Finite Group Databasebeta

← #13161143

Subgroups — #13161143

Order 1728.

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

normal subgroups only

This group has 872 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#11723987C211yesmaximal, characteristic
2#11723413C211yesmaximal, characteristic
3C3×#63662C211yesmaximal, characteristic, complemented
4C32×#835C211yesmaximal, characteristic, complemented
5#11722055C211yesmaximal, characteristic, complemented
6#11722054C211yesmaximal, characteristic, complemented
7#11722024C211yesmaximal, characteristic
8#10592513—31nomaximal
9#10591561—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
12#88859—21no—
13C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
14#88857—21no—
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88804—21no—
20Dic3×C3⋊Dic3—21no—
21#64155—31no—
22#64155—34no—
23#64155—64no—
24#64101—31no—
25#63849—34no—
26C32×#103—31no—
27#63662D311yescharacteristic, complemented
28#63647—31no—
29#63645—31no—
30#63635—31no—
31C3×#835—34no—
32C3×#835D314yescomplemented
33#63575—34no—
34#63573—34no—
35#63557—34no—
36C22×C3⋊(C3⋊D3)D411yescharacteristic
37C33×Q8—21no—
38C33×Q8—21no—
39C2×C33×C4—21no—
40C2×C33×C4C2311yescharacteristic
41C2×C3⋊(C3⋊Dic3)—21no—
42C2×C3⋊(C3⋊Dic3)D411yescharacteristic
43C4×C3⋊(C3⋊D3)—21no—
44C4×C3⋊(C3⋊D3)—21no—
45C2×C3×C3⋊Dic3—21no—
46C2×C32×Dic3—21no—
47C3×C3⋊(C3⋊C8)—21no—
48C32×C3⋊C8—21no—
49#4815—91no—
50#4814—94no—
51C2×C32×Q8—24no—
52C2×C32×Q8C2×D314yes—
53C2×C32×Q8C2×D311yescharacteristic
54C2×C4×C3⋊D3—31no—
55C2×C4×C3⋊D3—34no—
56C2×C4×C3⋊D3—64no—
57C2×C32×C8—31no—
58C32×C4⋊C4—31no—
59#3752—61no—
60#3752—61no—
61#3752—64no—
62#3752—68no—
63C4⋊(C3⋊Dic3)—31no—
64C4⋊(C3⋊Dic3)—34no—
65C4⋊(C3⋊Dic3)—64no—
66C4⋊(C3⋊Dic3)C2×D311yescharacteristic
67#3750—61no—
68#3750—64no—
69#3750—68no—
70C4×C3⋊Dic3—61no—
71C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
72C3×C4⋊Dic3—34no—
73C3×C4⋊Dic3C2×D314yes—
74C2×C3×C3⋊C8—34no—
75C2×C3×C3⋊C8C2×D314yes—
76#3722—64no—
77Dic32—64no—
78C22×C33C2×D411yescharacteristic
79C2×C3⋊(C3⋊D3)—21no—
80C2×C3⋊(C3⋊D3)—21no—
81C2×C3⋊(C3⋊D3)—21no—
82C33×C4—41no—
83C33×C4#4111yescharacteristic
84C33×C4#4111yescharacteristic
85C3⋊(C3⋊Dic3)—21no—
86C3⋊(C3⋊Dic3)—41no—
87C3×C3⋊Dic3—41no—
88C32×Dic3—41no—
89#946—94no—
90#946—91no—
91#946—184no—
92#892—94no—
93#892—91no—
94C3×#103—34no—
95C3×#103—91no—
96#835—34no—
97#835C3⋊D311yescharacteristic, complemented
98(C2×D3)⋊C8—94no—
99(C2×D3)⋊C8—91no—
100#818—94no—
101#817—91no—
102#808—94no—
103#807—91no—
104C22×C3⋊D3—31no—
105C22×C3⋊D3—34no—
106C22×C3⋊D3—64no—
107C32×Q8—21no—
108C32×Q8—21no—
109C32×Q8—24no—
110C32×Q8—24no—
111C32×Q8—44no—
112C32×Q8—44no—
113C2×C32×C4—28no—
114C2×C32×C4—24no—
115C2×C32×C4—21no—
116C2×C32×C4—24no—
117C2×C32×C4—61no—
118C2×C32×C4C22×D314yes—
119C2×C32×C4C22×D311yescharacteristic
120C2×C3⋊Dic3—21no—
121C2×C3⋊Dic3—31no—
122C2×C3⋊Dic3—34no—
123C2×C3⋊Dic3—64no—
124C2×C3⋊Dic3—68no—
125C2×C3⋊Dic3—64no—
126C2×C3⋊Dic3—61no—
127C4×C3⋊D3—61no—
128C4×C3⋊D3—61no—
129C4×C3⋊D3—64no—
130C4×C3⋊D3—64no—
131C4×C3⋊D3—124no—
132C4×C3⋊D3—124no—
133C2×C3×Dic3—24no—
134C2×C3×Dic3—64no—
135C32×C8—61no—
136C3⋊(C3⋊C8)—21no—
137C3×C3⋊C8—24no—
138C3×C3⋊C8—64no—
139#485—271noSylow
140C2×C33#12311yescharacteristic
141C2×C33#13511yescharacteristic
142C2×C33#13711yescharacteristic
143C3⋊(C3⋊D3)—41no—
144C2×C3×Q8—24no—
145C2×C3×Q8D3214yes—
146C2×C3×Q8C2×C3⋊D311yescharacteristic
147C2×C4×D3—94no—
148C2×C4×D3—91no—
149C2×C4×D3—184no—
150C2×C3×C8—34no—
151C2×C3×C8—91no—
152C3×C4⋊C4—34no—
153C3×C4⋊C4—91no—
154(C2×D3)⋊C4—181no—
155(C2×D3)⋊C4—184no—
156(C2×D3)⋊C4—184no—
157(C2×D3)⋊C4—181no—
158(C2×D3)⋊C4—188no—
159C4⋊Dic3—34no—
160C4⋊Dic3—94no—
161C4⋊Dic3—91no—
162C4⋊Dic3—184no—
163C4⋊Dic3C2×C3⋊D311yescharacteristic
164Dic3⋊C4—184no—
165Dic3⋊C4—181no—
166Dic3⋊C4—188no—
167C4×Dic3—181no—
168C4×Dic3—184no—
169C2×C3⋊C8—34no—
170C2×C3⋊C8C2×C3⋊D311yescharacteristic
171C22×C32—24no—
172C22×C32D3×D414yes—
173C22×C32D3×D411yescharacteristic
174C2×C3⋊D3—68no—
175C2×C3⋊D3—61no—
176C2×C3⋊D3—64no—
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—48no—
186C32×C4—44no—
187C32×C4—41no—
188C32×C4—121no—
189C32×C4#23514yes—
190C32×C4#23511yescharacteristic
191C32×C4#23714yes—
192C32×C4#23711yescharacteristic
193C3⋊Dic3—41no—
194C3⋊Dic3—61no—
195C3⋊Dic3—64no—
196C3⋊Dic3—124no—
197C3⋊Dic3—128no—
198C3⋊Dic3—121no—
199C3⋊Dic3—124no—
200C3×Dic3—44no—
201C3×Dic3—124no—
202C22⋊Q8—271no—
203#117—271no—
204#107—271no—
205#106—271no—
206#103—31no—
207#103—91no—
208C22⋊C8—271no—
209C33#48511yescharacteristic, complemented, Sylow
210C22×D3—94no—
211C22×D3—91no—
212C22×D3—184no—
213C3×Q8—24no—
214C3×Q8—24no—
215C3×Q8—21no—
216C3×Q8—21no—
217C3×Q8—44no—
218C3×Q8—44no—
219C2×C3×C4—28no—
220C2×C3×C4—24no—
221C2×C3×C4—21no—
222C2×C3×C4—24no—
223C2×C3×C4—64no—
224C2×C3×C4—181no—
225C2×C3×C4C2×D3214yes—
226C2×C3×C4C22×C3⋊D311yescharacteristic
227C2×Dic3—21no—
228C2×Dic3—64no—
229C2×Dic3—94no—
230C2×Dic3—91no—
231C2×Dic3—184no—
232C2×Dic3—188no—
233C2×Dic3—184no—
234C2×Dic3—181no—
235C4×D3—184no—
236C4×D3—184no—
237C4×D3—181no—
238C4×D3—181no—
239C4×D3—364no—
240C4×D3—364no—
241C3×C8—64no—
242C3×C8—181no—
243C3⋊C8—21no—
244C3⋊C8—64no—
245C2×C32—24no—
246C2×C32—24no—
247C2×C32—24no—
248C2×C32#88314yes—
249C2×C32#88311yescharacteristic
250C2×C32#91514yes—
251C2×C32#91611yescharacteristic
252C2×C32#91811yescharacteristic
253C2×C32#91914yes—
254C3⋊D3—128no—
255C3⋊D3—124no—
256C3⋊D3—121no—
257C2×Q8D3×C3⋊D311yescharacteristic
258C22×C4—271no—
259C2×C8—31no—
260C2×C8—91no—
261C4⋊C4—31no—
262C4⋊C4—91no—
263C4⋊C4—271no—
264C4⋊C4—541no—
265C22⋊C4—541no—
266C22⋊C4—541no—
267C42—541no—
268C22×C3—24no—
269C22×C3#381114yes—
270C22×C3D4×C3⋊D311yescharacteristic
271C2×D3—188no—
272C2×D3—181no—
273C2×D3—181no—
274C2×D3—181no—
275C2×D3—184no—
276C2×D3—184no—
277C2×D3—184no—
278C2×D3—364no—
279C2×D3—364no—
280C3×C4—24no—
281C3×C4—24no—
282C3×C4—48no—
283C3×C4—41no—
284C3×C4—44no—
285C3×C4—124no—
286C3×C4—361no—
287C3×C4#380414yes—
288C3×C4#380414yes—
289C3×C4#382811yescharacteristic
290C3×C4#383011yescharacteristic
291Dic3—41no—
292Dic3—124no—
293Dic3—181no—
294Dic3—184no—
295Dic3—364no—
296Dic3—368no—
297Dic3—361no—
298Dic3—364no—
299C32—24no—
300C32#481414yescomplemented
301C32#481511yescharacteristic, complemented
302C23—271no—
303Q8—21no—
304Q8—21no—
305C2×C4—21no—
306C2×C4—61no—
307C2×C4—181no—
308C2×C4—271no—
309C2×C4—541no—
310C2×C4—541no—
311C2×C4—541no—
312C2×C4C2×D3×C3⋊D311yescharacteristic
313C8—61no—
314C8—181no—
315C2×C3—24no—
316C2×C3—24no—
317C2×C3—24no—
318C2×C3#6395014yes—
319C2×C3#6395114yes—
320C2×C3#6397514yes—
321C2×C3#6409211yescharacteristic
322C2×C3#6412411yescharacteristic
323C2×C3#6412811yescharacteristic
324D3—368no—
325D3—361no—
326D3—364no—
327C22—541no—
328C22—541no—
329C22—541no—
330C22#8903911yescharacteristic
331C4—41no—
332C4—121no—
333C4—361no—
334C4—541no—
335C4—1081no—
336C4#8902811yescharacteristic
337C4#8903111yescharacteristic
338C3—24no—
339C3#1059156114yescomplemented
340C3#1059251311yescharacteristic, complemented
341C2—1081no—
342C2#1172358811yescharacteristic
343C2#1172358911yescharacteristic
344C2#1172362811yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC22×C3⋊(C3⋊D3)D4
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#485
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
3. C3×#63662C2semidirect
4. C32×#835C2semidirect
5. #11722055C2semidirect
6. #11722054C2semidirect
27. #63662D3semidirect
32. C3×#835D3semidirect
97. #835C3⋊D3semidirect
209. C33#485semidirect
300. C32#4814semidirect
301. C32#4815semidirect
339. C3#10591561semidirect
340. C3#10592513semidirect

Maximal subgroups

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

subgroupindex
#117239872
#117234132
C3×#636622
C32×#8352
#117220552
#117220542
#117220242
#105925133
#105915613