Finite Group Databasebeta

← C2×#11722044 #13160375

Subgroups — C2×#11722044 #13160375

Order 1728.

1292 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#235C211yesmaximal, characteristic
2C2×#88855C211yesmaximal, characteristic, complemented
3C2×C3×C4×C3⋊Dic3C211yesmaximal, characteristic, complemented
4#11722044C214yesmaximal, direct product factor
5C2×#63664—31nomaximal
6C2×#63569—34nomaximal
7C22×C33×C4C2211yescharacteristic
8C22×C3×C3⋊Dic3—21no—
9C2×C32×#67—21no—
10C32×#235—21no—
11C32×#235—21no—
12C32×#235C412yescomplemented
13C32×#235C2212yes—
14C2×C32×D12C411yescharacteristic, complemented
15C2×C32×C4×D3—21no—
16C2×C32×Dic6C411yescharacteristic, complemented
17#88855—21no—
18#88855C2212yescomplemented
19C2×C3⋊(C3⋊(C3⋊C8))—21no—
20C3×C4×C3⋊Dic3—21no—
21C3×C4×C3⋊Dic3C2212yescomplemented
22C2×C32×#41—31no—
23C2×C3×#235D314yes—
24C2×C4×C3⋊Dic3D311yescharacteristic, complemented
25C2×#3748—31no—
26C2×#3748—34no—
27C2×#3748—64no—
28C2×C3×C4×Dic3—34no—
29#63664—34no—
30#63569—316no—
31C23×C33D411yescharacteristic
32C22×C32×D3—21no—
33C2×C33×C4C2×C411yescharacteristic
34C2×C33×C4C2×C411yescharacteristic
35C2×C33×C4D411yescharacteristic
36C2×C33×C4D411yescharacteristic
37C2×C33×C4D411yescharacteristic
38C2×C33×C4C2311yescharacteristic
39C2×C3×C3⋊Dic3—24no—
40C2×C3×C3⋊Dic3—22no—
41C32×#67—22no—
42C32×#67—22no—
43C2×C32×Dic3—21no—
44C32×D12—21no—
45C32×D12C2×C412yescomplemented
46C32×C4×D3—22no—
47C32×C4×D3—22no—
48C32×Dic6—21no—
49C32×Dic6C2×C412yescomplemented
50C3⋊(C3⋊(C3⋊C8))—22no—
51C2×#837—94no—
52C2×#805—91no—
53C32×#41—32no—
54C32×#41—32no—
55C32×#41—61no—
56C32×#41—61no—
57C2×C32×Q8—31no—
58C2×C32×D4—31no—
59C2×C32×D4—61no—
60C22×C32×C4—24no—
61C22×C32×C4—61no—
62C22×C32×C4C2×D311yescharacteristic
63C22×C32×C4C2×D314yes—
64C22×C3⋊Dic3—21no—
65C2×C3×#67—24no—
66C22×C3×Dic3—64no—
67C3×#235—24no—
68C3×#235—24no—
69C3×#235Dic318yescomplemented
70C3×#235C2×D318yes—
71C2×C3×D12Dic314yescomplemented
72C2×C3×C4×D3—24no—
73C2×C3×Dic6Dic314yescomplemented
74C4×C3⋊Dic3—21no—
75C4×C3⋊Dic3C2×D312yescomplemented
76#3748—32no—
77#3748—38no—
78#3748—68no—
79#3748—68no—
80#3748—61no—
81#3748—64no—
82C2×C3⋊(C3⋊C8)—68no—
83C2×C3⋊(C3⋊C8)—61no—
84C2×C3⋊(C3⋊C8)—64no—
85C3×C4×Dic3—38no—
86C3×C4×Dic3—64no—
87C22×C33—22no—
88C22×C33C22⋊C411yescharacteristic
89C22×C33C22⋊C411yescharacteristic
90C22×C33C2×D411yescharacteristic
91C2×C32×D3—22no—
92C2×C32×D3—42no—
93C33×C4C22⋊C411yescharacteristic
94C33×C4C22⋊C411yescharacteristic
95C33×C4C22×C411yescharacteristic
96C33×C4C2×D411yescharacteristic
97C3×C3⋊Dic3—24no—
98C32×Dic3—22no—
99C2×C3×#41—34no—
100C2×#235C3⋊D311yescharacteristic
101C2×C3×C42—91no—
102C2×C4×Dic3—34no—
103C2×#208—94no—
104C2×#208—91no—
105C2×#208—184no—
106#837—916no—
107#805—94no—
108C23×C32—24no—
109C23×C32—61no—
110C23×C32#6714yes—
111C23×C32#6711yescharacteristic
112C22×C3×D3—24no—
113C32×Q8—32no—
114C32×Q8—61no—
115C32×D4—32no—
116C32×D4—62no—
117C32×D4—62no—
118C32×D4—61no—
119C2×C32×C4—24no—
120C2×C32×C4—24no—
121C2×C32×C4—24no—
122C2×C32×C4—24no—
123C2×C32×C4—24no—
124C2×C32×C4—24no—
125C2×C32×C4—61no—
126C2×C32×C4—62no—
127C2×C32×C4—62no—
128C2×C32×C4C4×D311yescharacteristic
129C2×C32×C4C4×D311yescharacteristic
130C2×C32×C4D1211yescharacteristic
131C2×C32×C4D1211yescharacteristic
132C2×C32×C4C2×Dic314yes—
133C2×C32×C4C2×Dic314yes—
134C2×C32×C4#6714yes—
135C2×C32×C4#6714yes—
136C2×C32×C4#6714yes—
137C2×C32×C4#6711yescharacteristic
138C2×C32×C4C22×D314yes—
139C2×C32×C4C22×D311yescharacteristic
140C2×C3⋊Dic3—24no—
141C2×C3⋊Dic3—22no—
142C3×#67—28no—
143C3×#67—28no—
144C2×C3×Dic3—24no—
145C2×C3×Dic3—616no—
146C2×C3×Dic3—68no—
147C3×D12—24no—
148C3×D12C2×Dic318yescomplemented
149C3×C4×D3—28no—
150C3×C4×D3—28no—
151C3×Dic6—24no—
152C3×Dic6C2×Dic318yescomplemented
153C3⋊(C3⋊C8)—616no—
154C3⋊(C3⋊C8)—62no—
155C3⋊(C3⋊C8)—68no—
156C2×#104—271noSylow
157C2×C33—21no—
158C2×C33—21no—
159C2×C33#10412yescomplemented
160C2×C33C2×C22⋊C411yescharacteristic
161C32×D3—42no—
162C3×#41—38no—
163C3×#41—38no—
164C3×#41—64no—
165C3×#41—64no—
166C2×C3×Q8—34no—
167C2×C3×D4—34no—
168C2×C3×D4—64no—
169C22×C3×C4—24no—
170C22×C3×C4—64no—
171C22×C3×C4—181no—
172C22×C3×C4D3214yes—
173C22×C3×C4C2×C3⋊D311yescharacteristic
174C2×#67—21no—
175C22×Dic3—64no—
176#235—21no—
177#235—21no—
178#235C3⋊Dic312yescomplemented
179#235C2×C3⋊D312yes—
180C2×D12C3⋊Dic311yescharacteristic, complemented
181C2×C4×D3—21no—
182C2×Dic6C3⋊Dic311yescharacteristic, complemented
183C3×C42—92no—
184C3×C42—181no—
185C4×Dic3—38no—
186C4×Dic3—64no—
187#208—98no—
188#208—92no—
189#208—188no—
190#208—188no—
191#208—184no—
192#208—181no—
193C2×C3⋊C8—188no—
194C2×C3⋊C8—184no—
195C2×C3⋊C8—181no—
196C22×C32—22no—
197C22×C32—24no—
198C22×C32—24no—
199C22×C32—24no—
200C22×C32—28no—
201C22×C32—48no—
202C22×C32—62no—
203C22×C32—122no—
204C22×C32(C2×D3)⋊C411yescharacteristic
205C22×C32(C2×D3)⋊C411yescharacteristic
206C22×C32#21714yes—
207C22×C32#21714yes—
208C22×C32C2×#6714yes—
209C22×C32C2×#6711yescharacteristic
210C2×C3×D3—28no—
211C2×C3×D3—48no—
212C32×C4—24no—
213C32×C4—24no—
214C32×C4—24no—
215C32×C4—24no—
216C32×C4—62no—
217C32×C4(C2×D3)⋊C411yescharacteristic
218C32×C4(C2×D3)⋊C411yescharacteristic
219C32×C4#21714yes—
220C32×C4#21714yes—
221C32×C4C2×C4×D311yescharacteristic
222C32×C4C2×D1211yescharacteristic
223C32×C4C22×Dic314yes—
224C32×C4C2×#6714yes—
225C3⋊Dic3—24no—
226C3×Dic3—28no—
227C3×Dic3—616no—
228C2×#41—31no—
229C2×M16—271no—
230C2×C42—91no—
231#104—274no—
232C33C2×#10411yescharacteristic, complemented, Sylow
233C23×C3—24no—
234C23×C3—64no—
235C23×C3#62514yes—
236C23×C3#63811yescharacteristic
237C22×D3—21no—
238C3×Q8—38no—
239C3×Q8—64no—
240C3×D4—38no—
241C3×D4—68no—
242C3×D4—68no—
243C3×D4—64no—
244C2×C3×C4—24no—
245C2×C3×C4—24no—
246C2×C3×C4—24no—
247C2×C3×C4—24no—
248C2×C3×C4—24no—
249C2×C3×C4—24no—
250C2×C3×C4—64no—
251C2×C3×C4—68no—
252C2×C3×C4—68no—
253C2×C3×C4—184no—
254C2×C3×C4—182no—
255C2×C3×C4D3×Dic314yes—
256C2×C3×C4D3×Dic314yes—
257C2×C3×C4#62514yes—
258C2×C3×C4#62614yes—
259C2×C3×C4#62614yes—
260C2×C3×C4C2×C3⋊Dic311yescharacteristic
261C2×C3×C4C2×C3⋊Dic311yescharacteristic
262C2×C3×C4#63811yescharacteristic
263C2×C3×C4#63811yescharacteristic
264C2×C3×C4#63811yescharacteristic
265C2×C3×C4C2×D3214yes—
266C2×C3×C4C22×C3⋊D311yescharacteristic
267#67—22no—
268#67—22no—
269C2×Dic3—21no—
270C2×Dic3—616no—
271C2×Dic3—68no—
272D12—21no—
273D12C2×C3⋊Dic312yescomplemented
274C4×D3—22no—
275C4×D3—22no—
276Dic6—21no—
277Dic6C2×C3⋊Dic312yescomplemented
278C3⋊C8—1816no—
279C3⋊C8—188no—
280C3⋊C8—182no—
281C2×C32—21no—
282C2×C32—21no—
283C2×C32—28no—
284C2×C32—24no—
285C2×C32—24no—
286C2×C32—24no—
287C2×C32—44no—
288C2×C32—44no—
289C2×C32—122no—
290C2×C32#80512yescomplemented
291C2×C32#83718yescomplemented
292C2×C32C2×(C2×D3)⋊C411yescharacteristic
293C2×C32C2×#21714yes—
294C3×D3—48no—
295#41—32no—
296#41—32no—
297#41—61no—
298#41—61no—
299C2×Q8—31no—
300C2×D4—31no—
301C2×D4—61no—
302C22×C4—61no—
303C22×C4—181no—
304C22×C4D3×C3⋊D311yescharacteristic
305M16—272no—
306M16—541no—
307C2×C8—541no—
308C42—92no—
309C42—181no—
310C22×C3—28no—
311C22×C3—22no—
312C22×C3—24no—
313C22×C3—24no—
314C22×C3—24no—
315C22×C3—48no—
316C22×C3—68no—
317C22×C3—128no—
318C22×C3(C2×D3)⋊Dic314yes—
319C22×C3(C2×D3)⋊Dic314yes—
320C22×C3#375711yescharacteristic
321C22×C3#375711yescharacteristic
322C22×C3C2×#62514yes—
323C22×C3C2×#63811yescharacteristic
324C2×D3—22no—
325C2×D3—42no—
326C3×C4—24no—
327C3×C4—24no—
328C3×C4—24no—
329C3×C4—24no—
330C3×C4—68no—
331C3×C4—184no—
332C3×C4(C2×D3)⋊Dic314yes—
333C3×C4(C2×D3)⋊Dic314yes—
334C3×C4#375711yescharacteristic
335C3×C4#375711yescharacteristic
336C3×C4C2×D3×Dic314yes—
337C3×C4C2×#62614yes—
338C3×C4C22×C3⋊Dic311yescharacteristic
339C3×C4C2×#63811yescharacteristic
340Dic3—22no—
341Dic3—616no—
342C32—24no—
343C32C2×#80511yescharacteristic, complemented
344C32C2×#83714yescomplemented
345C23—61no—
346C23#629011yescharacteristic
347Q8—32no—
348Q8—61no—
349D4—32no—
350D4—62no—
351D4—62no—
352D4—61no—
353C2×C4—61no—
354C2×C4—62no—
355C2×C4—62no—
356C2×C4—184no—
357C2×C4—182no—
358C2×C4D3×C3⋊Dic311yescharacteristic
359C2×C4D3×C3⋊Dic311yescharacteristic
360C2×C4#629011yescharacteristic
361C2×C4#629111yescharacteristic
362C2×C4#629111yescharacteristic
363C2×C4C2×D3×C3⋊D311yescharacteristic
364C8—542no—
365C2×C3—24no—
366C2×C3—24no—
367C2×C3—21no—
368C2×C3—21no—
369C2×C3—28no—
370C2×C3—24no—
371C2×C3—44no—
372C2×C3—44no—
373C2×C3—128no—
374C2×C3#6356918yescomplemented
375C2×C3#6366412yescomplemented
376C2×C3C2×(C2×D3)⋊Dic314yes—
377C2×C3C2×#375711yescharacteristic
378D3—42no—
379C22—22no—
380C22—62no—
381C22—122no—
382C22(C2×D3)⋊(C3⋊Dic3)11yescharacteristic
383C22(C2×D3)⋊(C3⋊Dic3)11yescharacteristic
384C22C2×#629011yescharacteristic
385C4—62no—
386C4—184no—
387C4(C2×D3)⋊(C3⋊Dic3)11yescharacteristic
388C4(C2×D3)⋊(C3⋊Dic3)11yescharacteristic
389C4C2×D3×C3⋊Dic311yescharacteristic
390C4C2×#629111yescharacteristic
391C3—24no—
392C3C2×#6356914yescomplemented
393C3C2×#6366411yescharacteristic, complemented
394C2—21no—
395C2—21no—
396C2—122no—
397C2#1172204412yesdirect product factor
398C2C2×(C2×D3)⋊(C3⋊Dic3)11yescharacteristic
subgroupG/H
centerC2×C4#6291
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC22×C33×C4C22
socleC22×C33C22⋊C4
prime radicalC2×C32×#235C2
cosocleC2×C33×C4C23
hypercenterC22×C4D3×C3⋊D3
hypocenterC33C2×#104
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×#88855C2semidirect
3. C2×C3×C4×C3⋊Dic3C2semidirect
4. #11722044C2direct (N × H)
12. C32×#235C4semidirect
14. C2×C32×D12C4semidirect
16. C2×C32×Dic6C4semidirect
18. #88855C22semidirect
21. C3×C4×C3⋊Dic3C22semidirect
24. C2×C4×C3⋊Dic3D3semidirect
45. C32×D12C2×C4semidirect
49. C32×Dic6C2×C4semidirect
69. C3×#235Dic3semidirect
71. C2×C3×D12Dic3semidirect
73. C2×C3×Dic6Dic3semidirect
75. C4×C3⋊Dic3C2×D3semidirect
148. C3×D12C2×Dic3semidirect
152. C3×Dic6C2×Dic3semidirect
159. C2×C33#104semidirect
178. #235C3⋊Dic3semidirect
180. C2×D12C3⋊Dic3semidirect
182. C2×Dic6C3⋊Dic3semidirect
232. C33C2×#104semidirect
273. D12C2×C3⋊Dic3semidirect
277. Dic6C2×C3⋊Dic3semidirect
290. C2×C32#805semidirect
291. C2×C32#837semidirect
343. C32C2×#805semidirect
344. C32C2×#837semidirect
374. C2×C3#63569semidirect
375. C2×C3#63664semidirect
392. C3C2×#63569semidirect
393. C3C2×#63664semidirect
397. C2#11722044direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×#2352
C2×#888552
C2×C3×C4×C3⋊Dic32
#117220442
C2×#636643
C2×#635693