Finite Group Databasebeta

← #13159637

Subgroups — #13159637

Order 1728.

1016 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 1016 conjugacy classes of subgroups — too many to draw. Only its 176 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
1C2×#88850C211yesmaximal, characteristic, complemented
2C4×#6291C211yesmaximal, characteristic
3C2×D3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
4C3×#63641C211yesmaximal, characteristic, complemented
5C32×(C2×D3)⋊C8C211yesmaximal, characteristic, complemented
6#11722024C211yesmaximal, characteristic, complemented
7#11722020C211yesmaximal, characteristic, complemented
8#10592429—31nomaximal
9#10590907—34nomaximal
10C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
11C2×C32×C4×D3C2211yescharacteristic
12C2×#6291C411yescharacteristic
13C2×C33×C8C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15#88850—22no—
16C3×C4×C3⋊Dic3C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88807C411yescharacteristic
20#88804C411yescharacteristic
21(C2×D3)⋊(C3⋊Dic3)C411yescharacteristic
22D3×C3⋊(C3⋊C8)—22no—
23C4×#638—31no—
24C22×C3⋊(C3⋊C8)—31no—
25C2×#3743—31no—
26C2×#3743—34no—
27C2×#3743—64no—
28C4×#626—34no—
29C2×D3×C3⋊C8—34no—
30C32×C22⋊C8—31no—
31#63647—31no—
32#63642—31no—
33#63641D311yescharacteristic, complemented
34C3×(C2×D3)⋊C8D314yescomplemented
35C3×Dic3⋊C8—34no—
36#63557—34no—
37#63554—34no—
38C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
39C22×C32×D3C2×C411yescharacteristic
40C2×C33×C4C2311yescharacteristic
41C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
42C4×C3⋊(C3⋊D3)—21no—
43C4×C3⋊(C3⋊D3)—21no—
44C2×C3×C3⋊Dic3C2×C411yescharacteristic
45C2×C3×C3⋊Dic3C2×C411yescharacteristic
46C2×C32×Dic3C2×C411yescharacteristic
47C32×C4×D3—21no—
48C32×C4×D3—21no—
49#6291—22no—
50C33×C8—21no—
51C3⋊(C3⋊(C3⋊C8))D412yescomplemented
52C3×C3⋊(C3⋊C8)—21no—
53C32×C3⋊C8—21no—
54#4836—91no—
55#4730—94no—
56C22×C32×C4—31no—
57C2×#638—31no—
58C2×C4×C3⋊D3—31no—
59C2×C4×C3⋊D3—34no—
60C2×C4×C3⋊D3—64no—
61C2×C3×C4×D3C2×D314yes—
62C2×#626—34no—
63C2×C32×C8—24no—
64C2×C32×C8—31no—
65C2×C32×C8C2×D314yes—
66C2×C32×C8C2×D311yescharacteristic
67#3757—31no—
68#3752—31no—
69#3750—31no—
70C4×C3⋊Dic3C2×D311yescharacteristic
71C2×C3⋊(C3⋊C8)—31no—
72C2×C3⋊(C3⋊C8)—34no—
73C2×C3⋊(C3⋊C8)—62no—
74C2×C3⋊(C3⋊C8)—64no—
75C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
76#3743—62no—
77#3743—68no—
78#3743—616no—
79C3×C4×Dic3—34no—
80C2×C3×C3⋊C8—34no—
81C2×C3×C3⋊C8C2×D314yes—
82Dic3⋊Dic3—34no—
83#3722—34no—
84(C2×D3)⋊Dic3—34no—
85D3×C3⋊C8—68no—
86C22×C33C22×C411yescharacteristic
87C2×C3⋊(C3⋊D3)—21no—
88C2×C3⋊(C3⋊D3)—21no—
89C2×C3⋊(C3⋊D3)—21no—
90C2×C32×D3—21no—
91C2×C32×D3—21no—
92C2×C32×D3M1612yescomplemented
93C33×C4C2×D411yescharacteristic
94C33×C4#4111yescharacteristic
95C3⋊(C3⋊Dic3)—21no—
96C3×C3⋊Dic3—21no—
97C3×C3⋊Dic3—21no—
98C32×Dic3—21no—
99C4×#67—94no—
100C22×C3⋊C8—94no—
101C2×ΓL(1,𝔽25)—94no—
102C2×ΓL(1,𝔽25)—91no—
103C2×ΓL(1,𝔽25)—184no—
104C2×D3×C8—91no—
105C4×D12—91no—
106C3×C4⋊C8—91no—
107C3×C22⋊C8—34no—
108(C2×D3)⋊C8—94no—
109(C2×D3)⋊C8—91no—
110(C2×D3)⋊C8C3⋊D311yescharacteristic, complemented
111#815—94no—
112Dic3⋊C8—34no—
113#803—91no—
114C23×C32—31no—
115C22×C3⋊D3—31no—
116C22×C3⋊D3—34no—
117C22×C3⋊D3—64no—
118C22×C3×D3C4×D314yes—
119C2×C32×C4—24no—
120C2×C32×C4—31no—
121C2×C32×C4—61no—
122C2×C32×C4—61no—
123C2×C32×C4C22×D314yes—
124C2×C32×C4C22×D311yescharacteristic
125#638—62no—
126C2×C3⋊Dic3—31no—
127C2×C3⋊Dic3—34no—
128C2×C3⋊Dic3—64no—
129C2×C3⋊Dic3C4×D311yescharacteristic
130C2×C3⋊Dic3C4×D311yescharacteristic
131C4×C3⋊D3—68no—
132C4×C3⋊D3—68no—
133C4×C3⋊D3—64no—
134C4×C3⋊D3—61no—
135C4×C3⋊D3—64no—
136C4×C3⋊D3—61no—
137C2×C3×Dic3—34no—
138C2×C3×Dic3—34no—
139C2×C3×Dic3C4×D314yes—
140C3×C4×D3—24no—
141C3×C4×D3—24no—
142#626—68no—
143C32×C8—28no—
144C32×C8—24no—
145C32×C8—21no—
146C32×C8—61no—
147C3⋊(C3⋊C8)—21no—
148C3⋊(C3⋊C8)—32no—
149C3⋊(C3⋊C8)—38no—
150C3⋊(C3⋊C8)—68no—
151C3×C3⋊C8—24no—
152C3×C3⋊C8—64no—
153#435—271noSylow
154C2×C33C4×D411yescharacteristic
155C2×C33C2×M1611yescharacteristic
156C2×C33#13111yescharacteristic
157C3⋊(C3⋊D3)—41no—
158C32×D3—22no—
159C22×C3×C4—34no—
160C2×#67—94no—
161C2×D12—91no—
162C2×C4×D3—94no—
163C2×C4×D3—91no—
164C2×C4×D3—184no—
165C2×C4×D3C2×C3⋊D311yescharacteristic
166C2×C3×C8—24no—
167C2×C3×C8—34no—
168C2×C3×C8—91no—
169C2×C3×C8D3214yes—
170C2×C3×C8C2×C3⋊D311yescharacteristic
171C3×C42—91no—
172#217—94no—
173(C2×D3)⋊C4—94no—
174(C2×D3)⋊C4—91no—
175(C2×D3)⋊C4—91no—
176C4⋊Dic3—91no—
177Dic3⋊C4—94no—
178C4×Dic3—34no—
179C2×C3⋊C8—34no—
180C2×C3⋊C8—94no—
181C2×C3⋊C8—91no—
182C2×C3⋊C8—184no—
183C2×C3⋊C8—188no—
184C2×C3⋊C8C2×C3⋊D311yescharacteristic
185ΓL(1,𝔽25)—188no—
186ΓL(1,𝔽25)—182no—
187ΓL(1,𝔽25)—1816no—
188D3×C8—182no—
189C22×C32—24no—
190C22×C32—32no—
191C22×C32—61no—
192C22×C32—61no—
193C22×C32C2×C4×D314yes—
194C22×C32C2×C4×D311yescharacteristic
195C2×C3⋊D3—68no—
196C2×C3⋊D3—64no—
197C2×C3⋊D3—61no—
198C2×C3⋊D3—61no—
199C2×C3⋊D3—61no—
200C2×C3⋊D3—64no—
201C2×C3⋊D3—64no—
202C2×C3⋊D3—124no—
203C2×C3⋊D3—124no—
204C2×C3×D3—24no—
205C2×C3×D3—24no—
206C2×C3×D3ΓL(1,𝔽25)18yescomplemented
207C32×C4—24no—
208C32×C4—24no—
209C32×C4—61no—
210C32×C4D3×D414yes—
211C32×C4D3×D411yescharacteristic
212C32×C4#23714yes—
213C32×C4#23911yescharacteristic
214C3⋊Dic3—21no—
215C3⋊Dic3—21no—
216C3⋊Dic3—68no—
217C3⋊Dic3—64no—
218C3⋊Dic3—61no—
219C3×Dic3—24no—
220C3×Dic3—64no—
221C3×Dic3—64no—
222C2×M16—271no—
223C22×C8—271no—
224C4×D4—271no—
225C4⋊C8—91no—
226C22⋊C8—31no—
227C22⋊C8—271no—
228#97—271no—
229C33#43511yescharacteristic, complemented, Sylow
230C23×C3—34no—
231C22×D3—94no—
232C22×D3—91no—
233C22×D3—184no—
234C22×D3C4×C3⋊D311yescharacteristic
235C2×C3×C4—24no—
236C2×C3×C4—34no—
237C2×C3×C4—64no—
238C2×C3×C4—64no—
239C2×C3×C4—91no—
240C2×C3×C4—91no—
241C2×C3×C4C2×D3214yes—
242C2×C3×C4C22×C3⋊D311yescharacteristic
243#67—188no—
244C2×Dic3—34no—
245C2×Dic3—34no—
246C2×Dic3—94no—
247C2×Dic3—91no—
248C2×Dic3—184no—
249C2×Dic3C4×C3⋊D311yescharacteristic
250D12—182no—
251C4×D3—21no—
252C4×D3—21no—
253C4×D3—188no—
254C4×D3—188no—
255C4×D3—184no—
256C4×D3—184no—
257C4×D3—181no—
258C4×D3—181no—
259C3×C8—28no—
260C3×C8—24no—
261C3×C8—21no—
262C3×C8—64no—
263C3×C8—181no—
264C3⋊C8—21no—
265C3⋊C8—64no—
266C3⋊C8—98no—
267C3⋊C8—92no—
268C3⋊C8—188no—
269C2×C32—24no—
270C2×C32—24no—
271C2×C32—24no—
272C2×C32—62no—
273C2×C32#67⋊C414yes—
274C2×C32#89311yescharacteristic
275C2×C32C2×ΓL(1,𝔽25)14yes—
276C2×C32#90111yescharacteristic
277C2×C32D3×M1611yescharacteristic
278C2×C32#90714yes—
279C3⋊D3—128no—
280C3⋊D3—124no—
281C3⋊D3—121no—
282C3×D3—28no—
283C2×D4—271no—
284C22×C4—31no—
285C22×C4—271no—
286M16—542no—
287C2×C8—31no—
288C2×C8—91no—
289C2×C8—271no—
290C2×C8—542no—
291C2×C8D3×C3⋊D311yescharacteristic
292C4⋊C4—271no—
293C22⋊C4—271no—
294C22⋊C4—271no—
295C42—91no—
296C22×C3—24no—
297C22×C3—38no—
298C22×C3—64no—
299C22×C3—64no—
300C22×C3C4×D3214yes—
301C22×C3C2×C4×C3⋊D311yescharacteristic
302C2×D3—21no—
303C2×D3—21no—
304C2×D3—188no—
305C2×D3—184no—
306C2×D3—184no—
307C2×D3—181no—
308C2×D3—181no—
309C2×D3—181no—
310C2×D3—184no—
311C2×D3—364no—
312C2×D3—364no—
313C2×D3#374312yescomplemented
314C3×C4—24no—
315C3×C4—24no—
316C3×C4—64no—
317C3×C4—181no—
318C3×C4—181no—
319C3×C4#379614yes—
320C3×C4#381114yes—
321C3×C4D4×C3⋊D311yescharacteristic
322C3×C4#383011yescharacteristic
323Dic3—21no—
324Dic3—64no—
325Dic3—64no—
326Dic3—188no—
327Dic3—181no—
328Dic3—184no—
329C32—24no—
330C32#473014yescomplemented
331C32#483611yescharacteristic, complemented
332C23—31no—
333C23—271no—
334D4—542no—
335C2×C4—31no—
336C2×C4—61no—
337C2×C4—61no—
338C2×C4—91no—
339C2×C4—91no—
340C2×C4—271no—
341C2×C4—541no—
342C2×C4—541no—
343C2×C4C2×D3×C3⋊D311yescharacteristic
344C8—21no—
345C8—61no—
346C8—181no—
347C8—272no—
348C2×C3—24no—
349C2×C3—24no—
350C2×C3—24no—
351C2×C3—68no—
352C2×C3D3×ΓL(1,𝔽25)14yes—
353C2×C3#6380414yes—
354C2×C3#6388114yes—
355C2×C3#6409011yescharacteristic
356C2×C3C2×#374311yescharacteristic
357C2×C3#6411611yescharacteristic
358D3—22no—
359D3—368no—
360D3—361no—
361D3—364no—
362C22—32no—
363C22—61no—
364C22—61no—
365C22—541no—
366C22—541no—
367C22—541no—
368C22C4×D3×C3⋊D311yescharacteristic
369C4—61no—
370C4—181no—
371C4—181no—
372C4—541no—
373C4#8901511yescharacteristic
374C4#8903911yescharacteristic
375C3—24no—
376C3#1059090714yescomplemented
377C3#1059242911yescharacteristic, complemented
378C2—62no—
379C2—1081no—
380C2D3×#374311yescharacteristic
381C2#1172333311yescharacteristic
382C2#1172346211yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×D3×C3⋊D3
commutator, socleC22×C33C22×C4
FittingC2×C33×C8C22
prime radicalC2×#6291C4
cosocleC2×C33×C4C23
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#435
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. C2×#88850C2semidirect
3. C2×D3×C3⋊(C3⋊C8)C2semidirect
4. C3×#63641C2semidirect
5. C32×(C2×D3)⋊C8C2semidirect
6. #11722024C2semidirect
7. #11722020C2semidirect
33. #63641D3semidirect
34. C3×(C2×D3)⋊C8D3semidirect
51. C3⋊(C3⋊(C3⋊C8))D4semidirect
92. C2×C32×D3M16semidirect
110. (C2×D3)⋊C8C3⋊D3semidirect
206. C2×C3×D3ΓL(1,𝔽25)semidirect
229. C33#435semidirect
313. C2×D3#3743semidirect
330. C32#4730semidirect
331. C32#4836semidirect
376. C3#10590907semidirect
377. C3#10592429semidirect

Maximal subgroups

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

subgroupindex
C2×#888502
C4×#62912
C2×D3×C3⋊(C3⋊C8)2
C3×#636412
C32×(C2×D3)⋊C82
#117220242
#117220202
#105924293
#105909073