Finite Group Databasebeta

← C2×#11722826 #13170517

Subgroups — C2×#11722826 #13170517

Order 1728.

1298 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89042C211yesmaximal, characteristic
2C2×#88787C212yesmaximal, complemented
3C2×C3⋊D3×C3⋊C8C211yesmaximal, characteristic, complemented
4C2×C3×#3712C211yesmaximal, characteristic, complemented
5C2×C3×#3710C212yesmaximal, complemented
6#11722826C218yesmaximal, direct product factor
7C2×#63804—32nomaximal
8C2×#63803—31nomaximal
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C2×C3×C4×C3⋊D3C2212yes—
11C2×#6296C411yescharacteristic
12C2×#6295C411yescharacteristic
13C2×#6295C412yes—
14#89042C414yes—
15#89042C2214yes—
16C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18C2×C32×C3⋊C8C2212yes—
19#88787C2218yescomplemented
20C3⋊D3×C3⋊C8C2214yescomplemented
21C3×#3712C2214yescomplemented
22C3×#3710C2218yescomplemented
23C2×#3798—31no—
24C2×#3798—32no—
25C2×#3743—32no—
26C2×C8×C3⋊D3—31no—
27C2×C3×ΓL(1,𝔽25)—32no—
28C2×C3×D3×C8—32no—
29C2×C3×D3×C8—32no—
30C2×#3712D311yescharacteristic, complemented
31C2×#3711—32no—
32C2×#3711—32no—
33C2×#3711—62no—
34C2×#3710D312yescomplemented
35C2×D3×C3⋊C8—32no—
36C2×D3×C3⋊C8—61no—
37#63804—316no—
38#63803—38no—
39C22×C3×C3⋊D3C2×C411yescharacteristic
40C22×C3×C3⋊D3C2×C412yes—
41C2×C33×C4C2311yescharacteristic
42C2×C3×C3⋊Dic3C2×C411yescharacteristic
43C2×C3×C3⋊Dic3C2×C412yes—
44C3×C4×C3⋊D3C2×C412yes—
45C3×C4×C3⋊D3C2×C414yes—
46C3×C4×C3⋊D3C2314yes—
47C3×C4×C3⋊D3C2312yes—
48#6296C2×C414yes—
49#6295C2×C418yes—
50#6295C2×C414yes—
51C3⋊(C3⋊(C3⋊C8))C2312yes—
52C32×C3⋊C8C2312yes—
53C32×C3⋊C8C2314yes—
54C2×#907—91no—
55C2×#901—92no—
56C2×C4×C3⋊D3C2×D312yes—
57C2×C4×C3⋊D3C2×D311yescharacteristic
58C2×C3×C4×D3—32no—
59C2×C3×C4×D3—32no—
60C2×C3×C4×D3—32no—
61C2×C3×C4×D3—61no—
62C2×C3×C4×D3—62no—
63C2×#627—31no—
64C2×#627—32no—
65C2×#626—32no—
66C2×#626—32no—
67C2×#626—32no—
68C2×#625—31no—
69C2×#625—32no—
70#3798—34no—
71#3798—38no—
72#3798—34no—
73#3798—38no—
74C2×C32×C8—31no—
75C2×C32×C8—32no—
76C2×C3⋊(C3⋊C8)—32no—
77C2×C3⋊(C3⋊C8)—31no—
78C2×C3⋊(C3⋊C8)—61no—
79C2×C3⋊(C3⋊C8)—62no—
80C2×C3⋊(C3⋊C8)—121no—
81#3743—38no—
82C8×C3⋊D3—34no—
83C2×C3×C3⋊C8—21no—
84C2×C3×C3⋊C8—22no—
85C2×C3×C3⋊C8C2×D312yes—
86C2×C3×C3⋊C8C2×D312yes—
87C2×C3×C3⋊C8C2×D312yes—
88C3×ΓL(1,𝔽25)—38no—
89C3×D3×C8—38no—
90C3×D3×C8—38no—
91#3712C2×D314yescomplemented
92#3711—38no—
93#3711—38no—
94#3711—68no—
95#3710C2×D318yescomplemented
96D3×C3⋊C8—38no—
97D3×C3⋊C8—64no—
98C22×C33C22×C411yescharacteristic
99C2×C3×C3⋊D3—22no—
100C2×C3×C3⋊D3—24no—
101C2×C3×C3⋊D3C22×C412yes—
102C2×C3×C3⋊D3C22×C414yes—
103C33×C4C22×C411yescharacteristic
104C33×C4C2411yescharacteristic
105C3×C3⋊Dic3C22×C414yes—
106C3×C3⋊Dic3C22×C412yes—
107C2×#235—92no—
108C2×#235—91no—
109C2×C3×M16—91no—
110C22×C3×C8—92no—
111C2×#208—92no—
112C22×C3⋊C8—91no—
113#907—98no—
114#901—916no—
115C2×ΓL(1,𝔽25)—32no—
116C2×ΓL(1,𝔽25)—92no—
117C2×ΓL(1,𝔽25)—92no—
118C2×ΓL(1,𝔽25)—182no—
119C2×D3×C8—32no—
120C2×D3×C8—32no—
121C2×D3×C8—92no—
122C2×D3×C8—181no—
123C22×C3⋊D3C4×D311yescharacteristic
124C22×C3⋊D3C4×D312yes—
125C22×C3×D3—32no—
126C22×C3×D3—32no—
127C22×C3×D3—32no—
128C22×C3×D3—61no—
129C22×C3×D3—62no—
130C2×C32×C4—21no—
131C2×C32×C4—22no—
132C2×C32×C4—41no—
133C2×C32×C4C22×D312yes—
134C2×C32×C4C22×D311yescharacteristic
135C2×C3⋊Dic3C4×D312yes—
136C2×C3⋊Dic3C4×D311yescharacteristic
137C4×C3⋊D3C4×D314yes—
138C4×C3⋊D3C4×D312yes—
139C4×C3⋊D3C22×D314yes—
140C4×C3⋊D3C22×D312yes—
141C2×C3×Dic3—32no—
142C2×C3×Dic3—32no—
143C2×C3×Dic3—32no—
144C2×C3×Dic3—61no—
145C2×C3×Dic3—62no—
146C3×C4×D3—34no—
147C3×C4×D3—34no—
148C3×C4×D3—34no—
149C3×C4×D3—34no—
150C3×C4×D3—34no—
151C3×C4×D3—34no—
152C3×C4×D3—62no—
153C3×C4×D3—64no—
154C3×C4×D3—62no—
155C3×C4×D3—64no—
156#627—34no—
157#627—38no—
158#626—38no—
159#626—38no—
160#626—38no—
161#625—34no—
162#625—38no—
163C32×C8—32no—
164C32×C8—34no—
165C3⋊(C3⋊C8)—34no—
166C3⋊(C3⋊C8)—32no—
167C3⋊(C3⋊C8)—62no—
168C3⋊(C3⋊C8)—64no—
169C3⋊(C3⋊C8)—122no—
170C3×C3⋊C8—22no—
171C3×C3⋊C8—24no—
172C3×C3⋊C8C22×D314yes—
173C3×C3⋊C8C22×D314yes—
174C3×C3⋊C8C22×D314yes—
175C2×#131—271noSylow
176C2×C33#13112yescomplemented
177C2×C33C23×C411yescharacteristic
178C3×C3⋊D3—22no—
179C3×C3⋊D3—24no—
180C22×C3×C4—91no—
181C22×C3×C4—92no—
182C2×#67—92no—
183C2×#67—92no—
184C2×#67—92no—
185#235—98no—
186#235—94no—
187#235—98no—
188#235—94no—
189C2×D12—92no—
190C2×D12—91no—
191C2×C4×D3—32no—
192C2×C4×D3—32no—
193C2×C4×D3—32no—
194C2×C4×D3—61no—
195C2×C4×D3—62no—
196C2×Dic6—92no—
197C2×Dic6—91no—
198C3×M16—94no—
199C2×C3×C8—32no—
200C2×C3×C8—32no—
201C2×C3×C8—32no—
202C2×C3×C8—61no—
203C2×C3×C8—62no—
204C2×C3×C8—98no—
205#208—98no—
206C2×C3⋊C8—92no—
207C2×C3⋊C8—91no—
208C2×C3⋊C8—94no—
209C2×C3⋊C8—181no—
210C2×C3⋊C8—182no—
211C2×C3⋊C8—361no—
212C2×C3⋊C8D3211yescharacteristic
213C2×C3⋊C8D3212yes—
214ΓL(1,𝔽25)—38no—
215ΓL(1,𝔽25)—98no—
216ΓL(1,𝔽25)—98no—
217ΓL(1,𝔽25)—188no—
218D3×C8—38no—
219D3×C8—38no—
220D3×C8—98no—
221D3×C8—184no—
222C22×C32—21no—
223C22×C32—22no—
224C22×C32—41no—
225C22×C32C2×C4×D312yes—
226C22×C32C2×C4×D311yescharacteristic
227C2×C3⋊D3—22no—
228C2×C3⋊D3—24no—
229C2×C3⋊D3C2×C4×D312yes—
230C2×C3⋊D3C2×C4×D314yes—
231C2×C3×D3—34no—
232C2×C3×D3—34no—
233C2×C3×D3—34no—
234C2×C3×D3—62no—
235C2×C3×D3—64no—
236C2×C3×D3—64no—
237C2×C3×D3—64no—
238C2×C3×D3—64no—
239C2×C3×D3—64no—
240C2×C3×D3—124no—
241C32×C4—21no—
242C32×C4—22no—
243C32×C4—21no—
244C32×C4—22no—
245C32×C4—41no—
246C32×C4—41no—
247C32×C4C2×C4×D312yes—
248C32×C4C2×C4×D311yescharacteristic
249C32×C4C23×D312yes—
250C32×C4C23×D311yescharacteristic
251C3⋊Dic3C2×C4×D314yes—
252C3⋊Dic3C2×C4×D312yes—
253C3×Dic3—34no—
254C3×Dic3—34no—
255C3×Dic3—34no—
256C3×Dic3—62no—
257C3×Dic3—64no—
258C2×#41—271no—
259#131—278no—
260C2×M16—91no—
261C2×M16—272no—
262C22×C8—92no—
263C22×C8—271no—
264C33C2×#13111yescharacteristic, complemented, Sylow
265C23×C3—91no—
266C23×C3—92no—
267C22×D3—32no—
268C22×D3—32no—
269C22×D3—32no—
270C22×D3—61no—
271C22×D3—62no—
272C2×C3×C4—21no—
273C2×C3×C4—22no—
274C2×C3×C4—41no—
275C2×C3×C4—91no—
276C2×C3×C4—92no—
277C2×C3×C4—92no—
278C2×C3×C4—94no—
279C2×C3×C4—92no—
280C2×C3×C4—94no—
281C2×C3×C4C2×D3211yescharacteristic
282C2×C3×C4C2×D3212yes—
283#67—98no—
284#67—98no—
285#67—98no—
286C2×Dic3—32no—
287C2×Dic3—32no—
288C2×Dic3—32no—
289C2×Dic3—61no—
290C2×Dic3—62no—
291D12—98no—
292D12—94no—
293C4×D3—34no—
294C4×D3—34no—
295C4×D3—34no—
296C4×D3—34no—
297C4×D3—34no—
298C4×D3—34no—
299C4×D3—62no—
300C4×D3—64no—
301C4×D3—62no—
302C4×D3—64no—
303Dic6—98no—
304Dic6—94no—
305C3×C8—34no—
306C3×C8—34no—
307C3×C8—34no—
308C3×C8—62no—
309C3×C8—64no—
310C3⋊C8—94no—
311C3⋊C8—92no—
312C3⋊C8—182no—
313C3⋊C8—184no—
314C3⋊C8—362no—
315C3⋊C8C2×D3212yes—
316C3⋊C8C2×D3214yes—
317C2×C32—21no—
318C2×C32—22no—
319C2×C32—22no—
320C2×C32—24no—
321C2×C32—42no—
322C2×C32—41no—
323C2×C32#90114yescomplemented
324C2×C32#90712yescomplemented
325C2×C32C22×C4×D312yes—
326C2×C32C22×C4×D311yescharacteristic
327C3⋊D3—22no—
328C3⋊D3—24no—
329C3×D3—64no—
330C3×D3—64no—
331C3×D3—64no—
332C3×D3—64no—
333C3×D3—124no—
334#41—274no—
335#41—274no—
336C2×Q8—271no—
337C2×D4—272no—
338C2×D4—271no—
339C22×C4—92no—
340C22×C4—91no—
341M16—94no—
342M16—278no—
343C2×C8—31no—
344C2×C8—32no—
345C2×C8—98no—
346C2×C8—274no—
347C2×C8—271no—
348C22×C3—21no—
349C22×C3—22no—
350C22×C3—41no—
351C22×C3—92no—
352C22×C3—94no—
353C22×C3—182no—
354C22×C3—184no—
355C22×C3C4×D3211yescharacteristic
356C22×C3C4×D3212yes—
357C2×D3—34no—
358C2×D3—34no—
359C2×D3—34no—
360C2×D3—62no—
361C2×D3—64no—
362C2×D3—64no—
363C2×D3—64no—
364C2×D3—64no—
365C2×D3—64no—
366C2×D3—124no—
367C3×C4—21no—
368C3×C4—22no—
369C3×C4—21no—
370C3×C4—22no—
371C3×C4—41no—
372C3×C4—41no—
373C3×C4—92no—
374C3×C4—94no—
375C3×C4C4×D3211yescharacteristic
376C3×C4C4×D3212yes—
377C3×C4C22×D3211yescharacteristic
378C3×C4C22×D3212yes—
379Dic3—34no—
380Dic3—34no—
381Dic3—34no—
382Dic3—62no—
383Dic3—64no—
384C32—21no—
385C32—22no—
386C32—41no—
387C32C2×#90112yescomplemented
388C32C2×#90711yescharacteristic, complemented
389C23—91no—
390C23—92no—
391Q8—274no—
392D4—278no—
393D4—274no—
394C2×C4—92no—
395C2×C4—91no—
396C2×C4—94no—
397C2×C4—92no—
398C2×C4—94no—
399C2×C4—92no—
400C2×C4D3311yescharacteristic
401C8—32no—
402C8—34no—
403C8—272no—
404C2×C3—22no—
405C2×C3—24no—
406C2×C3—21no—
407C2×C3—22no—
408C2×C3—42no—
409C2×C3—41no—
410C2×C3—182no—
411C2×C3—184no—
412C2×C3#6380312yescomplemented
413C2×C3#6380414yescomplemented
414C2×C3C2×C4×D3211yescharacteristic
415C2×C3C2×C4×D3212yes—
416D3—64no—
417D3—64no—
418D3—64no—
419D3—64no—
420D3—124no—
421C22—92no—
422C22—94no—
423C22—182no—
424C22—184no—
425C22#8894911yescharacteristic
426C4—94no—
427C4—92no—
428C4#8894911yescharacteristic
429C4C2×D3311yescharacteristic
430C3—21no—
431C3—22no—
432C3—41no—
433C3C2×#6380311yescharacteristic, complemented
434C3C2×#6380412yescomplemented
435C2—182no—
436C2—184no—
437C2#1172282612yesdirect product factor
438C2C2×#8894911yescharacteristic
subgroupG/H
center, hypercenterC2×C4D33
commutatorC2×C33C23×C4
FrattiniC4C2×D33
FittingC2×C33×C4C23
socleC22×C33C22×C4
prime radicalC2×#89042C2
cosocleC33×C4C24
hypocenterC33C2×#131
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×#88787C2semidirect
3. C2×C3⋊D3×C3⋊C8C2semidirect
4. C2×C3×#3712C2semidirect
5. C2×C3×#3710C2semidirect
6. #11722826C2direct (N × H)
19. #88787C22semidirect
20. C3⋊D3×C3⋊C8C22semidirect
21. C3×#3712C22semidirect
22. C3×#3710C22semidirect
30. C2×#3712D3semidirect
34. C2×#3710D3semidirect
91. #3712C2×D3semidirect
95. #3710C2×D3semidirect
176. C2×C33#131semidirect
264. C33C2×#131semidirect
323. C2×C32#901semidirect
324. C2×C32#907semidirect
387. C32C2×#901semidirect
388. C32C2×#907semidirect
412. C2×C3#63803semidirect
413. C2×C3#63804semidirect
433. C3C2×#63803semidirect
434. C3C2×#63804semidirect
437. C2#11722826direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#890422
C2×#887872
C2×C3⋊D3×C3⋊C82
C2×C3×#37122
C2×C3×#37102
#117228262
C2×#638043
C2×#638033