Finite Group Databasebeta

← #13159382

Subgroups — #13159382

Order 1728.

1012 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 1012 conjugacy classes of subgroups — too many to draw. Only its 166 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#11723501C211yesmaximal, characteristic
3C2×#88796C211yesmaximal, characteristic
4C3×#63636C211yesmaximal, characteristic
5C32×#808C211yesmaximal, characteristic, complemented
6#11722041C211yesmaximal, characteristic, complemented
7#11722018C211yesmaximal, characteristic, complemented
8#10592551—31nomaximal
9#10590460—34nomaximal
10C2×#6308C2211yescharacteristic
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×Dic3×C3⋊D3—21no—
13C33×C4⋊C4C2211yescharacteristic
14#88857—21no—
15C4×C3⋊(C3⋊Dic3)—21no—
16C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20#88803—21no—
21#88796C414yescomplemented
22C2×#3744—31no—
23#64102—31no—
24#64096—31no—
25#64096—34no—
26#64096—64no—
27#63898—34no—
28C2×#3716—34no—
29C32×#106—31no—
30C8×C3⋊Dic3—31no—
31#63637—31no—
32#63636D311yescharacteristic
33C3×#809—34no—
34C3×#808D314yescomplemented
35#63567—34no—
36#63553—34no—
37C22×C3×C3⋊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
45C3×C4⋊(C3⋊D3)—21no—
46C3×C4⋊(C3⋊D3)C2×C412yes—
47C2×C32×Dic3—21no—
48Dic3×C3⋊D3—42no—
49C3×C3⋊(C3⋊C8)—21no—
50C32×C3⋊C8C2×C412yescomplemented
51#4852—94no—
52#4758—91no—
53C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
54C2×C4×C3⋊D3—61no—
55C2×#634—31no—
56C2×#634—34no—
57C2×#634—64no—
58C2×C3×D12—34no—
59C2×D3×Dic3—64no—
60C2×C32×C8—31no—
61C32×C4⋊C4—24no—
62C32×C4⋊C4—31no—
63C32×C4⋊C4C2×D314yes—
64C32×C4⋊C4C2×D311yescharacteristic
65#3752—61no—
66#3750—61no—
67#3750—64no—
68#3750—68no—
69C4×C3⋊Dic3—31no—
70C4×C3⋊Dic3—34no—
71C4×C3⋊Dic3—61no—
72C4×C3⋊Dic3—64no—
73C4×C3⋊Dic3—64no—
74C4×C3⋊Dic3—68no—
75C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
76#3744—34no—
77C3×C4⋊Dic3C2×D314yes—
78C2×C3×C3⋊C8—34no—
79C2×C3×C3⋊C8C2×D314yes—
80(C2×D3)⋊Dic3—64no—
81#3716—316no—
82C22×C33C2×D411yescharacteristic
83C2×C3×C3⋊D3—22no—
84C2×C3×C3⋊D3—41no—
85C2×C3×C3⋊D3—41no—
86C33×C4—41no—
87C33×C4C22×C411yescharacteristic
88C33×C4#4111yescharacteristic
89C3⋊(C3⋊Dic3)—22no—
90C3⋊(C3⋊Dic3)—21no—
91C3⋊(C3⋊Dic3)SD1612yescomplemented
92C32×Dic3—41no—
93D4×Dic3—91no—
94C2×#214—91no—
95C2×#204—94no—
96#893—94no—
97#887—94no—
98#887—91no—
99#887—184no—
100C3×#106—34no—
101C3×#102—91no—
102#816—91no—
103C8×Dic3—94no—
104C8×Dic3—91no—
105#810—94no—
106#809—34no—
107#808C3⋊D311yescharacteristic, complemented
108C22×C3⋊D3—21no—
109C22×C3×D3—64no—
110C2×C32×C4—28no—
111C2×C32×C4—24no—
112C2×C32×C4—21no—
113C2×C32×C4—24no—
114C2×C32×C4—61no—
115C2×C32×C4C22×D314yes—
116C2×C32×C4C22×D311yescharacteristic
117C2×C3⋊Dic3—31no—
118C2×C3⋊Dic3—31no—
119C2×C3⋊Dic3—34no—
120C2×C3⋊Dic3—34no—
121C2×C3⋊Dic3—64no—
122C2×C3⋊Dic3—64no—
123C2×C3⋊Dic3—68no—
124C2×C3⋊Dic3—64no—
125C2×C3⋊Dic3—61no—
126C4⋊(C3⋊D3)—21no—
127C4⋊(C3⋊D3)C4×D312yes—
128C4×C3⋊D3—122no—
129#634—32no—
130#634—38no—
131#634—68no—
132#634—68no—
133#634—61no—
134#634—64no—
135C2×C3×Dic3—24no—
136C3×D12—38no—
137C3×D12—64no—
138D3×Dic3—128no—
139C32×C8—32no—
140C3⋊(C3⋊C8)—21no—
141C3×C3⋊C8—64no—
142C3×C3⋊C8C4×D318yescomplemented
143C4×SD16—271noSylow
144C2×C33C4×D411yescharacteristic
145C2×C33C2×SD1611yescharacteristic
146C2×C33#13511yescharacteristic
147C3×C3⋊D3—42no—
148C2×C3×D4—91no—
149C22×Dic3—181no—
150C2×D12—34no—
151C2×C4×D3—184no—
152C2×Dic6—94no—
153C2×Dic6—91no—
154C2×Dic6—184no—
155C2×C3×C8—34no—
156C2×C3×C8—91no—
157C3×C4⋊C4—24no—
158C3×C4⋊C4—34no—
159C3×C4⋊C4D3214yes—
160C3×C4⋊C4C2×C3⋊D311yescharacteristic
161#217—181no—
162#214—94no—
163(C2×D3)⋊C4—184no—
164C4⋊Dic3C2×C3⋊D311yescharacteristic
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—916no—
177C22×C32—24no—
178C22×C32D3×D414yes—
179C22×C32D3×D411yescharacteristic
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—24no—
187C32×C4—24no—
188C32×C4—48no—
189C32×C4—41no—
190C32×C4—44no—
191C32×C4—121no—
192C32×C4C2×C4×D314yes—
193C32×C4C2×C4×D311yescharacteristic
194C32×C4#23711yescharacteristic
195C32×C4#23914yes—
196C3⋊Dic3—32no—
197C3⋊Dic3—38no—
198C3⋊Dic3—68no—
199C3⋊Dic3—616no—
200C3⋊Dic3—62no—
201C3⋊Dic3—68no—
202C3⋊Dic3—61no—
203C3⋊Dic3—64no—
204C3⋊Dic3—124no—
205C3×Dic3—44no—
206C2×SD16—271no—
207C4×Q8—271no—
208C4×D4—271no—
209#106—31no—
210#103—271no—
211#102—91no—
212C4×C8—271no—
213C33C4×SD1611yescharacteristic, complemented, Sylow
214C23×C3—181no—
215C22×D3—64no—
216C3×D4—92no—
217C3×D4—181no—
218C2×C3×C4—28no—
219C2×C3×C4—24no—
220C2×C3×C4—21no—
221C2×C3×C4—24no—
222C2×C3×C4—64no—
223C2×C3×C4C2×D3214yes—
224C2×C3×C4C22×C3⋊D311yescharacteristic
225C2×Dic3—21no—
226C2×Dic3—94no—
227C2×Dic3—91no—
228C2×Dic3—94no—
229C2×Dic3—91no—
230C2×Dic3—184no—
231C2×Dic3—184no—
232C2×Dic3—188no—
233C2×Dic3—184no—
234C2×Dic3—181no—
235C2×Dic3—362no—
236D12—38no—
237D12—64no—
238C4×D3—368no—
239Dic6—98no—
240Dic6—92no—
241Dic6—188no—
242Dic6—188no—
243Dic6—184no—
244Dic6—181no—
245C3×C8—38no—
246C3×C8—181no—
247C3⋊C8—64no—
248C3⋊C8C4×C3⋊D312yescomplemented
249C2×C32—24no—
250C2×C32—24no—
251C2×C32—24no—
252C2×C32#67⋊C411yescharacteristic
253C2×C32#89314yes—
254C2×C32#91211yescharacteristic
255C2×C32D3×SD1614yes—
256C2×C32D3×SD1611yescharacteristic
257C2×C32#91614yes—
258C3⋊D3—42no—
259C3×D3—128no—
260C2×Q8—271no—
261C2×D4—91no—
262C22×C4—541no—
263SD16—274no—
264C2×C8—31no—
265C2×C8—91no—
266C4⋊C4—31no—
267C4⋊C4—541no—
268C4⋊C4D3×C3⋊D311yescharacteristic
269C22⋊C4—541no—
270C42—271no—
271C42—541no—
272C22×C3—24no—
273C22×C3—182no—
274C22×C3—361no—
275C22×C3—361no—
276C22×C3#381114yes—
277C22×C3D4×C3⋊D311yescharacteristic
278C2×D3—68no—
279C2×D3—124no—
280C2×D3—124no—
281C3×C4—24no—
282C3×C4—24no—
283C3×C4—48no—
284C3×C4—41no—
285C3×C4—44no—
286C3×C4—124no—
287C3×C4#379614yes—
288C3×C4C4×D3214yes—
289C3×C4C2×C4×C3⋊D311yescharacteristic
290C3×C4#383211yescharacteristic
291Dic3—41no—
292Dic3—92no—
293Dic3—98no—
294Dic3—188no—
295Dic3—1816no—
296Dic3—182no—
297Dic3—181no—
298Dic3—188no—
299Dic3—184no—
300Dic3—364no—
301C32—24no—
302C32#475811yescharacteristic, complemented
303C32#485214yescomplemented
304C23—181no—
305Q8—272no—
306Q8—541no—
307D4—92no—
308D4—181no—
309C2×C4—21no—
310C2×C4—61no—
311C2×C4—271no—
312C2×C4—271no—
313C2×C4—541no—
314C2×C4—1082no—
315C2×C4C2×D3×C3⋊D311yescharacteristic
316C8—32no—
317C8—181no—
318C2×C3—24no—
319C2×C3—24no—
320C2×C3—24no—
321C2×C3—362no—
322C2×C3#6388114yes—
323C2×C3#6393614yes—
324C2×C3#6394014yes—
325C2×C3#6410211yescharacteristic
326C2×C3SD16×C3⋊D311yescharacteristic
327C2×C3#6412511yescharacteristic
328D3—128no—
329C22—182no—
330C22—361no—
331C22—361no—
332C22#8903911yescharacteristic
333C4—41no—
334C4—121no—
335C4—272no—
336C4—541no—
337C4—542no—
338C4#8901711yescharacteristic
339C4C4×D3×C3⋊D311yescharacteristic
340C3—24no—
341C3#1059046014yescomplemented
342C3#1059255111yescharacteristic, complemented
343C2—362no—
344C2#1172347211yescharacteristic
345C2#1172356511yescharacteristic
346C2#1172357111yescharacteristic
subgroupG/H
centerC22#89039
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
cosocleC2×C33×C4C23
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33C4×SD16
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
5. C32×#808C2semidirect
6. #11722041C2semidirect
7. #11722018C2semidirect
21. #88796C4semidirect
34. C3×#808D3semidirect
44. #6308C2×C4semidirect
50. C32×C3⋊C8C2×C4semidirect
91. C3⋊(C3⋊Dic3)SD16semidirect
107. #808C3⋊D3semidirect
142. C3×C3⋊C8C4×D3semidirect
213. C33C4×SD16semidirect
248. C3⋊C8C4×C3⋊D3semidirect
302. C32#4758semidirect
303. C32#4852semidirect
341. C3#10590460semidirect
342. C3#10592551semidirect

Maximal subgroups

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

subgroupindex
#117239282
#117235012
C2×#887962
C3×#636362
C32×#8082
#117220412
#117220182
#105925513
#105904603