Finite Group Databasebeta

← #13158984

Subgroups — #13158984

Order 1728.

1000 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 1000 conjugacy classes of subgroups — too many to draw. Only its 186 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×#6308C211yesmaximal, characteristic, complemented
2C3×C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic
3C2×#88796C211yesmaximal, characteristic
4C32×#803C211yesmaximal, characteristic, complemented
5#11722055C211yesmaximal, characteristic, complemented
6#11722042C211yesmaximal, characteristic, complemented
7#11722034C211yesmaximal, characteristic
8#10592409—31nomaximal
9#10590360—34nomaximal
10C2×#6308C2211yescharacteristic
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×C4×C3⋊D3—21no—
13C33×C42C2211yescharacteristic
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15#88857—21no—
16C4×C3⋊(C3⋊Dic3)—21no—
17C3×#3752—21no—
18C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21#88796C414yescomplemented
22C2×#3744—31no—
23C4×C4⋊(C3⋊D3)D311yescharacteristic
24C4×#634—31no—
25C4×#634—34no—
26C4×#634—64no—
27C3×C4×D12—34no—
28C2×#3716—34no—
29C32×#97—31no—
30#63648—31no—
31#63645—31no—
32#63643—31no—
33C3×#803D314yescomplemented
34#63575—34no—
35#63567—34no—
36#63563—34no—
37C22×C3×C3⋊D3—21no—
38C2×C33×C4D411yescharacteristic
39C2×C33×C4D411yescharacteristic
40C2×C33×C4C2311yescharacteristic
41C2×C3⋊(C3⋊Dic3)—21no—
42C2×C3⋊(C3⋊Dic3)—21no—
43#6308—21no—
44#6308C2×C412yescomplemented
45C2×C3×C3⋊Dic3—21no—
46C3×C4⋊(C3⋊D3)—21no—
47C3×C4⋊(C3⋊D3)C2×C412yes—
48C3×C4×C3⋊D3—42no—
49C32×C3⋊C8—21no—
50C32×C3⋊C8C2×C412yescomplemented
51#5018—91no—
52#4710—94no—
53C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
54C2×C4×C3⋊D3—21no—
55C2×#634—31no—
56C2×#634—34no—
57C2×#634—64no—
58C2×C3×D12—34no—
59C2×C3×C4×D3—64no—
60C2×C32×C8—31no—
61C2×C32×C8—31no—
62C32×C42—24no—
63C32×C42C2×D314yes—
64C32×C42C2×D311yescharacteristic
65#3752—21no—
66C4⋊(C3⋊Dic3)—31no—
67C4⋊(C3⋊Dic3)—34no—
68C4⋊(C3⋊Dic3)—64no—
69C4⋊(C3⋊Dic3)C2×D311yescharacteristic
70#3750—68no—
71#3750—64no—
72#3750—61no—
73C4×C3⋊Dic3—68no—
74C4×C3⋊Dic3—64no—
75C4×C3⋊Dic3—61no—
76#3744—34no—
77C3×(C2×D3)⋊C4—64no—
78C3×C4⋊Dic3—34no—
79C2×C3×C3⋊C8C2×D314yes—
80C2×C3×C3⋊C8C2×D314yes—
81#3716—316no—
82C22×C33C2×D411yescharacteristic
83C2×C3×C3⋊D3—22no—
84C2×C3×C3⋊D3—41no—
85C2×C3×C3⋊D3—41no—
86C33×C4—21no—
87C33×C4—21no—
88C33×C4C22×C411yescharacteristic
89C33×C4#4111yescharacteristic
90C3⋊(C3⋊Dic3)—22no—
91C3⋊(C3⋊Dic3)—41no—
92C3×C3⋊Dic3—41no—
93C3×C4×D4—91no—
94C2×#214—91no—
95C2×#204—94no—
96C4×D12—34no—
97C4×Dic6—94no—
98C4×Dic6—91no—
99C4×Dic6—184no—
100C3×#97—34no—
101#832—91no—
102#821—94no—
103#818—94no—
104#816—94no—
105#810—91no—
106#807—91no—
107#803C3⋊D311yescharacteristic, complemented
108C22×C3⋊D3—21no—
109C22×C3×D3—64no—
110C2×C32×C4—24no—
111C2×C32×C4—24no—
112C2×C32×C4—24no—
113C2×C32×C4D1214yes—
114C2×C32×C4D1214yes—
115C2×C32×C4#6711yescharacteristic
116C2×C32×C4#6711yescharacteristic
117C2×C32×C4C22×D314yes—
118C2×C32×C4C22×D311yescharacteristic
119C2×C3⋊Dic3—21no—
120C2×C3⋊Dic3—68no—
121C2×C3⋊Dic3—64no—
122C2×C3⋊Dic3—61no—
123C2×C3⋊Dic3—68no—
124C2×C3⋊Dic3—64no—
125C2×C3⋊Dic3—61no—
126C4⋊(C3⋊D3)—21no—
127C4⋊(C3⋊D3)C4×D312yes—
128C4×C3⋊D3—42no—
129#634—32no—
130#634—38no—
131#634—68no—
132#634—68no—
133#634—64no—
134#634—61no—
135C2×C3×Dic3—64no—
136C3×D12—38no—
137C3×D12—64no—
138C3×C4×D3—128no—
139C32×C8—32no—
140C32×C8—61no—
141C3×C3⋊C8—24no—
142C3×C3⋊C8C4×D318yescomplemented
143#440—271noSylow
144C2×C33C4×D411yescharacteristic
145C2×C33AGL(1,ℤ/8)11yescharacteristic
146C2×C33#13711yescharacteristic
147C3×C3⋊D3—42no—
148C2×C3×D4—91no—
149C22×C3×C4—181no—
150C2×D12—34no—
151C2×C4×D3—64no—
152C2×Dic6—94no—
153C2×Dic6—91no—
154C2×Dic6—184no—
155C2×C3×C8—34no—
156C2×C3×C8—34no—
157C3×C4⋊C4—91no—
158C3×C22⋊C4—181no—
159C3×C42—24no—
160C3×C42D3214yes—
161C3×C42C2×C3⋊D311yescharacteristic
162#214—94no—
163(C2×D3)⋊C4—64no—
164C4⋊Dic3—34no—
165C4⋊Dic3—94no—
166C4⋊Dic3—91no—
167C4⋊Dic3—184no—
168Dic3⋊C4—188no—
169Dic3⋊C4—184no—
170Dic3⋊C4—181no—
171C4×Dic3—188no—
172C4×Dic3—184no—
173C4×Dic3—181no—
174C2×C3⋊C8C2×C3⋊D311yescharacteristic
175C2×C3⋊C8C2×C3⋊D311yescharacteristic
176#204—916no—
177C22×C32—24no—
178C22×C32C2×D1214yes—
179C22×C32C2×#6711yescharacteristic
180C2×C3⋊D3—22no—
181C2×C3⋊D3—41no—
182C2×C3⋊D3—41no—
183C2×C3×D3—68no—
184C2×C3×D3—124no—
185C2×C3×D3—124no—
186C32×C4—28no—
187C32×C4—24no—
188C32×C4—21no—
189C32×C4—21no—
190C32×C4—24no—
191C32×C4—24no—
192C32×C4—24no—
193C32×C4—44no—
194C32×C4C2×C4×D314yes—
195C32×C4C2×C4×D311yescharacteristic
196C32×C4#23514yes—
197C32×C4#23511yescharacteristic
198C3⋊Dic3—41no—
199C3⋊Dic3—616no—
200C3⋊Dic3—68no—
201C3⋊Dic3—62no—
202C3⋊Dic3—128no—
203C3⋊Dic3—124no—
204C3⋊Dic3—121no—
205C3×Dic3—124no—
206C2×SD16—271no—
207C4×Q8—271no—
208C4×D4—91no—
209#107—271no—
210#103—271no—
211#102—271no—
212#97—31no—
213C33#44011yescharacteristic, complemented, Sylow
214C23×C3—181no—
215C22×D3—64no—
216C3×D4—92no—
217C3×D4—181no—
218C2×C3×C4—24no—
219C2×C3×C4—24no—
220C2×C3×C4—24no—
221C2×C3×C4—181no—
222C2×C3×C4—362no—
223C2×C3×C4#62614yes—
224C2×C3×C4#62614yes—
225C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
226C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
227C2×C3×C4C2×D3214yes—
228C2×C3×C4C22×C3⋊D311yescharacteristic
229C2×Dic3—64no—
230C2×Dic3—188no—
231C2×Dic3—184no—
232C2×Dic3—181no—
233C2×Dic3—188no—
234C2×Dic3—184no—
235C2×Dic3—181no—
236D12—38no—
237D12—64no—
238C4×D3—128no—
239Dic6—98no—
240Dic6—92no—
241Dic6—188no—
242Dic6—188no—
243Dic6—184no—
244Dic6—181no—
245C3×C8—38no—
246C3×C8—64no—
247C3⋊C8—21no—
248C3⋊C8C4×C3⋊D312yescomplemented
249C2×C32—24no—
250C2×C32—24no—
251C2×C32—24no—
252C2×C32C4×D1214yes—
253C2×C32#90814yes—
254C2×C32#90914yes—
255C2×C32C4×#6711yescharacteristic
256C2×C32#93211yescharacteristic
257C2×C32#95111yescharacteristic
258C3⋊D3—42no—
259C3×D3—128no—
260C2×Q8—271no—
261C2×D4—91no—
262C22×C4—181no—
263SD16—274no—
264C2×C8—31no—
265C2×C8—31no—
266C4⋊C4—91no—
267C4⋊C4—271no—
268C4⋊C4—541no—
269C22⋊C4—181no—
270C42—541no—
271C42D3×C3⋊D311yescharacteristic
272C22×C3—24no—
273C22×C3—182no—
274C22×C3—361no—
275C22×C3—361no—
276C22×C3C2×#62614yes—
277C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
278C2×D3—68no—
279C2×D3—124no—
280C2×D3—124no—
281C3×C4—28no—
282C3×C4—24no—
283C3×C4—24no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—24no—
287C3×C4—24no—
288C3×C4—44no—
289C3×C4—361no—
290C3×C4#379814yes—
291C3×C4C4×D3214yes—
292C3×C4C2×C4×C3⋊D311yescharacteristic
293C3×C4#382811yescharacteristic
294Dic3—124no—
295Dic3—1816no—
296Dic3—182no—
297Dic3—188no—
298Dic3—368no—
299Dic3—364no—
300Dic3—361no—
301C32—24no—
302C32#471014yescomplemented
303C32#501811yescharacteristic, complemented
304C23—181no—
305Q8—272no—
306Q8—541no—
307D4—92no—
308D4—181no—
309C2×C4—181no—
310C2×C4—362no—
311C2×C4—541no—
312C2×C4—541no—
313C2×C4#629211yescharacteristic
314C2×C4#629211yescharacteristic
315C2×C4C2×D3×C3⋊D311yescharacteristic
316C8—32no—
317C8—61no—
318C2×C3—24no—
319C2×C3—24no—
320C2×C3—24no—
321C2×C3—362no—
322C2×C3#6383014yes—
323C2×C3#6383114yes—
324C2×C3C4×#62614yes—
325C2×C3C4×C4⋊(C3⋊D3)11yescharacteristic
326C2×C3#6411711yescharacteristic
327C2×C3#6411811yescharacteristic
328D3—128no—
329C22—182no—
330C22—361no—
331C22—361no—
332C22C2×#629211yescharacteristic
333C4—21no—
334C4—21no—
335C4—361no—
336C4—542no—
337C4—1081no—
338C4#8902011yescharacteristic
339C4C4×D3×C3⋊D311yescharacteristic
340C3—24no—
341C3#1059036014yescomplemented
342C3#1059240911yescharacteristic, complemented
343C2—362no—
344C2#1172337011yescharacteristic
345C2#1172338811yescharacteristic
346C2C4×#629211yescharacteristic
subgroupG/H
centerC22C2×#6292
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C42C22
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
cosocleC2×C33×C4C23
hypercenterC42D3×C3⋊D3
hypocenterC33#440
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C4×#6308C2semidirect
4. C32×#803C2semidirect
5. #11722055C2semidirect
6. #11722042C2semidirect
21. #88796C4semidirect
33. C3×#803D3semidirect
44. #6308C2×C4semidirect
50. C32×C3⋊C8C2×C4semidirect
107. #803C3⋊D3semidirect
142. C3×C3⋊C8C4×D3semidirect
213. C33#440semidirect
248. C3⋊C8C4×C3⋊D3semidirect
302. C32#4710semidirect
303. C32#5018semidirect
341. C3#10590360semidirect
342. C3#10592409semidirect

Maximal subgroups

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

subgroupindex
C4×#63082
C3×C4×C4⋊(C3⋊D3)2
C2×#887962
C32×#8032
#117220552
#117220422
#117220342
#105924093
#105903603