Finite Group Databasebeta

← #13159389

Subgroups — #13159389

Order 1728.

1000 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 1000 conjugacy classes of subgroups — too many to draw. Only its 164 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#11723928C211yesmaximal, characteristic, complemented
2#11723499C211yesmaximal, characteristic
3C2×#88794C211yesmaximal, characteristic
4C3×#63634C211yesmaximal, characteristic, complemented
5C32×#809C211yesmaximal, characteristic
6#11722042C211yesmaximal, characteristic, complemented
7#11722021C211yesmaximal, characteristic, complemented
8#10592456—31nomaximal
9#10590461—34nomaximal
10C2×#6308C2211yescharacteristic
11C2×C32×D12C2211yescharacteristic
12C2×D3×C3⋊Dic3—21no—
13C33×C4⋊C4C2211yescharacteristic
14#88857—21no—
15C4×C3⋊(C3⋊Dic3)—21no—
16C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20(C2×D3)⋊(C3⋊Dic3)—21no—
21#88794C414yescomplemented
22D4×C3⋊Dic3—31no—
23C2×#3754—31no—
24#64096—31no—
25#64096—34no—
26#64096—64no—
27#63898—34no—
28C2×#3716—34no—
29C32×#102—31no—
30#63643—31no—
31#63642—31no—
32#63634D311yescharacteristic, complemented
33C3×#809D314yes—
34C3×#807—34no—
35#63567—34no—
36#63555—34no—
37C22×C32×D3—21no—
38C2×C33×C4—21no—
39C2×C33×C4C2311yescharacteristic
40C2×C3⋊(C3⋊Dic3)—21no—
41C2×C3⋊(C3⋊Dic3)D411yescharacteristic
42C2×C3⋊(C3⋊Dic3)D411yescharacteristic
43#6308—21no—
44#6308C2×C412yescomplemented
45C2×C3×C3⋊Dic3—21no—
46C32×D12—21no—
47C32×D12C2×C412yes—
48D3×C3⋊Dic3—42no—
49C3×C3⋊(C3⋊C8)C2×C412yescomplemented
50C32×C3⋊C8—21no—
51#4889—91no—
52#4757—94no—
53C2×C32×D4—31no—
54C22×C3⋊Dic3—61no—
55C2×#634—31no—
56C2×#634—34no—
57C2×#634—64no—
58C2×C3×D12C2×D314yes—
59C2×D3×Dic3—64no—
60C2×C32×C8—31no—
61C32×C4⋊C4—24no—
62C32×C4⋊C4C2×D314yes—
63C32×C4⋊C4C2×D311yescharacteristic
64#3757—61no—
65#3754—34no—
66C4⋊(C3⋊Dic3)C2×D311yescharacteristic
67#3750—61no—
68#3750—64no—
69#3750—68no—
70C4×C3⋊Dic3—31no—
71C4×C3⋊Dic3—34no—
72C4×C3⋊Dic3—61no—
73C4×C3⋊Dic3—64no—
74C4×C3⋊Dic3—64no—
75C4×C3⋊Dic3—68no—
76C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
77C3×C4⋊Dic3—34no—
78C2×C3×C3⋊C8—34no—
79C2×C3×C3⋊C8C2×D314yes—
80(C2×D3)⋊Dic3—64no—
81#3716—316no—
82C22×C33C2×D411yescharacteristic
83C2×C32×D3—22no—
84C2×C32×D3—41no—
85C2×C32×D3—41no—
86C33×C4—41no—
87C33×C4C22×C411yescharacteristic
88C33×C4#4111yescharacteristic
89C3⋊(C3⋊Dic3)—22no—
90C3⋊(C3⋊Dic3)—21no—
91C3⋊(C3⋊Dic3)—21no—
92C3×C3⋊Dic3—41no—
93D4×Dic3—94no—
94C2×#214—94no—
95C2×#204—91no—
96#893—91no—
97#887—94no—
98#887—91no—
99#887—184no—
100C3×#107—91no—
101C3×#102—34no—
102#816—94no—
103#815—94no—
104#815—91no—
105#810—91no—
106#809C3⋊D311yescharacteristic
107#807—34no—
108C23×C32—61no—
109C22×C3×D3—24no—
110C32×D4—32no—
111C32×D4—61no—
112C2×C32×C4—28no—
113C2×C32×C4—24no—
114C2×C32×C4—21no—
115C2×C32×C4—24no—
116C2×C32×C4C22×D314yes—
117C2×C32×C4C22×D311yescharacteristic
118C2×C3⋊Dic3—21no—
119C2×C3⋊Dic3—31no—
120C2×C3⋊Dic3—31no—
121C2×C3⋊Dic3—34no—
122C2×C3⋊Dic3—34no—
123C2×C3⋊Dic3—64no—
124C2×C3⋊Dic3—64no—
125C2×C3⋊Dic3—68no—
126C2×C3⋊Dic3—64no—
127C2×C3⋊Dic3—61no—
128C2×C3⋊Dic3—122no—
129#634—32no—
130#634—38no—
131#634—68no—
132#634—68no—
133#634—64no—
134#634—61no—
135C2×C3×Dic3—64no—
136C3×D12—24no—
137C3×D12C4×D318yes—
138D3×Dic3—128no—
139C32×C8—61no—
140C3⋊(C3⋊C8)C4×D312yescomplemented
141C3×C3⋊C8—24no—
142C3×C3⋊C8—38no—
143#440—271noSylow
144C2×C33C4×D411yescharacteristic
145C2×C33AGL(1,ℤ/8)11yescharacteristic
146C2×C33#13711yescharacteristic
147C32×D3—42no—
148C2×C3×D4—34no—
149C22×Dic3—184no—
150C2×D12C2×C3⋊D311yescharacteristic
151C2×C4×D3—181no—
152C2×Dic6—94no—
153C2×Dic6—91no—
154C2×Dic6—184no—
155C2×C3×C8—34no—
156C2×C3×C8—91no—
157C3×C4⋊C4—24no—
158C3×C4⋊C4—91no—
159C3×C4⋊C4D3214yes—
160C3×C4⋊C4C2×C3⋊D311yescharacteristic
161#217—184no—
162#214—916no—
163(C2×D3)⋊C4—181no—
164C4⋊Dic3—34no—
165Dic3⋊C4—184no—
166Dic3⋊C4—181no—
167Dic3⋊C4—188no—
168C4×Dic3—94no—
169C4×Dic3—91no—
170C4×Dic3—184no—
171C4×Dic3—184no—
172C4×Dic3—181no—
173C4×Dic3—188no—
174C2×C3⋊C8—34no—
175C2×C3⋊C8C2×C3⋊D311yescharacteristic
176#204—94no—
177C22×C32—24no—
178C22×C32—62no—
179C22×C32—121no—
180C22×C32—121no—
181C22×C32D3×D414yes—
182C22×C32D3×D411yescharacteristic
183C2×C3×D3—28no—
184C2×C3×D3—44no—
185C2×C3×D3—44no—
186C32×C4—24no—
187C32×C4—24no—
188C32×C4—48no—
189C32×C4—44no—
190C32×C4—41no—
191C32×C4C2×C4×D314yes—
192C32×C4C2×C4×D311yescharacteristic
193C32×C4#23714yes—
194C32×C4#23911yescharacteristic
195C3⋊Dic3—41no—
196C3⋊Dic3—616no—
197C3⋊Dic3—68no—
198C3⋊Dic3—68no—
199C3⋊Dic3—62no—
200C3⋊Dic3—61no—
201C3⋊Dic3—64no—
202C3⋊Dic3—61no—
203C3⋊Dic3—64no—
204C3⋊Dic3—124no—
205C3×Dic3—124no—
206C2×SD16—271no—
207C4×Q8—271no—
208C4×D4—271no—
209#107—91no—
210#103—271no—
211#102—31no—
212#97—271no—
213C33#44011yescharacteristic, complemented, Sylow
214C23×C3—64no—
215C22×D3—21no—
216C3×D4—38no—
217C3×D4—64no—
218C2×C3×C4—28no—
219C2×C3×C4—24no—
220C2×C3×C4—21no—
221C2×C3×C4—24no—
222C2×C3×C4—181no—
223C2×C3×C4C2×D3214yes—
224C2×C3×C4C22×C3⋊D311yescharacteristic
225C2×Dic3—64no—
226C2×Dic3—94no—
227C2×Dic3—94no—
228C2×Dic3—91no—
229C2×Dic3—91no—
230C2×Dic3—184no—
231C2×Dic3—184no—
232C2×Dic3—188no—
233C2×Dic3—184no—
234C2×Dic3—181no—
235C2×Dic3—368no—
236D12—21no—
237D12C4×C3⋊D312yes—
238C4×D3—362no—
239Dic6—98no—
240Dic6—92no—
241Dic6—188no—
242Dic6—188no—
243Dic6—184no—
244Dic6—181no—
245C3×C8—64no—
246C3×C8—92no—
247C3⋊C8—21no—
248C3⋊C8—38no—
249C2×C32—24no—
250C2×C32—24no—
251C2×C32—24no—
252C2×C32—122no—
253C2×C32#67⋊C414yes—
254C2×C32#89311yescharacteristic
255C2×C32#91114yes—
256C2×C32#91111yescharacteristic
257C2×C32#91514yes—
258C2×C32#91811yescharacteristic
259C3×D3—48no—
260C2×Q8—271no—
261C2×D4—31no—
262C22×C4—541no—
263SD16—274no—
264C2×C8—31no—
265C2×C8—91no—
266C4⋊C4—91no—
267C4⋊C4—541no—
268C4⋊C4D3×C3⋊D311yescharacteristic
269C22⋊C4—541no—
270C42—271no—
271C42—541no—
272C22×C3—24no—
273C22×C3—68no—
274C22×C3—124no—
275C22×C3—124no—
276C22×C3#381114yes—
277C22×C3D4×C3⋊D311yescharacteristic
278C2×D3—22no—
279C2×D3—41no—
280C2×D3—41no—
281C3×C4—24no—
282C3×C4—24no—
283C3×C4—48no—
284C3×C4—41no—
285C3×C4—44no—
286C3×C4—361no—
287C3×C4#379614yes—
288C3×C4C4×D3214yes—
289C3×C4C2×C4×C3⋊D311yescharacteristic
290C3×C4#383011yescharacteristic
291Dic3—124no—
292Dic3—1816no—
293Dic3—188no—
294Dic3—182no—
295Dic3—181no—
296Dic3—181no—
297Dic3—188no—
298Dic3—184no—
299Dic3—184no—
300Dic3—364no—
301C32—24no—
302C32#475714yescomplemented
303C32#488911yescharacteristic, complemented
304C23—61no—
305Q8—272no—
306Q8—541no—
307D4—32no—
308D4—61no—
309C2×C4—21no—
310C2×C4—181no—
311C2×C4—271no—
312C2×C4—271no—
313C2×C4—541no—
314C2×C4—1082no—
315C2×C4C2×D3×C3⋊D311yescharacteristic
316C8—61no—
317C8—92no—
318C2×C3—24no—
319C2×C3—24no—
320C2×C3—24no—
321C2×C3—128no—
322C2×C3#6388114yes—
323C2×C3#6392714yes—
324C2×C3#6395114yes—
325C2×C3#6409011yescharacteristic
326C2×C3#6412011yescharacteristic
327C2×C3#6412411yescharacteristic
328D3—42no—
329C22—62no—
330C22—121no—
331C22—121no—
332C22#8903911yescharacteristic
333C4—41no—
334C4—361no—
335C4—541no—
336C4—541no—
337C4—542no—
338C4#8901511yescharacteristic
339C4C4×D3×C3⋊D311yescharacteristic
340C3—24no—
341C3#1059046114yescomplemented
342C3#1059245611yescharacteristic, complemented
343C2—122no—
344C2#1172346211yescharacteristic
345C2#1172354811yescharacteristic
346C2#1172358811yescharacteristic
subgroupG/H
centerC22#89039
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
cosocleC2×C33×C4C23
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#440
LCSG ▷ 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
1. #11723928C2semidirect
4. C3×#63634C2semidirect
6. #11722042C2semidirect
7. #11722021C2semidirect
21. #88794C4semidirect
32. #63634D3semidirect
44. #6308C2×C4semidirect
49. C3×C3⋊(C3⋊C8)C2×C4semidirect
140. C3⋊(C3⋊C8)C4×D3semidirect
213. C33#440semidirect
302. C32#4757semidirect
303. C32#4889semidirect
341. C3#10590461semidirect
342. C3#10592456semidirect

Maximal subgroups

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

subgroupindex
#117239282
#117234992
C2×#887942
C3×#636342
C32×#8092
#117220422
#117220212
#105924563
#105904613