Finite Group Databasebeta

← #13158936

Subgroups — #13158936

Order 1728.

988 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723913C211yesmaximal, characteristic
2C2×#88788C211yesmaximal, characteristic
3C2×#88785C211yesmaximal, characteristic
4C3×C4⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
5C32×C4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
6#11722050C211yesmaximal, characteristic, complemented
7#11722050C211yesmaximal, characteristic, complemented
8#10592538—31nomaximal
9#10590332—34nomaximal
10C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
11C33×C42C2211yescharacteristic
12#88859C411yescharacteristic
13#88859C411yescharacteristic
14#88857C411yescharacteristic
15#88857C411yescharacteristic
16C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21#88788—22no—
22#88785—22no—
23C2×#3743—31no—
24C2×C8×C3⋊D3—31no—
25#64081—31no—
26#64081—34no—
27#64081—64no—
28C2×#3712—34no—
29C2×#3710—34no—
30C32×C4⋊C8—31no—
31#63641—31no—
32#63641—31no—
33C4⋊(C3⋊(C3⋊C8))D311yescharacteristic, complemented
34C3×C4⋊(C3⋊C8)—34no—
35C3×C4⋊(C3⋊C8)D314yescomplemented
36#63572—34no—
37#63572—34no—
38C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
39C2×C33×C4C2×C411yescharacteristic
40C2×C33×C4C2×C411yescharacteristic
41C2×C33×C4C2311yescharacteristic
42C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
43C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
44C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
45C4×C3⋊(C3⋊D3)D412yes—
46C4×C3⋊(C3⋊D3)Q812yes—
47C3×C3⋊(C3⋊C8)—21no—
48C3×C3⋊(C3⋊C8)—21no—
49C32×C3⋊C8—21no—
50C32×C3⋊C8—21no—
51#4839—94no—
52#4839—91no—
53C2×C4×C3⋊D3—31no—
54C2×C4×C3⋊D3—34no—
55C2×C4×C3⋊D3—64no—
56C2×C32×C8—31no—
57C2×C32×C8—31no—
58C32×C42—24no—
59C32×C42C2×D314yes—
60C32×C42C2×D311yescharacteristic
61#3752—31no—
62#3752—31no—
63#3752—34no—
64#3752—34no—
65#3752—64no—
66#3752—64no—
67#3750—31no—
68#3750—31no—
69#3750—34no—
70#3750—34no—
71#3750—64no—
72#3750—64no—
73C4×C3⋊Dic3—31no—
74C4×C3⋊Dic3—34no—
75C4×C3⋊Dic3—64no—
76C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
77C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
78#3743—62no—
79C8×C3⋊D3—62no—
80C2×C3×C3⋊C8—34no—
81C2×C3×C3⋊C8—34no—
82C2×C3×C3⋊C8C2×D314yes—
83C2×C3×C3⋊C8C2×D314yes—
84#3712—68no—
85#3710—68no—
86C22×C33C22×C411yescharacteristic
87C2×C3⋊(C3⋊D3)—21no—
88C2×C3⋊(C3⋊D3)—21no—
89C2×C3⋊(C3⋊D3)C4⋊C412yes—
90C33×C4—21no—
91C33×C4—21no—
92C33×C4C2×D411yescharacteristic
93C33×C4C2×Q811yescharacteristic
94C3⋊(C3⋊Dic3)—21no—
95C3⋊(C3⋊Dic3)—21no—
96C3⋊(C3⋊Dic3)C4⋊C412yes—
97C2×ΓL(1,𝔽25)—94no—
98C2×ΓL(1,𝔽25)—91no—
99C2×D3×C8—94no—
100C2×D3×C8—91no—
101#872—94no—
102#872—91no—
103#872—184no—
104C3×C4⋊C8—34no—
105C3×C4⋊C8—91no—
106Dic3⋊C8—94no—
107Dic3⋊C8—94no—
108Dic3⋊C8—91no—
109Dic3⋊C8—91no—
110C4⋊(C3⋊C8)—34no—
111C4⋊(C3⋊C8)C3⋊D311yescharacteristic, complemented
112C22×C3⋊D3—31no—
113C22×C3⋊D3—34no—
114C22×C3⋊D3—64no—
115C2×C32×C4—24no—
116C2×C32×C4—24no—
117C2×C32×C4—24no—
118C2×C32×C4C4×D314yes—
119C2×C32×C4C4×D314yes—
120C2×C32×C4C4×D311yescharacteristic
121C2×C32×C4C4×D311yescharacteristic
122C2×C32×C4C22×D314yes—
123C2×C32×C4C22×D311yescharacteristic
124C2×C3⋊Dic3—31no—
125C2×C3⋊Dic3—31no—
126C2×C3⋊Dic3—31no—
127C2×C3⋊Dic3—34no—
128C2×C3⋊Dic3—34no—
129C2×C3⋊Dic3—34no—
130C2×C3⋊Dic3—64no—
131C2×C3⋊Dic3—64no—
132C2×C3⋊Dic3—64no—
133C4×C3⋊D3—32no—
134C4×C3⋊D3—32no—
135C4×C3⋊D3—38no—
136C4×C3⋊D3—38no—
137C4×C3⋊D3—68no—
138C4×C3⋊D3—68no—
139C32×C8—61no—
140C32×C8—61no—
141C3⋊(C3⋊C8)—21no—
142C3⋊(C3⋊C8)—21no—
143C3×C3⋊C8—24no—
144C3×C3⋊C8—24no—
145C3×C3⋊C8—64no—
146C3×C3⋊C8—64no—
147#424—271noSylow
148C2×C33C2×C4⋊C411yescharacteristic
149C2×C33#13111yescharacteristic
150C2×C33#13111yescharacteristic
151C3⋊(C3⋊D3)—22no—
152C2×C4×D3—94no—
153C2×C4×D3—91no—
154C2×C4×D3—184no—
155C2×C3×C8—34no—
156C2×C3×C8—34no—
157C2×C3×C8—91no—
158C2×C3×C8—91no—
159C3×C42—24no—
160C3×C42D3214yes—
161C3×C42C2×C3⋊D311yescharacteristic
162(C2×D3)⋊C4—94no—
163(C2×D3)⋊C4—94no—
164(C2×D3)⋊C4—91no—
165(C2×D3)⋊C4—91no—
166(C2×D3)⋊C4—184no—
167(C2×D3)⋊C4—184no—
168Dic3⋊C4—94no—
169Dic3⋊C4—94no—
170Dic3⋊C4—91no—
171Dic3⋊C4—91no—
172Dic3⋊C4—184no—
173Dic3⋊C4—184no—
174C4×Dic3—94no—
175C4×Dic3—91no—
176C4×Dic3—184no—
177C2×C3⋊C8—34no—
178C2×C3⋊C8—34no—
179C2×C3⋊C8C2×C3⋊D311yescharacteristic
180C2×C3⋊C8C2×C3⋊D311yescharacteristic
181ΓL(1,𝔽25)—182no—
182ΓL(1,𝔽25)—188no—
183D3×C8—182no—
184D3×C8—188no—
185C22×C32—24no—
186C22×C32C2×C4×D314yes—
187C22×C32C2×C4×D311yescharacteristic
188C2×C3⋊D3—32no—
189C2×C3⋊D3—38no—
190C2×C3⋊D3—68no—
191C2×C3⋊D3—61no—
192C2×C3⋊D3—61no—
193C2×C3⋊D3—64no—
194C2×C3⋊D3—64no—
195C2×C3⋊D3—124no—
196C2×C3⋊D3—124no—
197C32×C4—21no—
198C32×C4—21no—
199C32×C4—24no—
200C32×C4—24no—
201C32×C4—24no—
202C32×C4—24no—
203C32×C4—44no—
204C32×C4—44no—
205C32×C4D3×D414yes—
206C32×C4D3×D411yescharacteristic
207C32×C4D3×Q814yes—
208C32×C4D3×Q811yescharacteristic
209C3⋊Dic3—32no—
210C3⋊Dic3—38no—
211C3⋊Dic3—68no—
212C3⋊Dic3—61no—
213C3⋊Dic3—61no—
214C3⋊Dic3—64no—
215C3⋊Dic3—64no—
216C3⋊Dic3—124no—
217C3⋊Dic3—124no—
218C2×M16—271no—
219C22×C8—271no—
220#117—271no—
221C4⋊C8—31no—
222C4⋊C8—91no—
223C4⋊C8—271no—
224C4⋊C8—271no—
225C33#42411yescharacteristic, complemented, Sylow
226C22×D3—94no—
227C22×D3—91no—
228C22×D3—184no—
229C2×C3×C4—24no—
230C2×C3×C4—24no—
231C2×C3×C4—24no—
232C2×C3×C4#62414yes—
233C2×C3×C4#62414yes—
234C2×C3×C4C4×C3⋊D311yescharacteristic
235C2×C3×C4C4×C3⋊D311yescharacteristic
236C2×C3×C4C2×D3214yes—
237C2×C3×C4C22×C3⋊D311yescharacteristic
238C2×Dic3—94no—
239C2×Dic3—91no—
240C2×Dic3—94no—
241C2×Dic3—94no—
242C2×Dic3—91no—
243C2×Dic3—91no—
244C2×Dic3—184no—
245C2×Dic3—184no—
246C2×Dic3—184no—
247C4×D3—98no—
248C4×D3—98no—
249C4×D3—92no—
250C4×D3—92no—
251C4×D3—188no—
252C4×D3—188no—
253C3×C8—64no—
254C3×C8—64no—
255C3×C8—181no—
256C3×C8—181no—
257C3⋊C8—21no—
258C3⋊C8—21no—
259C3⋊C8—64no—
260C3⋊C8—64no—
261C2×C32—24no—
262C2×C32—24no—
263C2×C32—24no—
264C2×C32D3×C4⋊C414yes—
265C2×C32D3×C4⋊C411yescharacteristic
266C2×C32#90114yes—
267C2×C32#90111yescharacteristic
268C2×C32#90714yes—
269C2×C32#90711yescharacteristic
270C3⋊D3—62no—
271C3⋊D3—68no—
272C3⋊D3—128no—
273C22×C4—271no—
274M16—542no—
275C2×C8—31no—
276C2×C8—31no—
277C2×C8—91no—
278C2×C8—91no—
279C2×C8—542no—
280C4⋊C4—271no—
281C4⋊C4—271no—
282C22⋊C4—271no—
283C22⋊C4—271no—
284C42—271no—
285C42D3×C3⋊D311yescharacteristic
286C22×C3—24no—
287C22×C3C2×#62414yes—
288C22×C3C2×C4×C3⋊D311yescharacteristic
289C2×D3—92no—
290C2×D3—98no—
291C2×D3—188no—
292C2×D3—184no—
293C2×D3—184no—
294C2×D3—181no—
295C2×D3—181no—
296C2×D3—364no—
297C2×D3—364no—
298C3×C4—24no—
299C3×C4—24no—
300C3×C4—21no—
301C3×C4—21no—
302C3×C4—24no—
303C3×C4—24no—
304C3×C4—44no—
305C3×C4—44no—
306C3×C4#379714yes—
307C3×C4#380214yes—
308C3×C4D4×C3⋊D311yescharacteristic
309C3×C4Q8×C3⋊D311yescharacteristic
310Dic3—92no—
311Dic3—98no—
312Dic3—188no—
313Dic3—184no—
314Dic3—184no—
315Dic3—181no—
316Dic3—181no—
317Dic3—364no—
318Dic3—364no—
319C32—24no—
320C32#483914yescomplemented
321C32#483911yescharacteristic, complemented
322C23—271no—
323C2×C4—271no—
324C2×C4—272no—
325C2×C4—272no—
326C2×C4—271no—
327C2×C4—271no—
328C2×C4#628911yescharacteristic
329C2×C4#628911yescharacteristic
330C2×C4C2×D3×C3⋊D311yescharacteristic
331C8—61no—
332C8—61no—
333C8—181no—
334C8—181no—
335C2×C3—24no—
336C2×C3—24no—
337C2×C3—24no—
338C2×C3#6381714yes—
339C2×C3#6381714yes—
340C2×C3#6390014yes—
341C2×C3C4⋊C4×C3⋊D311yescharacteristic
342C2×C3#6411011yescharacteristic
343C2×C3#6411611yescharacteristic
344D3—182no—
345D3—188no—
346D3—368no—
347C22—272no—
348C22—541no—
349C22—541no—
350C22C2×#628911yescharacteristic
351C4—21no—
352C4—21no—
353C4—272no—
354C4—541no—
355C4—541no—
356C4#8901911yescharacteristic
357C4#8902611yescharacteristic
358C3—24no—
359C3#1059033214yescomplemented
360C3#1059253811yescharacteristic, complemented
361C2—542no—
362C2#1172335611yescharacteristic
363C2#1172335711yescharacteristic
364C2#1172350311yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×D3×C3⋊D3
commutator, socleC22×C33C22×C4
FittingC33×C42C22
prime radicalC22×C3⋊(C3⋊D3)C2×C4
cosocleC2×C33×C4C23
hypercenterC42D3×C3⋊D3
hypocenterC33#424
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
4. C3×C4⋊(C3⋊(C3⋊C8))C2semidirect
5. C32×C4⋊(C3⋊C8)C2semidirect
6. #11722050C2semidirect
7. #11722050C2semidirect
33. C4⋊(C3⋊(C3⋊C8))D3semidirect
35. C3×C4⋊(C3⋊C8)D3semidirect
111. C4⋊(C3⋊C8)C3⋊D3semidirect
225. C33#424semidirect
320. C32#4839semidirect
321. C32#4839semidirect
359. C3#10590332semidirect
360. C3#10592538semidirect

Maximal subgroups

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

subgroupindex
#117239132
C2×#887882
C2×#887852
C3×C4⋊(C3⋊(C3⋊C8))2
C32×C4⋊(C3⋊C8)2
#117220502
#105925383
#105903323