Finite Group Databasebeta

← C2×#11724591 #13181623

Subgroups — C2×#11724591 #13181623

Order 1728.

5420 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89080C211yesmaximal, characteristic, complemented
2C2×C3×Q8×C3⋊D3C211yesmaximal, characteristic, complemented
3C2×C32×D3×Q8C211yesmaximal, characteristic, complemented
4#11724591C2116yesmaximal, direct product factor
5C2×#89022C213yesmaximal, complemented
6C2×#89020C213yesmaximal, complemented
7C2×#89019C213yesmaximal, complemented
8C2×#89018C213yesmaximal, complemented
9C2×#64364—31nomaximal
10C2×#64316—34nomaximal
11C2×C33×Q8C2211yescharacteristic
12#89080C2218yescomplemented
13C2×C4⋊(C3⋊(C3⋊D3))C2213yes—
14C2×C4×C3⋊(C3⋊D3)C2213yes—
15C3×Q8×C3⋊D3C2218yescomplemented
16C2×C3×C4×C3⋊D3C2213yes—
17C2×C3×#634C2213yes—
18C32×D3×Q8C2218yescomplemented
19C2×C32×C4×D3C2213yes—
20C2×C32×Dic6C2213yes—
21C2×#6293C2213yes—
22C2×#6292C2213yes—
23C2×#6291C2213yes—
24C2×#6290C2211yescharacteristic
25C2×#6289C2216yes—
26#89022C22124yescomplemented
27#89020C22124yescomplemented
28#89019C22124yescomplemented
29#89018C22124yescomplemented
30C22×C32×Q8—31no—
31#64364—316no—
32C2×#3832—31no—
33C2×#3832—33no—
34C2×#3832—34no—
35C2×#3832—64no—
36C2×Q8×C3⋊D3—33no—
37C2×Q8×C3⋊D3D311yescharacteristic, complemented
38C2×#3828—33no—
39C2×#3828—33no—
40C2×C3×D3×Q8—34no—
41C2×C3×D3×Q8D314yescomplemented
42#64316—364no—
43C2×#3799—312no—
44C2×#3799—312no—
45C2×#3798—312no—
46C2×#3797—312no—
47C22×C3⋊(C3⋊D3)C2313yes—
48C22×C3×C3⋊D3C2311yescharacteristic
49C22×C32×D3C2311yescharacteristic
50C33×Q8C2314yes—
51C2×C33×C4C2313yes—
52C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
53C4⋊(C3⋊(C3⋊D3))C23112yes—
54C4×C3⋊(C3⋊D3)C23112yes—
55C2×C3×C3⋊Dic3C2313yes—
56C3×C4×C3⋊D3C23112yes—
57C3×#634C23112yes—
58C2×C32×Dic3C2313yes—
59C32×C4×D3C23112yes—
60C32×Dic6C23112yes—
61#6293C23112yes—
62#6292C23112yes—
63#6291C23112yes—
64#6290C2314yes—
65#6289C23124yes—
66C2×#1007—94no—
67C2×#1007—91no—
68C2×C32×Q8—24no—
69C2×C32×Q8—38no—
70C2×C32×Q8—33no—
71C2×C32×Q8C2×D314yes—
72C2×C32×Q8C2×D311yescharacteristic
73C22×C32×C4—33no—
74C2×#638—31no—
75C2×#638—33no—
76#3832—38no—
77#3832—332no—
78#3832—324no—
79#3832—632no—
80Q8×C3⋊D3—324no—
81Q8×C3⋊D3C2×D318yescomplemented
82#3828—324no—
83#3828—324no—
84C2×C4⋊(C3⋊D3)—33no—
85C2×C4⋊(C3⋊D3)—33no—
86C2×C4⋊(C3⋊D3)—312no—
87C2×C4⋊(C3⋊D3)—612no—
88C2×C4×C3⋊D3—33no—
89C2×C4×C3⋊D3—312no—
90C2×C4×C3⋊D3—36no—
91C2×C4×C3⋊D3—612no—
92C2×C4×C3⋊D3C2×D313yes—
93C2×#634—33no—
94C2×#634C2×D313yes—
95C3×D3×Q8—332no—
96C3×D3×Q8C2×D3132yescomplemented
97C2×C3×C4×D3—312no—
98C2×C3×C4×D3C2×D3112yes—
99C2×C3×Dic6—312no—
100C2×C3×Dic6C2×D3112yes—
101C2×#627—312no—
102C2×#626—312no—
103C2×#626—312no—
104C2×#625—34no—
105C2×#624—324no—
106#3799—396no—
107#3799—396no—
108#3798—396no—
109#3797—396no—
110C22×C33C2411yescharacteristic
111C2×C3⋊(C3⋊D3)—26no—
112C2×C3⋊(C3⋊D3)C2416yes—
113C2×C3×C3⋊D3—22no—
114C2×C3×C3⋊D3C2412yes—
115C2×C32×D3—22no—
116C2×C32×D3C2412yes—
117C33×C4C2416yes—
118C3⋊(C3⋊Dic3)C2412yes—
119C3×C3⋊Dic3C2416yes—
120C32×Dic3C2416yes—
121C22×C3×Q8—34no—
122C22×C3×Q8—91no—
123#1007—964no—
124#1007—916no—
125C2×#239—912no—
126C2×#239—94no—
127C2×#239—91no—
128C2×#239—93no—
129C2×#239—184no—
130C2×D3×Q8—34no—
131C2×D3×Q8—912no—
132C2×D3×Q8—93no—
133C2×D3×Q8C3⋊D311yescharacteristic, complemented
134C2×#235—912no—
135C2×#235—912no—
136C2×#235—93no—
137C2×#235—93no—
138C23×C32—31no—
139C22×C3⋊D3—33no—
140C22×C3⋊D3—312no—
141C22×C3⋊D3—612no—
142C22×C3⋊D3C22×D311yescharacteristic
143C22×C3×D3—34no—
144C22×C3×D3C22×D314yes—
145C32×Q8—216no—
146C32×Q8—312no—
147C32×Q8C22×D3116yes—
148C32×Q8C22×D314yes—
149C2×C32×C4—212no—
150C2×C32×C4—33no—
151C2×C32×C4—312no—
152C2×C32×C4C22×D3112yes—
153C2×C32×C4C22×D313yes—
154#638—34no—
155#638—312no—
156C2×C3⋊Dic3—31no—
157C2×C3⋊Dic3—34no—
158C2×C3⋊Dic3—64no—
159C2×C3⋊Dic3C22×D313yes—
160C4⋊(C3⋊D3)—312no—
161C4⋊(C3⋊D3)—312no—
162C4⋊(C3⋊D3)—348no—
163C4⋊(C3⋊D3)—648no—
164C4×C3⋊D3—312no—
165C4×C3⋊D3—324no—
166C4×C3⋊D3—348no—
167C4×C3⋊D3—648no—
168C4×C3⋊D3C22×D3112yes—
169#634—312no—
170#634C22×D3112yes—
171C2×C3×Dic3—312no—
172C2×C3×Dic3C22×D3112yes—
173C3×C4×D3—348no—
174C3×C4×D3C22×D3148yes—
175C3×Dic6—348no—
176C3×Dic6C22×D3148yes—
177#627—348no—
178#626—348no—
179#626—348no—
180#625—316no—
181#624—396no—
182C2×2−1+4—271noSylow
183C2×C332−1+412yescomplemented
184C2×C33C2511yescharacteristic
185C3⋊(C3⋊D3)—26no—
186C3×C3⋊D3—22no—
187C32×D3—22no—
188C2×C3×Q8—24no—
189C2×C3×Q8—332no—
190C2×C3×Q8—312no—
191C2×C3×Q8—98no—
192C2×C3×Q8—93no—
193C2×C3×Q8D3214yes—
194C2×C3×Q8C2×C3⋊D311yescharacteristic
195C22×C3×C4—312no—
196C22×C3×C4—93no—
197C2×#67—912no—
198C2×#67—94no—
199C2×#67—91no—
200C2×#67—93no—
201#239—996no—
202#239—932no—
203#239—98no—
204#239—924no—
205#239—1832no—
206D3×Q8—332no—
207D3×Q8—996no—
208D3×Q8—924no—
209D3×Q8C2×C3⋊D318yescomplemented
210#235—996no—
211#235—996no—
212#235—924no—
213#235—924no—
214C2×D12—912no—
215C2×D12—93no—
216C2×D12—912no—
217C2×D12—93no—
218C2×D12—1812no—
219C2×C4×D3—312no—
220C2×C4×D3—924no—
221C2×C4×D3—912no—
222C2×C4×D3—93no—
223C2×C4×D3—96no—
224C2×C4×D3—1812no—
225C2×C4×D3C2×C3⋊D313yes—
226C2×Dic6—312no—
227C2×Dic6—912no—
228C2×Dic6—93no—
229C2×Dic6C2×C3⋊D313yes—
230C22×C32—24no—
231C22×C32—32no—
232C22×C32—62no—
233C22×C32C23×D314yes—
234C22×C32C23×D311yescharacteristic
235C2×C3⋊D3—22no—
236C2×C3⋊D3—36no—
237C2×C3⋊D3—324no—
238C2×C3⋊D3—624no—
239C2×C3⋊D3—66no—
240C2×C3⋊D3—624no—
241C2×C3⋊D3—1224no—
242C2×C3⋊D3C23×D312yes—
243C2×C3×D3—28no—
244C2×C3×D3—38no—
245C2×C3×D3—68no—
246C2×C3×D3C23×D318yes—
247C32×C4—224no—
248C32×C4—36no—
249C32×C4C23×D3124yes—
250C32×C4C23×D316yes—
251C3⋊Dic3—32no—
252C3⋊Dic3—38no—
253C3⋊Dic3—68no—
254C3⋊Dic3C23×D316yes—
255C3×Dic3—324no—
256C3×Dic3C23×D3124yes—
2572−1+4—2716no—
258C2×#41—273no—
259C2×#41—273no—
260C2×#41—273no—
261C2×#41—271no—
262C22×Q8—31no—
263C22×Q8—91no—
264C22×Q8—273no—
265C33C2×2−1+411yescharacteristic, complemented, Sylow
266C23×C3—34no—
267C23×C3—91no—
268C22×D3—34no—
269C22×D3—912no—
270C22×D3—93no—
271C22×D3—1812no—
272C22×D3C22×C3⋊D311yescharacteristic
273C3×Q8—216no—
274C3×Q8—348no—
275C3×Q8—912no—
276C3×Q8C2×D32116yes—
277C3×Q8C22×C3⋊D314yes—
278C2×C3×C4—212no—
279C2×C3×C4—312no—
280C2×C3×C4—348no—
281C2×C3×C4—93no—
282C2×C3×C4—912no—
283C2×C3×C4C2×D32112yes—
284C2×C3×C4C22×C3⋊D313yes—
285#67—948no—
286#67—916no—
287#67—94no—
288#67—912no—
289C2×Dic3—312no—
290C2×Dic3—94no—
291C2×Dic3—91no—
292C2×Dic3—184no—
293C2×Dic3C22×C3⋊D313yes—
294D12—948no—
295D12—912no—
296D12—948no—
297D12—912no—
298D12—1848no—
299C4×D3—348no—
300C4×D3—996no—
301C4×D3—948no—
302C4×D3—912no—
303C4×D3—924no—
304C4×D3—1848no—
305C4×D3C22×C3⋊D3112yes—
306Dic6—348no—
307Dic6—948no—
308Dic6—912no—
309Dic6C22×C3⋊D3112yes—
310C2×C32—28no—
311C2×C32—24no—
312C2×C32—62no—
313C2×C32#100718yescomplemented
314C2×C32#100712yescomplemented
315C2×C32C24×D314yes—
316C2×C32C24×D311yescharacteristic
317C3⋊D3—22no—
318C3⋊D3—66no—
319C3⋊D3—624no—
320C3⋊D3—1224no—
321C3×D3—28no—
322C3×D3—68no—
323#41—2724no—
324#41—2724no—
325#41—2724no—
326#41—278no—
327C2×Q8—38no—
328C2×Q8—33no—
329C2×Q8—98no—
330C2×Q8—93no—
331C2×Q8—273no—
332C2×Q8—2724no—
333C2×Q8D3×C3⋊D311yescharacteristic
334C2×D4—273no—
335C2×D4—273no—
336C2×D4—271no—
337C2×D4—273no—
338C22×C4—33no—
339C22×C4—93no—
340C22×C4—276no—
341C22×C4—273no—
342C22×C3—24no—
343C22×C3—38no—
344C22×C3—68no—
345C22×C3—92no—
346C22×C3—182no—
347C22×C3C22×D3214yes—
348C22×C3C23×C3⋊D311yescharacteristic
349C2×D3—22no—
350C2×D3—38no—
351C2×D3—68no—
352C2×D3—924no—
353C2×D3—96no—
354C2×D3—1824no—
355C2×D3—1824no—
356C2×D3—186no—
357C2×D3—3624no—
358C2×D3C23×C3⋊D312yes—
359C3×C4—224no—
360C3×C4—324no—
361C3×C4—96no—
362C3×C4C22×D32124yes—
363C3×C4C23×C3⋊D316yes—
364Dic3—324no—
365Dic3—98no—
366Dic3—92no—
367Dic3—188no—
368Dic3C23×C3⋊D316yes—
369C32—24no—
370C32C2×#100714yescomplemented
371C32C2×#100711yescharacteristic, complemented
372C23—31no—
373C23—91no—
374C23—273no—
375Q8—312no—
376Q8—912no—
377Q8—2712no—
378Q8C2×D3×C3⋊D314yes—
379D4—2712no—
380D4—2712no—
381D4—274no—
382D4—2712no—
383C2×C4—33no—
384C2×C4—312no—
385C2×C4—93no—
386C2×C4—912no—
387C2×C4—2724no—
388C2×C4—271no—
389C2×C4—2712no—
390C2×C4C2×D3×C3⋊D313yes—
391C2×C3—28no—
392C2×C3—24no—
393C2×C3—68no—
394C2×C3—182no—
395C2×C3#6431618yescomplemented
396C2×C3#6436412yescomplemented
397C2×C3C23×D3214yes—
398C2×C3C24×C3⋊D311yescharacteristic
399D3—22no—
400D3—68no—
401D3—1824no—
402D3—186no—
403D3—3624no—
404C22—32no—
405C22—62no—
406C22—92no—
407C22—182no—
408C22—276no—
409C22—546no—
410C22C22×D3×C3⋊D311yescharacteristic
411C4—36no—
412C4—96no—
413C4—272no—
414C4C22×D3×C3⋊D316yes—
415C3—24no—
416C3C2×#6431614yescomplemented
417C3C2×#6436411yescharacteristic, complemented
418C2—62no—
419C2—182no—
420C2—546no—
421C2#1172459112yesdirect product factor
422C2C23×D3×C3⋊D311yescharacteristic
subgroupG/H
centerC22C22×D3×C3⋊D3
commutator, cosocleC2×C33C25
FrattiniC2C23×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C24
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33C2×2−1+4
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#89080C2semidirect
2. C2×C3×Q8×C3⋊D3C2semidirect
3. C2×C32×D3×Q8C2semidirect
4. #11724591C2direct (N × H)
5. C2×#89022C2semidirect
6. C2×#89020C2semidirect
7. C2×#89019C2semidirect
8. C2×#89018C2semidirect
12. #89080C22semidirect
15. C3×Q8×C3⋊D3C22semidirect
18. C32×D3×Q8C22semidirect
26. #89022C22semidirect
27. #89020C22semidirect
28. #89019C22semidirect
29. #89018C22semidirect
37. C2×Q8×C3⋊D3D3semidirect
41. C2×C3×D3×Q8D3semidirect
81. Q8×C3⋊D3C2×D3semidirect
96. C3×D3×Q8C2×D3semidirect
133. C2×D3×Q8C3⋊D3semidirect
183. C2×C332−1+4semidirect
209. D3×Q8C2×C3⋊D3semidirect
265. C33C2×2−1+4semidirect
313. C2×C32#1007semidirect
314. C2×C32#1007semidirect
370. C32C2×#1007semidirect
371. C32C2×#1007semidirect
395. C2×C3#64316semidirect
396. C2×C3#64364semidirect
416. C3C2×#64316semidirect
417. C3C2×#64364semidirect
421. C2#11724591direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#890802
C2×C3×Q8×C3⋊D32
C2×C32×D3×Q82
#117245912
C2×#890222
C2×#890202
C2×#890192
C2×#890182
C2×#643643
C2×#643163