Finite Group Databasebeta

← #13160915

Subgroups — #13160915

Order 1728.

1108 conjugacy classes of subgroups; only the 196 normal subgroups are shown.

normal subgroups only

This group has 1108 conjugacy classes of subgroups — too many to draw. Only its 196 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×#88860C211yesmaximal, characteristic
2C32×D4×Dic3C211yesmaximal, characteristic, complemented
3#11723484C211yesmaximal, characteristic, complemented
4C3×#63659C211yesmaximal, characteristic, complemented
5#11722056C211yesmaximal, characteristic, complemented
6#11722035C211yesmaximal, characteristic, complemented
7Dic3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8C4×#3753—31nomaximal
9#10591400—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C22×C32×Dic3—21no—
13C2×#6289—21no—
14#88860C414yescomplemented
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×#217—21no—
19C32×C4⋊Dic3C2211yescharacteristic
20C32×C4×Dic3C2211yescharacteristic
21#88804—21no—
22C32×C4×D4—31no—
23C2×#3753—31no—
24C2×#3753—34no—
25C2×#3753—64no—
26C4×C4⋊(C3⋊D3)—31no—
27C3×D4×Dic3D314yescomplemented
28#63894—34no—
29#63659D311yescharacteristic, complemented
30#63636—31no—
31#63634—31no—
32C4×C3⋊(C3⋊C8)—31no—
33C3×#832—34no—
34#63576—34no—
35#63564—34no—
36Dic3×C3⋊C8—34no—
37C23×C33—21no—
38C22×C3⋊(C3⋊D3)—21no—
39C33×D4—21no—
40C33×D4C2×C412yescomplemented
41C2×C33×C4C2311yescharacteristic
42C4⋊(C3⋊(C3⋊D3))—21no—
43C4⋊(C3⋊(C3⋊D3))C2×C412yescomplemented
44C2×C3×C3⋊Dic3—21no—
45C2×C32×Dic3—21no—
46C2×C32×Dic3—42no—
47C2×C32×Dic3D411yescharacteristic
48C2×C32×Dic3D411yescharacteristic
49#6289—42no—
50C3⋊(C3⋊(C3⋊C8))C2×C412yescomplemented
51C3×C3⋊(C3⋊C8)—21no—
52C4×#213—94no—
53#4756—91no—
54C2×C32×D4—24no—
55C2×C32×D4C2×D314yes—
56C2×C32×D4C2×D311yescharacteristic
57C22×C32×C4—61no—
58C2×C4⋊(C3⋊D3)—31no—
59C2×C4⋊(C3⋊D3)—34no—
60C2×C4⋊(C3⋊D3)—64no—
61C2×C4×C3⋊D3—61no—
62C22×C3×Dic3—24no—
63C2×#624—64no—
64C32×C4⋊C4—31no—
65C32×C22⋊C4—61no—
66C32×C42—31no—
67#3753—34no—
68#3753—316no—
69#3753—616no—
70#3752—61no—
71C4⋊(C3⋊Dic3)C2×D311yescharacteristic
72C2×C3⋊(C3⋊C8)—31no—
73C2×C3⋊(C3⋊C8)—34no—
74C2×C3⋊(C3⋊C8)—64no—
75C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
76C3×#217—24no—
77C3×C4⋊Dic3—34no—
78C3×C4⋊Dic3C2×D314yes—
79C3×C4×Dic3C2×D314yes—
80C2×C3×C3⋊C8—34no—
81#3722—64no—
82C22×C33—22no—
83C22×C33—41no—
84C22×C33—41no—
85C22×C33C2×D411yescharacteristic
86C2×C3⋊(C3⋊D3)—22no—
87C2×C3⋊(C3⋊D3)—41no—
88C2×C3⋊(C3⋊D3)—41no—
89C33×C4C22×C411yescharacteristic
90C33×C4#4111yescharacteristic
91C3×C3⋊Dic3—41no—
92C32×Dic3—21no—
93C32×Dic3—41no—
94C32×Dic3D812yescomplemented
95C3×C4×D4—34no—
96D4×Dic3C3⋊D311yescharacteristic, complemented
97C2×#213—94no—
98C2×#213—91no—
99C2×#213—184no—
100#893—91no—
101C4×D12—94no—
102C3×#102—91no—
103#832—34no—
104#821—91no—
105C8×Dic3—91no—
106#809—94no—
107#807—94no—
108#807—91no—
109C4×C3⋊C8—94no—
110C23×C32—21no—
111C23×C32—24no—
112C23×C32—44no—
113C22×C3⋊D3—61no—
114C22×C3⋊D3—64no—
115C22×C3⋊D3—124no—
116C32×D4—21no—
117C32×D4—28no—
118C32×D4—24no—
119C32×D4—44no—
120C32×D4C4×D318yescomplemented
121C32×D4C4×D312yescomplemented
122C2×C32×C4—24no—
123C2×C32×C4—31no—
124C2×C32×C4—31no—
125C2×C32×C4—61no—
126C2×C32×C4—122no—
127C2×C32×C4C22×D314yes—
128C2×C32×C4C22×D311yescharacteristic
129C2×C3⋊Dic3—21no—
130C4⋊(C3⋊D3)—32no—
131C4⋊(C3⋊D3)—38no—
132C4⋊(C3⋊D3)—68no—
133C4⋊(C3⋊D3)—61no—
134C4⋊(C3⋊D3)—64no—
135C4⋊(C3⋊D3)—124no—
136C4×C3⋊D3—122no—
137C2×C3×Dic3—24no—
138C2×C3×Dic3—48no—
139C2×C3×Dic3—64no—
140C2×C3×Dic3#6714yes—
141C2×C3×Dic3#6714yes—
142#624—128no—
143C3⋊(C3⋊C8)—21no—
144C3⋊(C3⋊C8)—32no—
145C3⋊(C3⋊C8)—38no—
146C3⋊(C3⋊C8)—68no—
147C3×C3⋊C8—64no—
148C4×D8—271noSylow
149C2×C33—42no—
150C2×C33C4×D411yescharacteristic
151C2×C33C2×D811yescharacteristic
152C2×C33#13511yescharacteristic
153C3⋊(C3⋊D3)—42no—
154C2×C3×D4—24no—
155C2×C3×D4D3214yes—
156C2×C3×D4C2×C3⋊D311yescharacteristic
157C22×C3×C4—64no—
158C22×Dic3—21no—
159C2×D12—94no—
160C2×D12—91no—
161C2×D12—184no—
162C2×C4×D3—184no—
163C2×C4×D3—181no—
164C2×C3×C8—91no—
165C3×C4⋊C4—34no—
166C3×C4⋊C4—91no—
167C3×C22⋊C4—64no—
168C3×C42—34no—
169#217—21no—
170#213—916no—
171#213—94no—
172#213—1816no—
173(C2×D3)⋊C4—184no—
174(C2×D3)⋊C4—181no—
175C4⋊Dic3—34no—
176C4⋊Dic3C2×C3⋊D311yescharacteristic
177C4×Dic3C2×C3⋊D311yescharacteristic
178C2×C3⋊C8—34no—
179C2×C3⋊C8—94no—
180C2×C3⋊C8—91no—
181C2×C3⋊C8—184no—
182C22×C32—22no—
183C22×C32—24no—
184C22×C32—28no—
185C22×C32—48no—
186C22×C32—41no—
187C22×C32—41no—
188C22×C32—44no—
189C22×C32—44no—
190C22×C32—84no—
191C22×C32—84no—
192C22×C32D3×D411yescharacteristic
193C22×C32C2×#6714yes—
194C2×C3⋊D3—62no—
195C2×C3⋊D3—68no—
196C2×C3⋊D3—128no—
197C2×C3⋊D3—121no—
198C2×C3⋊D3—121no—
199C2×C3⋊D3—124no—
200C2×C3⋊D3—124no—
201C2×C3⋊D3—244no—
202C2×C3⋊D3—244no—
203C32×C4—24no—
204C32×C4—24no—
205C32×C4—32no—
206C32×C4—61no—
207C32×C4—121no—
208C32×C4C2×C4×D314yes—
209C32×C4C2×C4×D311yescharacteristic
210C32×C4#23514yes—
211C32×C4#23711yescharacteristic
212C3⋊Dic3—41no—
213C3×Dic3—24no—
214C3×Dic3—44no—
215C3×Dic3—124no—
216C3×Dic3#21318yescomplemented
217C2×D8—271no—
218C4×D4—31no—
219C4×D4—271no—
220#107—271no—
221#102—91no—
222#102—271no—
223C4×C8—271no—
224C33C4×D811yescharacteristic, complemented, Sylow
225C23×C3—24no—
226C23×C3—21no—
227C23×C3—44no—
228C22×D3—184no—
229C22×D3—181no—
230C22×D3—364no—
231C3×D4—24no—
232C3×D4—21no—
233C3×D4—28no—
234C3×D4—44no—
235C3×D4#62418yescomplemented
236C3×D4C4×C3⋊D312yescomplemented
237C2×C3×C4—24no—
238C2×C3×C4—34no—
239C2×C3×C4—34no—
240C2×C3×C4—64no—
241C2×C3×C4—128no—
242C2×C3×C4—181no—
243C2×C3×C4C2×D3214yes—
244C2×C3×C4C22×C3⋊D311yescharacteristic
245C2×Dic3—21no—
246C2×Dic3—42no—
247C2×Dic3—64no—
248C2×Dic3#63811yescharacteristic
249C2×Dic3#63811yescharacteristic
250D12—98no—
251D12—92no—
252D12—188no—
253D12—184no—
254D12—181no—
255D12—364no—
256C4×D3—368no—
257C4×D3—362no—
258C3×C8—181no—
259C3⋊C8—64no—
260C3⋊C8—98no—
261C3⋊C8—92no—
262C3⋊C8—188no—
263C2×C32—24no—
264C2×C32—24no—
265C2×C32—24no—
266C2×C32—42no—
267C2×C32—48no—
268C2×C32—88no—
269C2×C32#67⋊C411yescharacteristic
270C2×C32D3×D811yescharacteristic
271C2×C32#91611yescharacteristic
272C2×C32C4×#6714yes—
273C2×C32C2×#21314yes—
274C2×C32#95014yes—
275C3⋊D3—122no—
276C3⋊D3—128no—
277C3⋊D3—248no—
278C2×D4—271no—
279C2×D4D3×C3⋊D311yescharacteristic
280C22×C4—61no—
281C22×C4—541no—
282D8—274no—
283C2×C8—91no—
284C2×C8—271no—
285C4⋊C4—31no—
286C4⋊C4—91no—
287C22⋊C4—61no—
288C22⋊C4—541no—
289C42—31no—
290C22×C3—22no—
291C22×C3—28no—
292C22×C3—24no—
293C22×C3—48no—
294C22×C3—41no—
295C22×C3—41no—
296C22×C3—44no—
297C22×C3—44no—
298C22×C3—84no—
299C22×C3—84no—
300C22×C3D3×#6714yes—
301C22×C3C2×#63811yescharacteristic
302C2×D3—182no—
303C2×D3—188no—
304C2×D3—368no—
305C2×D3—361no—
306C2×D3—361no—
307C2×D3—364no—
308C2×D3—364no—
309C2×D3—724no—
310C2×D3—724no—
311C3×C4—24no—
312C3×C4—24no—
313C3×C4—38no—
314C3×C4—64no—
315C3×C4—124no—
316C3×C4—361no—
317C3×C4#380414yes—
318C3×C4C2×#62414yes—
319C3×C4C2×C4×C3⋊D311yescharacteristic
320C3×C4#382811yescharacteristic
321Dic3—21no—
322Dic3—41no—
323Dic3—124no—
324Dic3#375312yescomplemented
325C32—24no—
326C32#475611yescharacteristic, complemented
327C32C4×#21314yescomplemented
328C23—21no—
329C23—541no—
330D4—21no—
331D4—272no—
332D4—541no—
333D4#628912yescomplemented
334C2×C4—31no—
335C2×C4—31no—
336C2×C4—61no—
337C2×C4—122no—
338C2×C4—181no—
339C2×C4—1082no—
340C2×C4C2×D3×C3⋊D311yescharacteristic
341C8—181no—
342C8—272no—
343C2×C3—24no—
344C2×C3—24no—
345C2×C3—24no—
346C2×C3—48no—
347C2×C3—42no—
348C2×C3—88no—
349C2×C3D3×#21314yes—
350C2×C3#6393514yes—
351C2×C3#6395214yes—
352C2×C3C4×#63811yescharacteristic
353C2×C3C2×#375311yescharacteristic
354C2×C3#6415911yescharacteristic
355D3—368no—
356D3—362no—
357D3—728no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—542no—
362C22—1081no—
363C22—1081no—
364C22D3×#63811yescharacteristic
365C4—32no—
366C4—61no—
367C4—121no—
368C4—361no—
369C4#8903111yescharacteristic
370C4C2×#628911yescharacteristic
371C3—24no—
372C3#1059140014yescomplemented
373C3C4×#375311yescharacteristic, complemented
374C2—42no—
375C2—1082no—
376C2D3×#375311yescharacteristic
377C2#1172355811yescharacteristic
378C2#1172361811yescharacteristic
subgroupG/H
centerC22D3×#638
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radicalC2×#88860C2
cosocleC2×C33×C4C23
hypercenterC2×D4D3×C3⋊D3
hypocenterC33C4×D8
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C32×D4×Dic3C2semidirect
3. #11723484C2semidirect
4. C3×#63659C2semidirect
5. #11722056C2semidirect
6. #11722035C2semidirect
7. Dic3×C3⋊(C3⋊C8)C2semidirect
14. #88860C4semidirect
27. C3×D4×Dic3D3semidirect
29. #63659D3semidirect
40. C33×D4C2×C4semidirect
43. C4⋊(C3⋊(C3⋊D3))C2×C4semidirect
50. C3⋊(C3⋊(C3⋊C8))C2×C4semidirect
94. C32×Dic3D8semidirect
96. D4×Dic3C3⋊D3semidirect
120. C32×D4C4×D3semidirect
121. C32×D4C4×D3semidirect
216. C3×Dic3#213semidirect
224. C33C4×D8semidirect
235. C3×D4#624semidirect
236. C3×D4C4×C3⋊D3semidirect
324. Dic3#3753semidirect
326. C32#4756semidirect
327. C32C4×#213semidirect
333. D4#6289semidirect
372. C3#10591400semidirect
373. C3C4×#3753semidirect

Maximal subgroups

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

subgroupindex
C2×#888602
C32×D4×Dic32
#117234842
C3×#636592
#117220562
#117220352
Dic3×C3⋊(C3⋊C8)2
C4×#37533
#105914003