Finite Group Databasebeta

← C2×#11723335 #13176667

Subgroups — C2×#11723335 #13176667

Order 1728.

1868 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89022C211yesmaximal, characteristic, complemented
2C2×#89021C211yesmaximal, characteristic, complemented
3C2×#88851C211yesmaximal, characteristic, complemented
4C2×C3×#3744C211yesmaximal, characteristic, complemented
5C2×C32×D3×C8C211yesmaximal, characteristic, complemented
6C2×#88800C211yesmaximal, characteristic, complemented
7C2×#88791C211yesmaximal, characteristic, complemented
8#11723335C218yesmaximal, direct product factor
9C2×#64113—31nomaximal
10C2×#63806—34nomaximal
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×#6308C2211yescharacteristic
13C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
14C2×C3×#634C2211yescharacteristic
15C2×C32×C4×D3C2211yescharacteristic
16C2×#6291—21no—
17C2×#6290—21no—
18C2×#6289—21no—
19C2×Dic3×C3⋊D3—21no—
20#89022—22no—
21#89022C2214yescomplemented
22#89021—22no—
23#89021C2214yescomplemented
24C2×C33×C8C2211yescharacteristic
25#88851C2214yescomplemented
26C3×#3744C2214yescomplemented
27C2×C32×C3⋊C8C2211yescharacteristic
28C32×D3×C8C2214yescomplemented
29#88800C2214yescomplemented
30#88791C2214yescomplemented
31C2×#3828—31no—
32C2×#3828—31no—
33C2×#3799—34no—
34C2×#3795—34no—
35C22×C32×C8—31no—
36C2×#3746—31no—
37#64113—38no—
38C2×#3745—31no—
39C2×#3744—31no—
40C2×#3744—34no—
41C2×#3744—64no—
42C2×#3744D311yescharacteristic, complemented
43C2×C3×#204—34no—
44C2×C3×D3×C8D314yescomplemented
45C2×#3719—34no—
46C2×#3714—34no—
47#63806—332no—
48C22×C3⋊(C3⋊D3)—21no—
49C22×C3×C3⋊D3—21no—
50C22×C32×D3D411yescharacteristic
51C2×C33×C4C2311yescharacteristic
52C2×C3⋊(C3⋊Dic3)—21no—
53C4⋊(C3⋊(C3⋊D3))—21no—
54C4⋊(C3⋊(C3⋊D3))C2312yes—
55#6308—21no—
56#6308C2312yes—
57C2×C3×C3⋊Dic3—21no—
58C3×C4⋊(C3⋊D3)—21no—
59C3×C4⋊(C3⋊D3)C2312yes—
60C3×#634—21no—
61C3×#634C2312yes—
62C2×C32×Dic3D411yescharacteristic
63C32×C4×D3D412yes—
64C32×C4×D3C2312yes—
65#6291—24no—
66#6290—24no—
67#6289—24no—
68Dic3×C3⋊D3—24no—
69C33×C8C2312yes—
70C32×C3⋊C8C2312yes—
71C2×#916—91no—
72C2×#904—94no—
73C22×C32×C4—31no—
74C2×#638—61no—
75C2×#638—61no—
76#3828—34no—
77#3828—34no—
78#3828—62no—
79#3828—62no—
80C2×C4⋊(C3⋊D3)—31no—
81C2×C4⋊(C3⋊D3)—34no—
82C2×C4⋊(C3⋊D3)—64no—
83C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
84C2×C4×C3⋊D3—61no—
85C2×C4×C3⋊D3—61no—
86C2×#634—31no—
87C2×#634—34no—
88C2×#634—64no—
89C2×#634C2×D311yescharacteristic
90C2×C3×D12—34no—
91C2×C3×C4×D3C2×D314yes—
92C2×C3×Dic6—34no—
93C2×#626—64no—
94C2×#625—64no—
95C2×#624—64no—
96C2×D3×Dic3—64no—
97#3799—316no—
98#3799—68no—
99#3795—316no—
100#3795—68no—
101C2×C32×C8—24no—
102C2×C32×C8—31no—
103C2×C32×C8—34no—
104C2×C32×C8C2×D314yes—
105C2×C32×C8C2×D311yescharacteristic
106#3746—34no—
107#3745—34no—
108#3744—34no—
109#3744—316no—
110#3744—616no—
111#3744C2×D314yescomplemented
112C2×C3×C3⋊C8C2×D314yes—
113C3×#204—316no—
114C3×D3×C8C2×D3116yescomplemented
115#3719—316no—
116#3714—316no—
117C22×C33C2×D411yescharacteristic
118C2×C3⋊(C3⋊D3)—22no—
119C2×C3⋊(C3⋊D3)—42no—
120C2×C3×C3⋊D3—22no—
121C2×C3×C3⋊D3—42no—
122C2×C32×D3—22no—
123C2×C32×D3C2×D412yes—
124C33×C4C2×D411yescharacteristic
125C33×C4C2411yescharacteristic
126C3⋊(C3⋊Dic3)—22no—
127C3×C3⋊Dic3—22no—
128C32×Dic3C2×D412yes—
129C2×#239—91no—
130C2×#237—91no—
131C2×#235—94no—
132C2×#235—94no—
133C2×C3×SD16—91no—
134C22×C3×C8—34no—
135C2×#216—91no—
136C2×#213—91no—
137#916—98no—
138C2×Dic12—94no—
139#904—932no—
140C2×D24—94no—
141C2×#204—34no—
142C2×#204—94no—
143C2×#204—91no—
144C2×#204—184no—
145C2×D3×C8C3⋊D311yescharacteristic, complemented
146C23×C32—31no—
147C22×C3⋊D3—21no—
148C22×C3⋊D3—61no—
149C22×C3⋊D3—64no—
150C22×C3⋊D3—124no—
151C22×C3×D3—64no—
152C22×C3×D3D1214yes—
153C2×C32×C4—24no—
154C2×C32×C4—31no—
155C2×C32×C4—32no—
156C2×C32×C4—32no—
157C2×C32×C4C22×D314yes—
158C2×C32×C4C22×D311yescharacteristic
159#638—64no—
160#638—64no—
161C2×C3⋊Dic3—21no—
162C2×C3⋊Dic3—61no—
163C2×C3⋊Dic3—64no—
164C2×C3⋊Dic3—124no—
165C4⋊(C3⋊D3)—21no—
166C4⋊(C3⋊D3)—32no—
167C4⋊(C3⋊D3)—38no—
168C4⋊(C3⋊D3)—68no—
169C4⋊(C3⋊D3)—61no—
170C4⋊(C3⋊D3)—64no—
171C4⋊(C3⋊D3)—124no—
172C4⋊(C3⋊D3)C22×D312yes—
173C4×C3⋊D3—64no—
174C4×C3⋊D3—64no—
175#634—21no—
176#634—32no—
177#634—38no—
178#634—68no—
179#634—61no—
180#634—64no—
181#634—124no—
182#634C22×D312yes—
183C2×C3×Dic3—64no—
184C2×C3×Dic3D1214yes—
185C3×D12—38no—
186C3×D12—64no—
187C3×C4×D3D1218yes—
188C3×C4×D3C22×D318yes—
189C3×Dic6—38no—
190C3×Dic6—64no—
191#626—616no—
192#625—616no—
193#624—616no—
194D3×Dic3—616no—
195C32×C8—28no—
196C32×C8—32no—
197C32×C8C22×D318yes—
198C32×C8C22×D312yes—
199C3×C3⋊C8C22×D318yes—
200C2×#135—271noSylow
201C2×C33#13512yescomplemented
202C2×C33C22×D411yescharacteristic
203C3⋊(C3⋊D3)—42no—
204C3×C3⋊D3—42no—
205C32×D3—22no—
206C2×C3×Q8—91no—
207C2×C3×D4—91no—
208C22×C3×C4—34no—
209C2×#67—184no—
210C2×#67—184no—
211C2×#67—181no—
212C22×Dic3—181no—
213#239—94no—
214#239—182no—
215#237—94no—
216#237—182no—
217#235—916no—
218#235—916no—
219#235—188no—
220#235—188no—
221C2×D12—34no—
222C2×D12—94no—
223C2×D12—91no—
224C2×D12—184no—
225C2×D12—181no—
226C2×C4×D3—184no—
227C2×C4×D3—184no—
228C2×C4×D3—181no—
229C2×C4×D3C2×C3⋊D311yescharacteristic
230C2×Dic6—34no—
231C2×Dic6—94no—
232C2×Dic6—91no—
233C2×Dic6—184no—
234C3×SD16—94no—
235C2×C3×C8—24no—
236C2×C3×C8—34no—
237C2×C3×C8—316no—
238C2×C3×C8D3214yes—
239C2×C3×C8C2×C3⋊D311yescharacteristic
240#216—94no—
241#213—94no—
242C2×C3⋊C8C2×C3⋊D311yescharacteristic
243Dic12—916no—
244D24—916no—
245#204—316no—
246#204—916no—
247#204—94no—
248#204—1816no—
249D3×C8C2×C3⋊D314yescomplemented
250C22×C32—24no—
251C22×C32—32no—
252C22×C32—62no—
253C22×C32C2×D1214yes—
254C22×C32D3×D411yescharacteristic
255C2×C3⋊D3—22no—
256C2×C3⋊D3—42no—
257C2×C3⋊D3—62no—
258C2×C3⋊D3—68no—
259C2×C3⋊D3—128no—
260C2×C3⋊D3—122no—
261C2×C3⋊D3—128no—
262C2×C3⋊D3—248no—
263C2×C3×D3—28no—
264C2×C3×D3—68no—
265C2×C3×D3—128no—
266C2×C3×D3C2×D1218yes—
267C32×C4—24no—
268C32×C4—24no—
269C32×C4—32no—
270C32×C4C2×D1214yes—
271C32×C4D3×D411yescharacteristic
272C32×C4C23×D314yes—
273C32×C4C23×D311yescharacteristic
274C3⋊Dic3—22no—
275C3⋊Dic3—62no—
276C3⋊Dic3—68no—
277C3⋊Dic3—128no—
278C3×Dic3—68no—
279C3×Dic3C2×D1218yes—
280C2×#41—271no—
281C2×#41—271no—
282#135—278no—
283C2×Q16—271no—
284C2×SD16—91no—
285C2×SD16—271no—
286C2×D8—271no—
287C22×C8—31no—
288C33C2×#13511yescharacteristic, complemented, Sylow
289C23×C3—34no—
290C23×C3—181no—
291C22×D3—64no—
292C22×D3—184no—
293C22×D3—181no—
294C22×D3—364no—
295C22×D3C4⋊(C3⋊D3)11yescharacteristic
296C3×Q8—92no—
297C3×Q8—181no—
298C3×D4—92no—
299C3×D4—181no—
300C2×C3×C4—24no—
301C2×C3×C4—34no—
302C2×C3×C4—38no—
303C2×C3×C4—38no—
304C2×C3×C4—181no—
305C2×C3×C4C2×D3214yes—
306C2×C3×C4C22×C3⋊D311yescharacteristic
307#67—1816no—
308#67—1816no—
309#67—184no—
310C2×Dic3—64no—
311C2×Dic3—184no—
312C2×Dic3—181no—
313C2×Dic3—184no—
314C2×Dic3—364no—
315C2×Dic3C4⋊(C3⋊D3)11yescharacteristic
316D12—38no—
317D12—64no—
318D12—98no—
319D12—92no—
320D12—188no—
321D12—184no—
322D12—184no—
323D12—181no—
324D12—364no—
325C4×D3—1816no—
326C4×D3—1816no—
327C4×D3—184no—
328C4×D3C4⋊(C3⋊D3)12yes—
329C4×D3C22×C3⋊D312yes—
330Dic6—38no—
331Dic6—64no—
332Dic6—98no—
333Dic6—92no—
334Dic6—188no—
335Dic6—184no—
336Dic6—181no—
337Dic6—364no—
338C3×C8—28no—
339C3×C8—38no—
340C3×C8C2×D3218yes—
341C3×C8C22×C3⋊D312yes—
342C3⋊C8C22×C3⋊D312yes—
343C2×C32—24no—
344C2×C32—28no—
345C2×C32—62no—
346C2×C32#90418yescomplemented
347C2×C32#91612yescomplemented
348C2×C32C22×D1214yes—
349C2×C32C2×D3×D411yescharacteristic
350C3⋊D3—42no—
351C3⋊D3—122no—
352C3⋊D3—128no—
353C3⋊D3—248no—
354C3×D3—28no—
355C3×D3—128no—
356#41—274no—
357#41—274no—
358#41—542no—
359#41—542no—
360C2×Q8—91no—
361C2×Q8—271no—
362C2×D4—91no—
363C2×D4—271no—
364C2×D4—541no—
365C2×D4—541no—
366C22×C4—31no—
367C22×C4—541no—
368C22×C4—541no—
369Q16—274no—
370SD16—94no—
371SD16—274no—
372D8—274no—
373C2×C8—31no—
374C2×C8—34no—
375C2×C8D3×C3⋊D311yescharacteristic
376C22×C3—24no—
377C22×C3—38no—
378C22×C3—68no—
379C22×C3—182no—
380C22×C3—362no—
381C22×C3D3×D1214yes—
382C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
383C2×D3—22no—
384C2×D3—68no—
385C2×D3—128no—
386C2×D3—188no—
387C2×D3—182no—
388C2×D3—368no—
389C2×D3—368no—
390C2×D3—362no—
391C2×D3—728no—
392C2×D3C2×C4⋊(C3⋊D3)12yes—
393C3×C4—24no—
394C3×C4—24no—
395C3×C4—38no—
396C3×C4—182no—
397C3×C4D3×D1214yes—
398C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
399C3×C4C22×D3214yes—
400C3×C4C23×C3⋊D311yescharacteristic
401Dic3—68no—
402Dic3—188no—
403Dic3—182no—
404Dic3—368no—
405Dic3C2×C4⋊(C3⋊D3)12yes—
406C32—24no—
407C32C2×#90414yescomplemented
408C32C2×#91611yescharacteristic, complemented
409C23—31no—
410C23—181no—
411C23—541no—
412Q8—92no—
413Q8—181no—
414Q8—272no—
415Q8—541no—
416D4—92no—
417D4—181no—
418D4—272no—
419D4—544no—
420D4—544no—
421D4—541no—
422C2×C4—31no—
423C2×C4—32no—
424C2×C4—32no—
425C2×C4—181no—
426C2×C4—544no—
427C2×C4—544no—
428C2×C4—541no—
429C2×C4C2×D3×C3⋊D311yescharacteristic
430C8—32no—
431C8C2×D3×C3⋊D312yes—
432C2×C3—24no—
433C2×C3—28no—
434C2×C3—68no—
435C2×C3—362no—
436C2×C3#6380618yescomplemented
437C2×C3#6411312yescomplemented
438C2×C3C2×D3×D1214yes—
439C2×C3C22×C4⋊(C3⋊D3)11yescharacteristic
440D3—22no—
441D3—128no—
442D3—362no—
443D3—368no—
444D3—728no—
445C22—32no—
446C22—62no—
447C22—182no—
448C22—362no—
449C22—542no—
450C22—1082no—
451C22D3×C4⋊(C3⋊D3)11yescharacteristic
452C4—32no—
453C4—182no—
454C4—542no—
455C4D3×C4⋊(C3⋊D3)11yescharacteristic
456C4C22×D3×C3⋊D311yescharacteristic
457C3—24no—
458C3C2×#6380614yescomplemented
459C3C2×#6411311yescharacteristic, complemented
460C2—62no—
461C2—362no—
462C2—1082no—
463C2#1172333512yesdirect product factor
464C2C2×D3×C4⋊(C3⋊D3)11yescharacteristic
subgroupG/H
centerC22D3×C4⋊(C3⋊D3)
commutator, cosocleC33×C4C24
FrattiniC4C22×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
hypercenterC2×C8D3×C3⋊D3
hypocenterC33C2×#135
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. C2×#89022C2semidirect
2. C2×#89021C2semidirect
3. C2×#88851C2semidirect
4. C2×C3×#3744C2semidirect
5. C2×C32×D3×C8C2semidirect
6. C2×#88800C2semidirect
7. C2×#88791C2semidirect
8. #11723335C2direct (N × H)
21. #89022C22semidirect
23. #89021C22semidirect
25. #88851C22semidirect
26. C3×#3744C22semidirect
28. C32×D3×C8C22semidirect
29. #88800C22semidirect
30. #88791C22semidirect
42. C2×#3744D3semidirect
44. C2×C3×D3×C8D3semidirect
111. #3744C2×D3semidirect
114. C3×D3×C8C2×D3semidirect
145. C2×D3×C8C3⋊D3semidirect
201. C2×C33#135semidirect
249. D3×C8C2×C3⋊D3semidirect
288. C33C2×#135semidirect
346. C2×C32#904semidirect
347. C2×C32#916semidirect
407. C32C2×#904semidirect
408. C32C2×#916semidirect
436. C2×C3#63806semidirect
437. C2×C3#64113semidirect
458. C3C2×#63806semidirect
459. C3C2×#64113semidirect
463. C2#11723335direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#890222
C2×#890212
C2×#888512
C2×C3×#37442
C2×C32×D3×C82
C2×#888002
C2×#887912
#117233352
C2×#641133
C2×#638063