Finite Group Databasebeta

← C2×#11722944 #13172275

Subgroups — C2×#11722944 #13172275

Order 1728.

1538 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6296C211yesmaximal, characteristic
2C2×#88807C212yesmaximal
3C2×Dic3×C3⋊Dic3C211yesmaximal, characteristic
4C2×C3×#3724C211yesmaximal, characteristic
5C2×C3×Dic32C212yesmaximal
6#11722944C218yesmaximal, direct product factor
7C2×C4×#627—31nomaximal
8C2×#63837—32nomaximal
9C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
10C22×C3×C3⋊Dic3C2212yes—
11C22×C3×C3⋊Dic3C2211yescharacteristic
12C22×C32×Dic3C2212yes—
13C22×C32×Dic3C2211yescharacteristic
14C2×#6296C418yescomplemented
15C2×#6296C2214yes—
16#88807C2218yes—
17Dic3×C3⋊Dic3C2214yes—
18C3×#3724C2214yes—
19C3×Dic32C2218yes—
20C22×#627—31no—
21C22×#627—32no—
22C2×#3750—32no—
23C2×C4×C3⋊Dic3—31no—
24C2×C3×Dic3⋊C4—32no—
25C2×C3×C4×Dic3—32no—
26C2×C3×C4×Dic3—32no—
27C2×#3724D311yescharacteristic
28C2×Dic3⋊Dic3—32no—
29C2×Dic3⋊Dic3—32no—
30C2×Dic3⋊Dic3—62no—
31C2×Dic32—32no—
32C2×Dic32—61no—
33C2×Dic32D312yes—
34C4×#627—38no—
35#63837—316no—
36C23×C33C2311yescharacteristic
37C2×C3⋊(C3⋊Dic3)—22no—
38C2×C3⋊(C3⋊Dic3)C2312yes—
39C2×C3×C3⋊Dic3C2×C418yes—
40C2×C3×C3⋊Dic3C2×C414yes—
41C2×C3×C3⋊Dic3C2314yes—
42C2×C3×C3⋊Dic3C2312yes—
43C2×C32×Dic3—24no—
44C2×C32×Dic3Q814yescomplemented
45C2×C32×Dic3C2314yes—
46C2×C32×Dic3C2312yes—
47#6296C2×C4116yescomplemented
48C2×#887—91no—
49C2×C4×Dic6—92no—
50C22×C32×C4—31no—
51C22×C32×C4—32no—
52C22×C3⋊Dic3—31no—
53C22×C3⋊Dic3—32no—
54C22×C3⋊Dic3—61no—
55C22×C3⋊Dic3—62no—
56C22×C3⋊Dic3—121no—
57C22×C3⋊Dic3C2×D311yescharacteristic
58C22×C3⋊Dic3C2×D312yes—
59C22×C3×Dic3—21no—
60C22×C3×Dic3—22no—
61C22×C3×Dic3—32no—
62C22×C3×Dic3—32no—
63C22×C3×Dic3—32no—
64C22×C3×Dic3—61no—
65C22×C3×Dic3—62no—
66C22×C3×Dic3C2×D312yes—
67C22×C3×Dic3C2×D312yes—
68C22×C3×Dic3C2×D312yes—
69C2×#627—38no—
70C2×#627—316no—
71C2×#627—38no—
72C2×#627—34no—
73#3750—38no—
74C4×C3⋊Dic3—34no—
75C3×Dic3⋊C4—38no—
76C3×C4×Dic3—38no—
77C3×C4×Dic3—38no—
78#3724C2×D314yes—
79Dic3⋊Dic3—38no—
80Dic3⋊Dic3—38no—
81Dic3⋊Dic3—68no—
82Dic32—38no—
83Dic32—64no—
84Dic32C2×D318yes—
85C22×C33C22×C412yes—
86C22×C33C2×Q812yes—
87C22×C33#4112yes—
88C22×C33C2411yescharacteristic
89C3⋊(C3⋊Dic3)—22no—
90C3×C3⋊Dic3C22×C418yes—
91C3×C3⋊Dic3C22×C414yes—
92C32×Dic3—24no—
93C32×Dic3C2×Q814yescomplemented
94C22×Dic6—92no—
95C22×Dic6—91no—
96C2×C3×C4⋊C4—91no—
97C2×C3×C42—92no—
98C2×C4⋊Dic3—92no—
99C2×Dic3⋊C4—32no—
100C2×Dic3⋊C4—92no—
101C2×Dic3⋊C4—92no—
102C2×Dic3⋊C4—182no—
103C2×C4×Dic3—32no—
104C2×C4×Dic3—32no—
105C2×C4×Dic3—92no—
106C2×C4×Dic3—91no—
107C2×C4×Dic3—181no—
108#887—98no—
109C4×Dic6—916no—
110C23×C32—22no—
111C23×C32—21no—
112C23×C32—41no—
113C23×C32C22×D312yes—
114C23×C32C22×D311yescharacteristic
115C2×C32×C4—34no—
116C2×C32×C4—32no—
117C2×C32×C4—34no—
118C2×C32×C4—64no—
119C2×C3⋊Dic3—34no—
120C2×C3⋊Dic3—32no—
121C2×C3⋊Dic3—62no—
122C2×C3⋊Dic3—64no—
123C2×C3⋊Dic3—62no—
124C2×C3⋊Dic3—64no—
125C2×C3⋊Dic3—64no—
126C2×C3⋊Dic3—122no—
127C2×C3⋊Dic3—124no—
128C2×C3⋊Dic3—124no—
129C2×C3⋊Dic3C4×D318yes—
130C2×C3⋊Dic3C4×D314yes—
131C2×C3⋊Dic3C22×D314yes—
132C2×C3⋊Dic3C22×D312yes—
133C2×C3×Dic3—24no—
134C2×C3×Dic3—24no—
135C2×C3×Dic3—24no—
136C2×C3×Dic3—22no—
137C2×C3×Dic3—24no—
138C2×C3×Dic3—38no—
139C2×C3×Dic3—38no—
140C2×C3×Dic3—38no—
141C2×C3×Dic3—34no—
142C2×C3×Dic3—34no—
143C2×C3×Dic3—34no—
144C2×C3×Dic3—44no—
145C2×C3×Dic3—62no—
146C2×C3×Dic3—64no—
147C2×C3×Dic3—64no—
148C2×C3×Dic3—68no—
149C2×C3×Dic3Dic618yescomplemented
150C2×C3×Dic3C22×D314yes—
151C2×C3×Dic3C22×D314yes—
152C2×C3×Dic3C22×D314yes—
153#627—332no—
154#627—316no—
155C2×C4×Q8—271noSylow
156C2×C33C4×Q814yescomplemented
157C2×C33C23×C411yescharacteristic
158C2×C33C22×Q811yescharacteristic
159C2×C33C2×#4111yescharacteristic
160C22×C3×C4—32no—
161C22×C3×C4—32no—
162C22×C3×C4—32no—
163C22×C3×C4—61no—
164C22×C3×C4—62no—
165C22×C3×C4—91no—
166C22×C3×C4—92no—
167C22×Dic3—32no—
168C22×Dic3—32no—
169C22×Dic3—32no—
170C22×Dic3—61no—
171C22×Dic3—62no—
172C22×Dic3—91no—
173C22×Dic3—92no—
174C22×Dic3—181no—
175C22×Dic3—182no—
176C22×Dic3—361no—
177C22×Dic3D3212yes—
178C22×Dic3D3211yescharacteristic
179C2×Dic6—98no—
180C2×Dic6—94no—
181C2×Dic6—916no—
182C2×Dic6—98no—
183C3×C4⋊C4—94no—
184C3×C42—98no—
185C4⋊Dic3—98no—
186Dic3⋊C4—38no—
187Dic3⋊C4—98no—
188Dic3⋊C4—98no—
189Dic3⋊C4—188no—
190C4×Dic3—38no—
191C4×Dic3—38no—
192C4×Dic3—98no—
193C4×Dic3—94no—
194C4×Dic3—184no—
195C22×C32—22no—
196C22×C32—21no—
197C22×C32—24no—
198C22×C32—22no—
199C22×C32—24no—
200C22×C32—22no—
201C22×C32—24no—
202C22×C32—22no—
203C22×C32—42no—
204C22×C32—42no—
205C22×C32—42no—
206C22×C32—41no—
207C22×C32C2×Dic614yes—
208C22×C32C2×C4×D314yes—
209C22×C32C2×C4×D312yes—
210C22×C32#23514yes—
211C22×C32#23712yes—
212C22×C32D3×Q812yes—
213C22×C32C23×D312yes—
214C22×C32C23×D311yescharacteristic
215C32×C4—34no—
216C32×C4—64no—
217C3⋊Dic3—62no—
218C3⋊Dic3—64no—
219C3⋊Dic3—64no—
220C3⋊Dic3—124no—
221C3⋊Dic3—124no—
222C3⋊Dic3C2×C4×D318yes—
223C3⋊Dic3C2×C4×D314yes—
224C3×Dic3—24no—
225C3×Dic3—24no—
226C3×Dic3—24no—
227C3×Dic3—38no—
228C3×Dic3—38no—
229C3×Dic3—38no—
230C3×Dic3—44no—
231C3×Dic3—64no—
232C3×Dic3—68no—
233C3×Dic3C2×Dic618yescomplemented
234C22×Q8—271no—
235C4×Q8—278no—
236C2×C4⋊C4—91no—
237C2×C4⋊C4—272no—
238C2×C42—92no—
239C2×C42—271no—
240C33C2×C4×Q811yescharacteristic, complemented, Sylow
241C23×C3—21no—
242C23×C3—22no—
243C23×C3—41no—
244C23×C3C2×D3212yes—
245C23×C3C2×D3211yescharacteristic
246C2×C3×C4—34no—
247C2×C3×C4—38no—
248C2×C3×C4—34no—
249C2×C3×C4—34no—
250C2×C3×C4—62no—
251C2×C3×C4—64no—
252C2×C3×C4—64no—
253C2×C3×C4—64no—
254C2×C3×C4—64no—
255C2×C3×C4—92no—
256C2×C3×C4—94no—
257C2×C3×C4—98no—
258C2×C3×C4—94no—
259C2×C3×C4—124no—
260C2×Dic3—24no—
261C2×Dic3—34no—
262C2×Dic3—34no—
263C2×Dic3—34no—
264C2×Dic3—38no—
265C2×Dic3—38no—
266C2×Dic3—38no—
267C2×Dic3—64no—
268C2×Dic3—68no—
269C2×Dic3—62no—
270C2×Dic3—64no—
271C2×Dic3—92no—
272C2×Dic3—94no—
273C2×Dic3—182no—
274C2×Dic3—184no—
275C2×Dic3—184no—
276C2×Dic3—182no—
277C2×Dic3—184no—
278C2×Dic3—362no—
279C2×Dic3—364no—
280C2×Dic3—364no—
281C2×Dic3#62714yescomplemented
282C2×Dic3C2×D3214yes—
283C2×Dic3C2×D3212yes—
284Dic6—932no—
285Dic6—916no—
286C2×C32—22no—
287C2×C32—21no—
288C2×C32—21no—
289C2×C32—22no—
290C2×C32—22no—
291C2×C32—21no—
292C2×C32—28no—
293C2×C32—24no—
294C2×C32—44no—
295C2×C32—41no—
296C2×C32—41no—
297C2×C32—41no—
298C2×C32C4×Dic618yescomplemented
299C2×C32#88714yescomplemented
300C2×C32C22×Dic612yes—
301C2×C32C22×C4×D312yes—
302C2×C32C22×C4×D311yescharacteristic
303C2×C32C2×#23512yes—
304C2×C32C2×#23711yescharacteristic
305C2×C32C2×D3×Q811yescharacteristic
306C2×Q8—274no—
307C2×Q8—278no—
308C22×C4—31no—
309C22×C4—32no—
310C22×C4—92no—
311C22×C4—91no—
312C22×C4—271no—
313C4⋊C4—94no—
314C4⋊C4—278no—
315C42—98no—
316C42—274no—
317C22×C3—21no—
318C22×C3—22no—
319C22×C3—24no—
320C22×C3—22no—
321C22×C3—22no—
322C22×C3—24no—
323C22×C3—22no—
324C22×C3—24no—
325C22×C3—41no—
326C22×C3—42no—
327C22×C3—42no—
328C22×C3—42no—
329C22×C3D3×Dic614yes—
330C22×C3#379812yes—
331C22×C3C4×D3214yes—
332C22×C3C4×D3212yes—
333C22×C3#380414yes—
334C22×C3C2×#62712yes—
335C22×C3C22×D3212yes—
336C22×C3C22×D3211yescharacteristic
337C3×C4—38no—
338C3×C4—64no—
339C3×C4—64no—
340C3×C4—64no—
341C3×C4—94no—
342C3×C4—98no—
343C3×C4—124no—
344Dic3—24no—
345Dic3—38no—
346Dic3—38no—
347Dic3—38no—
348Dic3—64no—
349Dic3—68no—
350Dic3—184no—
351Dic3—182no—
352Dic3—184no—
353Dic3—364no—
354Dic3—364no—
355Dic3C2×#62714yescomplemented
356C32—21no—
357C32—22no—
358C32—41no—
359C32C2×C4×Dic612yescomplemented
360C32C2×#88711yescharacteristic, complemented
361C23D3311yescharacteristic
362Q8—2716no—
363C2×C4—32no—
364C2×C4—34no—
365C2×C4—34no—
366C2×C4—64no—
367C2×C4—94no—
368C2×C4—92no—
369C2×C4—98no—
370C2×C4—94no—
371C2×C4—272no—
372C2×C4—542no—
373C2×C3—22no—
374C2×C3—21no—
375C2×C3—21no—
376C2×C3—22no—
377C2×C3—21no—
378C2×C3—22no—
379C2×C3—28no—
380C2×C3—24no—
381C2×C3—41no—
382C2×C3—41no—
383C2×C3—41no—
384C2×C3—44no—
385C2×C3#6383718yescomplemented
386C2×C3C4×#62714yescomplemented
387C2×C3C2×D3×Dic612yes—
388C2×C3C2×#379811yescharacteristic
389C2×C3C2×C4×D3211yescharacteristic
390C2×C3C2×C4×D3212yes—
391C2×C3C2×#380412yes—
392C2×C3C22×#62711yescharacteristic
393C22#8894912yes—
394C22D3×#62712yes—
395C22#8895912yes—
396C22C2×D3311yescharacteristic
397C4—34no—
398C4—64no—
399C4—98no—
400C4—94no—
401C4—542no—
402C3—22no—
403C3—21no—
404C3—41no—
405C3C2×#6383712yescomplemented
406C3C2×C4×#62711yescharacteristic, complemented
407C2#1172294414yesdirect product factor
408C2C2×#8894911yescharacteristic
409C2C2×D3×#62711yescharacteristic
410C2C2×#8895911yescharacteristic
subgroupG/H
center, hypercenterC23D33
commutatorC2×C33C23×C4
FrattiniC22C2×D33
Fitting, socle, prime radicalC23×C33C23
cosocleC22×C33C24
hypocenterC33C2×C4×Q8
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
6. #11722944C2direct (N × H)
14. C2×#6296C4semidirect
44. C2×C32×Dic3Q8semidirect
47. #6296C2×C4semidirect
93. C32×Dic3C2×Q8semidirect
149. C2×C3×Dic3Dic6semidirect
156. C2×C33C4×Q8semidirect
233. C3×Dic3C2×Dic6semidirect
240. C33C2×C4×Q8semidirect
281. C2×Dic3#627semidirect
298. C2×C32C4×Dic6semidirect
299. C2×C32#887semidirect
355. Dic3C2×#627semidirect
359. C32C2×C4×Dic6semidirect
360. C32C2×#887semidirect
385. C2×C3#63837semidirect
386. C2×C3C4×#627semidirect
405. C3C2×#63837semidirect
406. C3C2×C4×#627semidirect
407. C2#11722944direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#62962
C2×#888072
C2×Dic3×C3⋊Dic32
C2×C3×#37242
C2×C3×Dic322
#117229442
C2×C4×#6273
C2×#638373