yes—
180C3⋊Dic3—32no—
181C3⋊Dic3—38no—
182C3⋊Dic3—41no—
183C3⋊Dic3—68no—
184C3⋊Dic3—61no—
185C3⋊Dic3—64no—
186C3⋊Dic3—124no—
187C3×Dic3—44no—
188C3×Dic3—44no—
189C3×Dic3—124no—
190C4⋊Q8—271no—
191C4×D4—271no—
192#107—31no—
193#107—271no—
194C4⋊C8—271no—
195#102—91no—
196#102—271no—
197C33#47411yescharacteristic, complemented, Sylow
198C23×C3—181no—
199C22×D3—64no—
200C3×D4—92no—
201C3×D4—181no—
202C2×C3×C4—24no—
203C2×C3×C4—64no—
204C2×C3×C4—64no—
205C2×C3×C4—181no—
206C2×C3×C4C2×D3214yes—
207C2×C3×C4C22×C3⋊D311yescharacteristic
208C2×Dic3—21no—
209C2×Dic3—21no—
210C2×Dic3—64no—
211C2×Dic3—94no—
212C2×Dic3—91no—
213C2×Dic3—94no—
214C2×Dic3—91no—
215C2×Dic3—184no—
216C2×Dic3—184no—
217C2×Dic3—362no—
218D12—38no—
219D12—64no—
220C4×D3—368no—
221Dic6—362no—
222Dic6—368no—
223C3×C8—24no—
224C3×C8—21no—
225C3×C8—44no—
226C3⋊C8—184no—
227C3⋊C8—181no—
228C3⋊C8—364no—
229C2×C32—24no—
230C2×C32—24no—
231C2×C32—24no—
232C2×C32#87811yescharacteristic
233C2×C32#89614yes—
234C2×C32D3×D814yes—
235C2×C32D3×D811yescharacteristic
236C2×C32#91511yescharacteristic
237C2×C32#91814yes—
238C3⋊D3—42no—
239C3×D3—128no—
240C2×Q8—541no—
241C2×D4—91no—
242C22×C4—541no—
243C2×C8—271no—
244C2×C8D3×C3⋊D311yescharacteristic
245C4⋊C4—31no—
246C4⋊C4—31no—
247C4⋊C4—91no—
248C4⋊C4—541no—
249C22⋊C4—541no—
250C42—271no—
251C22×C3—24no—
252C22×C3—182no—
253C22×C3—361no—
254C22×C3—361no—
255C22×C3#380214yes—
256C22×C3D4×C3⋊D311yescharacteristic
257C2×D3—68no—
258C2×D3—124no—
259C2×D3—124no—
260C3×C4—24no—
261C3×C4—24no—
262C3×C4—124no—
263C3×C4—124no—
264C3×C4—361no—
265C3×C4D3×Dic614yes—
266C3×C4#380414yes—
267C3×C4#382811yescharacteristic
268C3×C4Q8×C3⋊D311yescharacteristic
269Dic3—41no—
270Dic3—41no—
271Dic3—98no—
272Dic3—92no—
273Dic3—124no—
274Dic3—188no—
275Dic3—184no—
276Dic3—181no—
277Dic3—364no—
278C32—24no—
279C32#475911yescharacteristic, complemented
280C32#489014yescomplemented
281C23—181no—
282Q8—1082no—
283D4—92no—
284D4—181no—
285C2×C4—61no—
286C2×C4—61no—
287C2×C4—181no—
288C2×C4—271no—
289C2×C4—271no—
290C2×C4—1082no—
291C2×C4C2×D3×C3⋊D311yescharacteristic
292C8—21no—
293C8—541no—
294C2×C3—24no—
295C2×C3—24no—
296C2×C3—24no—
297C2×C3—362no—
298C2×C3#6379714yes—
299C2×C3#6381114yes—
300C2×C3#6389914yes—
301C2×C3#6410511yescharacteristic
302C2×C3D8×C3⋊D311yescharacteristic
303C2×C3#6412711yescharacteristic
304D3—128no—
305C22—182no—
306C22—361no—
307C22—361no—
308C22#8902611yescharacteristic
309C4—121no—
310C4—121no—
311C4—272no—
312C4—361no—
313C4—541no—
314C4C3⋊D3×Dic611yescharacteristic
315C4#8903111yescharacteristic
316C3—24no—
317C3#1059077814yescomplemented
318C3#1059258911yescharacteristic, complemented
319C2—362no—
320C2#1172332211yescharacteristic
321C2#1172334611yescharacteristic
322C2#1172350211yescharacteristic
subgroupG/H
centerC22#89026
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#474
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
1. #11723536C2semidirect
3. #11722184C2semidirect
5. C32×#818C2semidirect
6. #11722056C2semidirect
33. C3×#818D3semidirect
42. C3×C4⋊(C3⋊D3)Q8semidirect
95. #818C3⋊D3semidirect
117. C4⋊(C3⋊D3)Dic6semidirect
197. C33#474semidirect
279. C32#4759semidirect
280. C32#4890semidirect
317. C3#10590778semidirect
318. C3#10592589semidirect

Maximal subgroups

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

subgroupindex
#117235362
#117235012
#117221842
C3×#636482
C32×#8182
#117220562
#117220332
#105925893
#105907783