Finite Group Databasebeta

← #13159303

Subgroups — #13159303

Order 1728.

1024 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1024 conjugacy classes of subgroups — too many to draw. Only its 146 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#11723934C211yesmaximal, characteristic
2#11723536C211yesmaximal, characteristic, complemented
3C3×#63634C211yesmaximal, characteristic, complemented
4C32×#807C211yesmaximal, characteristic, complemented
5#11722050C211yesmaximal, characteristic, complemented
6#11722036C211yesmaximal, characteristic
7#11722035C211yesmaximal, characteristic
8#10592589—31nomaximal
9#10590430—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C4×C3⋊(C3⋊D3)—21no—
12C2×#6293—21no—
13C33×C4⋊C4C2211yescharacteristic
14#88859—21no—
15C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20#88806—21no—
21#64102—31no—
22#64102—34no—
23#64102—64no—
24#64097—31no—
25#63919—34no—
26C32×#107—31no—
27#63648—31no—
28#63641—31no—
29#63636—31no—
30#63634D311yescharacteristic, complemented
31C3×#807—34no—
32C3×#807D314yescomplemented
33#63572—34no—
34#63564—34no—
35#63564—34no—
36C22×C3⋊(C3⋊D3)—21no—
37C2×C33×C4—21no—
38C2×C33×C4C2311yescharacteristic
39C2×C3⋊(C3⋊Dic3)D411yescharacteristic
40C2×C3⋊(C3⋊Dic3)D411yescharacteristic
41C4⋊(C3⋊(C3⋊D3))—21no—
42C4⋊(C3⋊(C3⋊D3))Q812yescomplemented
43C4×C3⋊(C3⋊D3)—42no—
44C2×C3×C3⋊Dic3—21no—
45C2×C32×Dic3—21no—
46#6293—42no—
47C3×C3⋊(C3⋊C8)—21no—
48C32×C3⋊C8—21no—
49#4890—94no—
50#4890—91no—
51C2×C4⋊(C3⋊D3)—31no—
52C2×C4⋊(C3⋊D3)—34no—
53C2×C4⋊(C3⋊D3)—64no—
54C2×C4×C3⋊D3—61no—
55C2×C4×C3⋊D3—64no—
56C2×C4×C3⋊D3—68no—
57C2×#634—61no—
58C2×#627—64no—
59C2×C32×C8—31no—
60C32×C4⋊C4—24no—
61C32×C4⋊C4—31no—
62C32×C4⋊C4C2×D314yes—
63C32×C4⋊C4C2×D311yescharacteristic
64#3752—61no—
65#3752—64no—
66#3752—68no—
67C4⋊(C3⋊Dic3)C2×D311yescharacteristic
68#3750—61no—
69C4×C3⋊Dic3—31no—
70C4×C3⋊Dic3—34no—
71C4×C3⋊Dic3—64no—
72C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
73C3×C4⋊Dic3—34no—
74C3×C4⋊Dic3C2×D314yes—
75C2×C3×C3⋊C8—34no—
76C2×C3×C3⋊C8C2×D314yes—
77#3724—64no—
78C22×C33C2×D411yescharacteristic
79C2×C3⋊(C3⋊D3)—22no—
80C2×C3⋊(C3⋊D3)—41no—
81C2×C3⋊(C3⋊D3)—41no—
82C33×C4—41no—
83C33×C4C2×Q811yescharacteristic
84C33×C4#4111yescharacteristic
85C3⋊(C3⋊Dic3)—21no—
86C3⋊(C3⋊Dic3)D812yes—
87C3×C3⋊Dic3—41no—
88C32×Dic3—41no—
89#893—94no—
90#893—91no—
91#893—184no—
92#888—94no—
93#888—91no—
94C3×#107—34no—
95C3×#107—91no—
96#821—94no—
97#821—91no—
98Dic3⋊C8—94no—
99Dic3⋊C8—91no—
100#809—94no—
101#809—91no—
102#807—34no—
103#807C3⋊D311yescharacteristic, complemented
104C22×C3⋊D3—68no—
105C22×C3⋊D3—64no—
106C22×C3⋊D3—61no—
107C2×C32×C4—28no—
108C2×C32×C4—24no—
109C2×C32×C4—21no—
110C2×C32×C4—24no—
111C2×C32×C4—61no—
112C2×C32×C4C22×D314yes—
113C2×C32×C4C22×D311yescharacteristic
114C2×C3⋊Dic3—21no—
115C2×C3⋊Dic3—31no—
116C2×C3⋊Dic3—31no—
117C2×C3⋊Dic3—34no—
118C2×C3⋊Dic3—34no—
119C2×C3⋊Dic3—64no—
120C2×C3⋊Dic3—64no—
121C4⋊(C3⋊D3)—32no—
122C4⋊(C3⋊D3)—38no—
123C4⋊(C3⋊D3)—68no—
124C4⋊(C3⋊D3)—68no—
125C4⋊(C3⋊D3)—64no—
126C4⋊(C3⋊D3)—61no—
127C4×C3⋊D3—122no—
128C4×C3⋊D3—128no—
129C4×C3⋊D3—1216no—
130#634—122no—
131C2×C3×Dic3—24no—
132C2×C3×Dic3—64no—
133#627—128no—
134C32×C8—61no—
135C3⋊(C3⋊C8)—21no—
136C3×C3⋊C8—24no—
137C3×C3⋊C8—64no—
138#474—271noSylow
139C2×C33C22⋊Q811yescharacteristic
140C2×C33C2×D811yescharacteristic
141C2×C33#13711yescharacteristic
142C3⋊(C3⋊D3)—42no—
143C2×D12—94no—
144C2×D12—91no—
145C2×D12—184no—
146C2×C4×D3—184no—
147C2×C4×D3—181no—
148C2×C4×D3—188no—
149C2×Dic6—181no—
150C2×Dic6—184no—
151C2×C3×C8—34no—
152C2×C3×C8—91no—
153C3×C4⋊C4—24no—
154C3×C4⋊C4—34no—
155C3×C4⋊C4—91no—
156C3×C4⋊C4D3214yes—
157C3×C4⋊C4C2×C3⋊D311yescharacteristic
158(C2×D3)⋊C4—184no—
159(C2×D3)⋊C4—181no—
160(C2×D3)⋊C4—188no—
161C4⋊Dic3—34no—
162C4⋊Dic3C2×C3⋊D311yescharacteristic
163Dic3⋊C4—181no—
164Dic3⋊C4—184no—
165C4×Dic3—94no—
166C4×Dic3—91no—
167C4×Dic3—184no—
168C2×C3⋊C8—34no—
169C2×C3⋊C8C2×C3⋊D311yescharacteristic
170C22×C32—24no—
171C22×C32D3×D414yes—
172C22×C32D3×D411yescharacteristic
173C2×C3⋊D3—616no—
174C2×C3⋊D3—68no—
175C2×C3⋊D3—62no—
176C2×C3⋊D3—128no—
177C2×C3⋊D3—128no—
178C2×C3⋊D3—124no—
179C2×C3⋊D3—121no—
180C2×C3⋊D3—124no—
181C2×C3⋊D3—121no—
182C32×C4—24no—
183C32×C4—24no—
184C32×C4—48no—
185C32×C4—44no—
186C32×C4—41no—
187C32×C4—121no—
188C32×C4#23514yes—
189C32×C4#23511yescharacteristic
190C32×C4D3×Q814yes—
191C32×C4D3×Q811yescharacteristic
192C3⋊Dic3—32no—
193C3⋊Dic3—38no—
194C3⋊Dic3—41no—
195C3⋊Dic3—68no—
196C3⋊Dic3—61no—
197C3⋊Dic3—64no—
198C3⋊Dic3—124no—
199C3×Dic3—44no—
200C3×Dic3—124no—
201C4⋊Q8—271no—
202C4×D4—271no—
203#107—31no—
204#107—91no—
205C4⋊C8—271no—
206#102—271no—
207#102—271no—
208C33#47411yescharacteristic, complemented, Sylow
209C22×D3—188no—
210C22×D3—184no—
211C22×D3—181no—
212C2×C3×C4—28no—
213C2×C3×C4—24no—
214C2×C3×C4—21no—
215C2×C3×C4—24no—
216C2×C3×C4—64no—
217C2×C3×C4—181no—
218C2×C3×C4C2×D3214yes—
219C2×C3×C4C22×C3⋊D311yescharacteristic
220C2×Dic3—21no—
221C2×Dic3—64no—
222C2×Dic3—94no—
223C2×Dic3—91no—
224C2×Dic3—94no—
225C2×Dic3—91no—
226C2×Dic3—184no—
227C2×Dic3—184no—
228D12—98no—
229D12—92no—
230D12—188no—
231D12—188no—
232D12—184no—
233D12—181no—
234C4×D3—362no—
235C4×D3—368no—
236C4×D3—3616no—
237Dic6—362no—
238Dic6—368no—
239C3×C8—64no—
240C3×C8—181no—
241C3⋊C8—21no—
242C3⋊C8—64no—
243C2×C32—24no—
244C2×C32—24no—
245C2×C32—24no—
246C2×C32#89614yes—
247C2×C32#89611yescharacteristic
248C2×C32D3×D814yes—
249C2×C32D3×D811yescharacteristic
250C2×C32#91814yes—
251C2×C32#91811yescharacteristic
252C3⋊D3—1216no—
253C3⋊D3—128no—
254C3⋊D3—122no—
255C2×Q8—541no—
256C2×D4—271no—
257C22×C4—541no—
258C2×C8—31no—
259C2×C8—91no—
260C4⋊C4—31no—
261C4⋊C4—91no—
262C4⋊C4—541no—
263C4⋊C4D3×C3⋊D311yescharacteristic
264C22⋊C4—541no—
265C42—271no—
266C22×C3—24no—
267C22×C3#381114yes—
268C22×C3D4×C3⋊D311yescharacteristic
269C2×D3—1816no—
270C2×D3—182no—
271C2×D3—188no—
272C2×D3—368no—
273C2×D3—368no—
274C2×D3—361no—
275C2×D3—361no—
276C2×D3—364no—
277C2×D3—364no—
278C3×C4—24no—
279C3×C4—24no—
280C3×C4—48no—
281C3×C4—41no—
282C3×C4—44no—
283C3×C4—124no—
284C3×C4—361no—
285C3×C4#379714yes—
286C3×C4#379814yes—
287C3×C4#382811yescharacteristic
288C3×C4Q8×C3⋊D311yescharacteristic
289Dic3—41no—
290Dic3—92no—
291Dic3—98no—
292Dic3—124no—
293Dic3—188no—
294Dic3—181no—
295Dic3—184no—
296Dic3—364no—
297C32—24no—
298C32#489014yescomplemented
299C32#489011yescharacteristic, complemented
300C23—541no—
301Q8—1082no—
302D4—272no—
303D4—541no—
304C2×C4—21no—
305C2×C4—61no—
306C2×C4—181no—
307C2×C4—271no—
308C2×C4—271no—
309C2×C4—1082no—
310C2×C4C2×D3×C3⋊D311yescharacteristic
311C8—61no—
312C8—181no—
313C2×C3—24no—
314C2×C3—24no—
315C2×C3—24no—
316C2×C3#6386514yes—
317C2×C3#6392614yes—
318C2×C3#6394514yes—
319C2×C3#6410511yescharacteristic
320C2×C3D8×C3⋊D311yescharacteristic
321C2×C3#6412711yescharacteristic
322D3—3616no—
323D3—362no—
324D3—368no—
325C22—542no—
326C22—1081no—
327C22—1081no—
328C22#8903911yescharacteristic
329C4—41no—
330C4—121no—
331C4—272no—
332C4—361no—
333C4—541no—
334C4#8901911yescharacteristic
335C4#8902011yescharacteristic
336C3—24no—
337C3#1059043014yescomplemented
338C3#1059258911yescharacteristic, complemented
339C2—1082no—
340C2#1172344211yescharacteristic
341C2#1172354711yescharacteristic
342C2#1172357811yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×C4⋊(C3⋊(C3⋊D3))C22
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#474
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #11723536C2semidirect
3. C3×#63634C2semidirect
4. C32×#807C2semidirect
5. #11722050C2semidirect
30. #63634D3semidirect
32. C3×#807D3semidirect
42. C4⋊(C3⋊(C3⋊D3))Q8semidirect
103. #807C3⋊D3semidirect
208. C33#474semidirect
298. C32#4890semidirect
299. C32#4890semidirect
337. C3#10590430semidirect
338. C3#10592589semidirect

Maximal subgroups

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

subgroupindex
#117239342
#117235362
C3×#636342
C32×#8072
#117220502
#117220362
#117220352
#105925893
#105904303