Finite Group Databasebeta

← C2×#11721074 #13150646

Subgroups — C2×#11721074 #13150646

Order 1728.

1059 conjugacy classes of subgroups.

The subgroup lattice has 455 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6229C211yesmaximal, characteristic, complemented
2C2×#3828⋊C3C211yesmaximal, characteristic
3C2×#3748⋊C3C211yesmaximal, characteristic, complemented
4C2×#88439C212yesmaximal, complemented
5#11721074C218yesmaximal, direct product factor
6C2×#88438C212yesmaximal, complemented
7C2×#63833—31nomaximal
8C2×#63827—31nomaximal
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C22×#1084—21no—
11C2×#6229—22no—
12C2×#6229—21no—
13C2×#6229C2212yes—
14C2×#6229C2214yescomplemented
15C2×#638⋊C3—21no—
16#3828⋊C3—22no—
17#3828⋊C3C2214yes—
18C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
19C2×C4×(C3⋊D3)⋊C3—21no—
20C2×#634⋊C3C2211yescharacteristic
21#3748⋊C3C2214yescomplemented
22C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
23#88439C2218yescomplemented
24#88438C2218yescomplemented
25C2×#3828D311yescharacteristic
26C2×C3×#235—31no—
27C22×C3×Dic6—31no—
28C22×C3×Dic6—31no—
29C22×C3×Dic6—61no—
30C2×#3748D311yescharacteristic, complemented
31C2×C3×#208—31no—
32C2×#3719—32no—
33C2×#3718—32no—
34#63833—38no—
35C2×#3717—32no—
36#63827—38no—
37C2×#3715—32no—
38C23×H(3,𝔽3)D411yescharacteristic
39C22×(C3⋊D3)⋊C3—21no—
40C2×C4×H(3,𝔽3)D411yescharacteristic
41C2×C4×H(3,𝔽3)D412yes—
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43C2×C4×H(3,𝔽3)C2312yes—
44C2×#1084—22no—
45C2×#1084—24no—
46#6229—22no—
47#6229—24no—
48#6229C2314yes—
49#638⋊C3—24no—
50C2×(C3⋊Dic3)⋊C3—21no—
51C4⋊((C3⋊D3)⋊C3)—21no—
52C4⋊((C3⋊D3)⋊C3)C2312yes—
53C4×(C3⋊D3)⋊C3—24no—
54#634⋊C3—21no—
55#634⋊C3C2312yes—
56(C3⋊(C3⋊C8))⋊C3C2314yes—
57C2×#951—91no—
58C2×#942—91no—
59C2×#909—91no—
60C22×C32×C4—21no—
61C22×C32×C4C2×D311yescharacteristic
62C22×C32×C4C2×D311yescharacteristic
63C2×#638—21no—
64#3828—22no—
65#3828C2×D314yes—
66C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
67C2×C4×C3⋊D3—21no—
68C2×#634C2×D311yescharacteristic
69C2×C3×#67—61no—
70C22×C3×Dic3—62no—
71C22×C3×Dic3—61no—
72C22×C3×Dic3—61no—
73C3×#235—34no—
74C3×#235—62no—
75C2×C3×D12—31no—
76C2×C3×C4×D3—61no—
77C2×C3×Dic6—32no—
78C2×C3×Dic6—34no—
79C2×C3×Dic6—32no—
80C2×C3×Dic6—34no—
81C2×C3×Dic6—31no—
82C2×C3×Dic6—62no—
83C2×C3×Dic6—64no—
84C2×C3×Dic6—64no—
85C2×C3×Dic6—62no—
86C2×C3×Dic6—61no—
87C2×C3×Dic6—62no—
88C2×C3×Dic6—61no—
89C2×C3×Dic6—62no—
90#3748C2×D314yescomplemented
91C2×C3⋊(C3⋊C8)C2×D312yes—
92C3×#208—34no—
93C2×C3×C3⋊C8—32no—
94#3719—38no—
95#3718—38no—
96#3717—38no—
97#3715—38no—
98C22×H(3,𝔽3)—22no—
99C22×H(3,𝔽3)C2×D412yes—
100C22×H(3,𝔽3)C2×D411yescharacteristic
101C2×(C3⋊D3)⋊C3—22no—
102C2×(C3⋊D3)⋊C3—42no—
103C4×H(3,𝔽3)C2×D411yescharacteristic
104C4×H(3,𝔽3)C2×D412yes—
105C4×H(3,𝔽3)C2411yescharacteristic
106#1084—24no—
107(C3⋊Dic3)⋊C3—22no—
108C2×C3×#41—91no—
109C22×C3×Q8—91no—
110C2×#235—31no—
111C2×#235—91no—
112C22×Dic6—31no—
113C22×Dic6—31no—
114C22×Dic6—61no—
115C2×C3×M16—91no—
116#951—98no—
117C2×#216—92no—
118C2×#216—92no—
119#942—98no—
120C2×#215—92no—
121C2×#214—92no—
122C2×#208—31no—
123C2×#208—91no—
124#909—98no—
125C2×Dic12—92no—
126C2×#204—92no—
127C23×C32—21no—
128C23×C32D1211yescharacteristic
129C23×C32#6711yescharacteristic
130C22×C3⋊D3—21no—
131C22×C3×D3—61no—
132C2×C32×C4—21no—
133C2×C32×C4—22no—
134C2×C32×C4—21no—
135C2×C32×C4—22no—
136C2×C32×C4D1211yescharacteristic
137C2×C32×C4D1212yes—
138C2×C32×C4#6711yescharacteristic
139C2×C32×C4#6712yes—
140C2×C32×C4C22×D311yescharacteristic
141C2×C32×C4C22×D312yes—
142C2×C32×C4C22×D311yescharacteristic
143C2×C32×C4C22×D312yes—
144#638—24no—
145C2×C3⋊Dic3—21no—
146C4⋊(C3⋊D3)—21no—
147C4⋊(C3⋊D3)C22×D312yes—
148C4×C3⋊D3—24no—
149#634—21no—
150#634C22×D312yes—
151C3×#67—64no—
152C2×C3×Dic3—64no—
153C2×C3×Dic3—68no—
154C2×C3×Dic3—61no—
155C2×C3×Dic3—64no—
156C2×C3×Dic3—62no—
157C2×C3×Dic3—64no—
158C2×C3×Dic3—62no—
159C3×D12—32no—
160C3×D12—61no—
161C3×C4×D3—64no—
162C3×Dic6—32no—
163C3×Dic6—34no—
164C3×Dic6—34no—
165C3×Dic6—64no—
166C3×Dic6—64no—
167C3×Dic6—68no—
168C3×Dic6—61no—
169C3×Dic6—62no—
170C3×Dic6—64no—
171C3×Dic6—62no—
172C3×Dic6—64no—
173C3⋊(C3⋊C8)C22×D314yes—
174C3×C3⋊C8—34no—
175C2×#137—271noSylow
176C2×H(3,𝔽3)—22no—
177C2×H(3,𝔽3)#13712yescomplemented
178C2×H(3,𝔽3)C22×D411yescharacteristic
179(C3⋊D3)⋊C3—42no—
180C3×#41—94no—
181C3×#41—182no—
182C2×C3×Q8—91no—
183C2×C3×Q8—92no—
184C2×C3×Q8—94no—
185C2×C3×Q8—181no—
186C2×C3×Q8—182no—
187C2×C3×D4—91no—
188C2×C3×D4—181no—
189C22×C3×C4—31no—
190C22×C3×C4—31no—
191C22×C3×C4—61no—
192C22×C3×C4—181no—
193C22×C3×C4—181no—
194C22×C3×C4D3211yescharacteristic
195C2×#67—61no—
196C2×#67—181no—
197C22×Dic3—61no—
198C22×Dic3—61no—
199C22×Dic3—62no—
200#235—34no—
201#235—62no—
202#235—94no—
203#235—182no—
204C2×D12—31no—
205C2×D12—91no—
206C2×C4×D3—61no—
207C2×C4×D3—181no—
208C2×Dic6—32no—
209C2×Dic6—34no—
210C2×Dic6—32no—
211C2×Dic6—34no—
212C2×Dic6—31no—
213C2×Dic6—62no—
214C2×Dic6—64no—
215C2×Dic6—61no—
216C2×Dic6—62no—
217C2×Dic6—61no—
218C2×Dic6—62no—
219C2×Dic6—64no—
220C2×Dic6—62no—
221C2×Dic6—91no—
222C3×M16—94no—
223C2×C3×C8—92no—
224#216—98no—
225#216—98no—
226#215—98no—
227#214—98no—
228#208—34no—
229#208—94no—
230C2×C3⋊C8—32no—
231C2×C3⋊C8—92no—
232Dic12—98no—
233#204—98no—
234C22×C32—24no—
235C22×C32—22no—
236C22×C32—22no—
237C22×C32—22no—
238C22×C32—21no—
239C22×C32C2×D1212yes—
240C22×C32C2×D1211yescharacteristic
241C22×C32C2×#6712yes—
242C22×C32C2×#6711yescharacteristic
243C2×C3⋊D3—22no—
244C2×C3⋊D3—42no—
245C2×C3×D3—62no—
246C2×C3×D3—122no—
247C32×C4—21no—
248C32×C4—21no—
249C32×C4—22no—
250C32×C4C2×D1211yescharacteristic
251C32×C4C2×D1212yes—
252C32×C4C2×#6711yescharacteristic
253C32×C4C2×#6712yes—
254C32×C4C23×D311yescharacteristic
255C32×C4C23×D311yescharacteristic
256C3⋊Dic3—22no—
257C3×Dic3—68no—
258C3×Dic3—62no—
259C3×Dic3—64no—
260C3×Dic3—64no—
261C2×#41—91no—
262C22×Q8—91no—
263#137—278no—
264C2×Q16—272no—
265C2×SD16—272no—
266C2×M16—91no—
267H(3,𝔽3)C2×#13711yescharacteristic, complemented, Sylow
268C23×C3—31no—
269C23×C3—31no—
270C23×C3—61no—
271C23×C3—181no—
272C23×C3#62611yescharacteristic
273C22×D3—61no—
274C22×D3—181no—
275C3×Q8—92no—
276C3×Q8—94no—
277C3×Q8—181no—
278C3×Q8—182no—
279C3×Q8—184no—
280C3×D4—92no—
281C3×D4—181no—
282C3×D4—184no—
283C2×C3×C4—31no—
284C2×C3×C4—32no—
285C2×C3×C4—31no—
286C2×C3×C4—32no—
287C2×C3×C4—31no—
288C2×C3×C4—32no—
289C2×C3×C4—31no—
290C2×C3×C4—32no—
291C2×C3×C4—62no—
292C2×C3×C4—61no—
293C2×C3×C4—62no—
294C2×C3×C4—61no—
295C2×C3×C4—184no—
296C2×C3×C4—181no—
297C2×C3×C4—184no—
298C2×C3×C4—182no—
299C2×C3×C4#62611yescharacteristic
300C2×C3×C4#62612yes—
301C2×C3×C4C2×D3211yescharacteristic
302C2×C3×C4C2×D3212yes—
303#67—64no—
304#67—184no—
305C2×Dic3—64no—
306C2×Dic3—64no—
307C2×Dic3—62no—
308C2×Dic3—62no—
309C2×Dic3—68no—
310C2×Dic3—64no—
311C2×Dic3—61no—
312C2×Dic3—181no—
313D12—32no—
314D12—61no—
315D12—92no—
316D12—181no—
317C4×D3—64no—
318C4×D3—184no—
319Dic6—34no—
320Dic6—34no—
321Dic6—32no—
322Dic6—64no—
323Dic6—62no—
324Dic6—64no—
325Dic6—64no—
326Dic6—62no—
327Dic6—68no—
328Dic6—64no—
329Dic6—61no—
330Dic6—92no—
331Dic6—181no—
332C3×C8—94no—
333C3⋊C8—34no—
334C3⋊C8—94no—
335C2×C32—24no—
336C2×C32—22no—
337C2×C32—22no—
338C2×C32—21no—
339C2×C32—22no—
340C2×C32#90912yescomplemented
341C2×C32#95112yescomplemented
342C2×C32C22×D1211yescharacteristic
343C2×C32C22×#6711yescharacteristic
344C3⋊D3—42no—
345C3×D3—122no—
346#41—94no—
347#41—182no—
348C2×Q8—92no—
349C2×Q8—91no—
350C2×Q8—94no—
351C2×Q8—182no—
352C2×Q8—181no—
353C2×D4—91no—
354C2×D4—181no—
355C22×C4—181no—
356C22×C4—181no—
357C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
358Q16—278no—
359SD16—278no—
360M16—94no—
361C2×C8—92no—
362C22×C3—22no—
363C22×C3—32no—
364C22×C3—32no—
365C22×C3—31no—
366C22×C3—31no—
367C22×C3—62no—
368C22×C3—62no—
369C22×C3—64no—
370C22×C3—62no—
371C22×C3—61no—
372C22×C3—182no—
373C22×C3—362no—
374C22×C3C2×#62612yes—
375C22×C3C2×#62611yescharacteristic
376C2×D3—62no—
377C2×D3—122no—
378C2×D3—182no—
379C2×D3—362no—
380C3×C4—31no—
381C3×C4—31no—
382C3×C4—31no—
383C3×C4—32no—
384C3×C4—32no—
385C3×C4—31no—
386C3×C4—61no—
387C3×C4—62no—
388C3×C4—61no—
389C3×C4—182no—
390C3×C4—184no—
391C3×C4C2×#62611yescharacteristic
392C3×C4C2×#62612yes—
393C3×C4C22×D3211yescharacteristic
394Dic3—64no—
395Dic3—64no—
396Dic3—68no—
397Dic3—62no—
398Dic3—182no—
399C32—21no—
400C32C2×#90911yescharacteristic, complemented
401C32C2×#95111yescharacteristic, complemented
402C23—181no—
403C23#620011yescharacteristic
404Q8—94no—
405Q8—92no—
406Q8—184no—
407Q8—182no—
408Q8—181no—
409D4—92no—
410D4—184no—
411D4—181no—
412C2×C4—184no—
413C2×C4—181no—
414C2×C4—182no—
415C2×C4—184no—
416C2×C4#620012yes—
417C2×C4#620011yescharacteristic
418C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
419C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
420C8—94no—
421C2×C3—22no—
422C2×C3—31no—
423C2×C3—31no—
424C2×C3—32no—
425C2×C3—32no—
426C2×C3—62no—
427C2×C3—62no—
428C2×C3—64no—
429C2×C3—61no—
430C2×C3—62no—
431C2×C3—362no—
432C2×C3#6383112yes—
433C2×C3C22×#62611yescharacteristic
434D3—122no—
435D3—362no—
436C22—22no—
437C22—182no—
438C22—362no—
439C22C2×#620012yes—
440C22C2×#620011yescharacteristic
441C4—182no—
442C4—184no—
443C4C2×#620012yes—
444C4C2×#620011yescharacteristic
445C4C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
446C3—31no—
447C3—31no—
448C3—61no—
449C3C2×#6383111yescharacteristic
450C2—22no—
451C2—362no—
452C2#1172107412yesdirect product factor
453C2C22×#620011yescharacteristic
subgroupG/H
centerC22C2×#6200
commutator, cosocleC4×H(3,𝔽3)C24
FrattiniC3×C4C22×D32
FittingC22×C4×H(3,𝔽3)C22
socleC22×C3C2×#626
prime radicalC2×#3828⋊C3C2
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#137
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×#6229C2semidirect
3. C2×#3748⋊C3C2semidirect
4. C2×#88439C2semidirect
5. #11721074C2direct (N × H)
6. C2×#88438C2semidirect
14. C2×#6229C22semidirect
21. #3748⋊C3C22semidirect
23. #88439C22semidirect
24. #88438C22semidirect
30. C2×#3748D3semidirect
90. #3748C2×D3semidirect
177. C2×H(3,𝔽3)#137semidirect
267. H(3,𝔽3)C2×#137semidirect
340. C2×C32#909semidirect
341. C2×C32#951semidirect
400. C32C2×#909semidirect
401. C32C2×#951semidirect
452. C2#11721074direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#62292
C2×#3828⋊C32
C2×#3748⋊C32
C2×#884392
#117210742
C2×#884382
C2×#638333
C2×#638273