Finite Group Databasebeta

← #13159520

Subgroups — #13159520

Order 1728.

860 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 860 conjugacy classes of subgroups — too many to draw. Only its 168 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
1C4×#6289C211yesmaximal, characteristic
2#11722190C211yesmaximal, characteristic
3C3×#63641C211yesmaximal, characteristic, complemented
4C32×#815C211yesmaximal, characteristic, complemented
5#11722051C211yesmaximal, characteristic, complemented
6#11722024C211yesmaximal, characteristic
7#11722019C211yesmaximal, characteristic, complemented
8#10592404—31nomaximal
9#10590665—34nomaximal
10C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
11C2×#6289C411yescharacteristic
12C2×#6289C411yescharacteristic
13C2×C33×C8C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×C4×C3⋊Dic3C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4×Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19Dic3×C3⋊Dic3C411yescharacteristic
20Dic3×C3⋊Dic3C411yescharacteristic
21C42×C3⋊D3—31no—
22C4×#624—34no—
23C32×#97—31no—
24#63647—31no—
25#63647—31no—
26#63647—34no—
27#63647—64no—
28#63641—31no—
29#63641D311yescharacteristic, complemented
30#63630—31no—
31C3×#815D314yescomplemented
32C3×Dic3⋊C8—34no—
33Dic3⋊(C3⋊C8)—34no—
34#63557—34no—
35#63554—34no—
36C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
37C2×C33×C4C2311yescharacteristic
38C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
39C4×C3⋊(C3⋊D3)—21no—
40C4×C3⋊(C3⋊D3)—21no—
41C2×C3×C3⋊Dic3C2×C411yescharacteristic
42C2×C3×C3⋊Dic3C2×C411yescharacteristic
43C2×C32×Dic3C2×C411yescharacteristic
44C2×C32×Dic3C2×C411yescharacteristic
45#6289—22no—
46#6289—22no—
47C33×C8—21no—
48C3⋊(C3⋊(C3⋊C8))—21no—
49C3×C3⋊(C3⋊C8)—21no—
50C32×C3⋊C8—21no—
51#4835—91no—
52#4705—94no—
53C2×C4×C3⋊D3—31no—
54C2×C4×C3⋊D3—34no—
55C2×C4×C3⋊D3—31no—
56C2×C4×C3⋊D3—31no—
57C2×C4×C3⋊D3—64no—
58C2×#624—34no—
59C2×#624—34no—
60C2×C32×C8—24no—
61C2×C32×C8—31no—
62C2×C32×C8C2×D314yes—
63C2×C32×C8C2×D311yescharacteristic
64C32×C42—31no—
65C4×C3⋊Dic3—31no—
66C4×C3⋊Dic3—31no—
67C4×C3⋊Dic3C2×D311yescharacteristic
68C2×C3⋊(C3⋊C8)—31no—
69C2×C3⋊(C3⋊C8)—34no—
70C2×C3⋊(C3⋊C8)—64no—
71C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
72C3×C4×Dic3—34no—
73C3×C4×Dic3C2×D314yes—
74C2×C3×C3⋊C8—34no—
75C2×C3×C3⋊C8C2×D314yes—
76Dic32—34no—
77Dic32—34no—
78C22×C33C22×C411yescharacteristic
79C2×C3⋊(C3⋊D3)—21no—
80C2×C3⋊(C3⋊D3)—21no—
81C2×C3⋊(C3⋊D3)M1612yes—
82C33×C4#4111yescharacteristic
83C33×C4#4111yescharacteristic
84C3⋊(C3⋊Dic3)—21no—
85C3×C3⋊Dic3—21no—
86C3×C3⋊Dic3M1612yes—
87C32×Dic3—21no—
88C32×Dic3—21no—
89C42×D3—94no—
90C42×D3—91no—
91C3×C4⋊C8—91no—
92C3×#97—34no—
93(C2×D3)⋊C8—94no—
94(C2×D3)⋊C8—94no—
95(C2×D3)⋊C8—91no—
96(C2×D3)⋊C8—91no—
97(C2×D3)⋊C8—184no—
98#815—91no—
99#815C3⋊D311yescharacteristic, complemented
100Dic3⋊C8—34no—
101Dic3⋊C8—94no—
102C4⋊(C3⋊C8)—91no—
103#803—94no—
104C22×C3⋊D3—31no—
105C22×C3⋊D3—34no—
106C22×C3⋊D3—64no—
107C2×C32×C4—24no—
108C2×C32×C4—31no—
109C2×C32×C4—31no—
110C2×C32×C4C22×D314yes—
111C2×C32×C4C22×D311yescharacteristic
112C2×C3⋊Dic3—31no—
113C2×C3⋊Dic3—34no—
114C2×C3⋊Dic3—64no—
115C2×C3⋊Dic3C4×D311yescharacteristic
116C2×C3⋊Dic3C4×D311yescharacteristic
117C4×C3⋊D3—62no—
118C4×C3⋊D3—61no—
119C4×C3⋊D3—61no—
120C4×C3⋊D3—62no—
121C4×C3⋊D3—64no—
122C4×C3⋊D3—64no—
123C4×C3⋊D3—124no—
124C4×C3⋊D3—124no—
125C2×C3×Dic3—34no—
126C2×C3×Dic3—34no—
127C2×C3×Dic3C4×D314yes—
128C2×C3×Dic3C4×D314yes—
129#624—68no—
130#624—68no—
131C32×C8—21no—
132C32×C8—24no—
133C32×C8—44no—
134C32×C8—61no—
135C3⋊(C3⋊C8)—21no—
136C3⋊(C3⋊C8)—61no—
137C3⋊(C3⋊C8)—64no—
138C3⋊(C3⋊C8)—124no—
139C3×C3⋊C8—24no—
140C3×C3⋊C8—64no—
141#432—271noSylow
142C2×C33#11711yescharacteristic
143C2×C33C2×M1611yescharacteristic
144C2×C33C2×M1611yescharacteristic
145C3⋊(C3⋊D3)—22no—
146C2×C4×D3—94no—
147C2×C4×D3—94no—
148C2×C4×D3—94no—
149C2×C4×D3—91no—
150C2×C4×D3—91no—
151C2×C4×D3—91no—
152C2×C4×D3—184no—
153C2×C3×C8—24no—
154C2×C3×C8—34no—
155C2×C3×C8—91no—
156C2×C3×C8D3214yes—
157C2×C3×C8C2×C3⋊D311yescharacteristic
158C3×C42—34no—
159C3×C42—91no—
160C4×Dic3—34no—
161C4×Dic3—94no—
162C4×Dic3—94no—
163C4×Dic3—91no—
164C4×Dic3—91no—
165C4×Dic3C2×C3⋊D311yescharacteristic
166C2×C3⋊C8—34no—
167C2×C3⋊C8—94no—
168C2×C3⋊C8—91no—
169C2×C3⋊C8—184no—
170C2×C3⋊C8C2×C3⋊D311yescharacteristic
171C22×C32—24no—
172C22×C32C2×C4×D314yes—
173C22×C32C2×C4×D311yescharacteristic
174C2×C3⋊D3—32no—
175C2×C3⋊D3—38no—
176C2×C3⋊D3—68no—
177C2×C3⋊D3—61no—
178C2×C3⋊D3—61no—
179C2×C3⋊D3—64no—
180C2×C3⋊D3—64no—
181C2×C3⋊D3—124no—
182C2×C3⋊D3—124no—
183C32×C4—24no—
184C32×C4—24no—
185C32×C4—61no—
186C32×C4—61no—
187C32×C4#23514yes—
188C32×C4#23514yes—
189C32×C4#23711yescharacteristic
190C32×C4#23911yescharacteristic
191C3⋊Dic3—21no—
192C3⋊Dic3—61no—
193C3⋊Dic3—64no—
194C3⋊Dic3—124no—
195C3⋊Dic3ΓL(1,𝔽25)12yes—
196C3×Dic3—24no—
197C3×Dic3—24no—
198C3×Dic3—38no—
199C3×Dic3—64no—
200C2×C42—271no—
201C4⋊C8—91no—
202C4⋊C8—271no—
203C22⋊C8—271no—
204C22⋊C8—271no—
205#97—31no—
206#97—271no—
207C33#43211yescharacteristic, complemented, Sylow
208C22×D3—94no—
209C22×D3—91no—
210C22×D3—184no—
211C2×C3×C4—24no—
212C2×C3×C4—34no—
213C2×C3×C4—34no—
214C2×C3×C4—91no—
215C2×C3×C4—91no—
216C2×C3×C4C2×D3214yes—
217C2×C3×C4C22×C3⋊D311yescharacteristic
218C2×Dic3—34no—
219C2×Dic3—34no—
220C2×Dic3—94no—
221C2×Dic3—91no—
222C2×Dic3—184no—
223C2×Dic3C4×C3⋊D311yescharacteristic
224C2×Dic3C4×C3⋊D311yescharacteristic
225C4×D3—182no—
226C4×D3—182no—
227C4×D3—188no—
228C4×D3—188no—
229C4×D3—184no—
230C4×D3—184no—
231C4×D3—181no—
232C4×D3—181no—
233C4×D3—364no—
234C4×D3—364no—
235C3×C8—24no—
236C3×C8—21no—
237C3×C8—44no—
238C3×C8—64no—
239C3×C8—181no—
240C3⋊C8—21no—
241C3⋊C8—64no—
242C3⋊C8—184no—
243C3⋊C8—181no—
244C3⋊C8—364no—
245C2×C32—24no—
246C2×C32—24no—
247C2×C32—24no—
248C2×C32#87214yes—
249C2×C32#89211yescharacteristic
250C2×C32C2×ΓL(1,𝔽25)11yescharacteristic
251C2×C32D3×M1614yes—
252C2×C32D3×M1614yes—
253C2×C32D3×M1611yescharacteristic
254C3⋊D3—62no—
255C3⋊D3—68no—
256C3⋊D3—128no—
257C22×C4—271no—
258C22×C4—271no—
259C22×C4—271no—
260C2×C8—31no—
261C2×C8—91no—
262C2×C8—271no—
263C2×C8D3×C3⋊D311yescharacteristic
264C42—31no—
265C42—91no—
266C42—271no—
267C42—271no—
268C22×C3—24no—
269C22×C3C4×D3214yes—
270C22×C3C2×C4×C3⋊D311yescharacteristic
271C2×D3—98no—
272C2×D3—92no—
273C2×D3—188no—
274C2×D3—184no—
275C2×D3—181no—
276C2×D3—184no—
277C2×D3—181no—
278C2×D3—364no—
279C2×D3—364no—
280C3×C4—24no—
281C3×C4—24no—
282C3×C4—64no—
283C3×C4—64no—
284C3×C4—92no—
285C3×C4—181no—
286C3×C4#379914yes—
287C3×C4#380414yes—
288C3×C4#382811yescharacteristic
289C3×C4#382811yescharacteristic
290Dic3—21no—
291Dic3—21no—
292Dic3—38no—
293Dic3—64no—
294Dic3—181no—
295Dic3—184no—
296Dic3—364no—
297C32—24no—
298C32#470514yescomplemented
299C32#483511yescharacteristic, complemented
300C23—271no—
301C2×C4—31no—
302C2×C4—31no—
303C2×C4—91no—
304C2×C4—91no—
305C2×C4—271no—
306C2×C4—542no—
307C2×C4—542no—
308C2×C4—541no—
309C2×C4—541no—
310C2×C4C2×D3×C3⋊D311yescharacteristic
311C8—21no—
312C8—61no—
313C8—181no—
314C8—541no—
315C2×C3—24no—
316C2×C3—24no—
317C2×C3—24no—
318C2×C3D3×ΓL(1,𝔽25)14yes—
319C2×C3#6379114yes—
320C2×C3#6383614yes—
321C2×C3#6408111yescharacteristic
322C2×C3M16×C3⋊D311yescharacteristic
323C2×C3M16×C3⋊D311yescharacteristic
324D3—182no—
325D3—188no—
326D3—368no—
327C22—272no—
328C22—541no—
329C22—541no—
330C22C4×D3×C3⋊D311yescharacteristic
331C4—61no—
332C4—61no—
333C4—92no—
334C4—181no—
335C4—541no—
336C4#8902211yescharacteristic
337C4#8903111yescharacteristic
338C3—24no—
339C3#1059066514yescomplemented
340C3#1059240411yescharacteristic, complemented
341C2—542no—
342C2C3⋊D3×ΓL(1,𝔽25)11yescharacteristic
343C2#1172331211yescharacteristic
344C2#1172346811yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×D3×C3⋊D3
commutator, socleC22×C33C22×C4
FittingC2×C33×C8C22
prime radicalC22×C3⋊(C3⋊D3)C2×C4
cosocleC2×C33×C4C23
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#432
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
3. C3×#63641C2semidirect
4. C32×#815C2semidirect
5. #11722051C2semidirect
7. #11722019C2semidirect
29. #63641D3semidirect
31. C3×#815D3semidirect
99. #815C3⋊D3semidirect
207. C33#432semidirect
298. C32#4705semidirect
299. C32#4835semidirect
339. C3#10590665semidirect
340. C3#10592404semidirect

Maximal subgroups

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

subgroupindex
C4×#62892
#117221902
C3×#636412
C32×#8152
#117220512
#117220242
#117220192
#105924043
#105906653