Finite Group Databasebeta

← C2×#11723342 #13176672

Subgroups — C2×#11723342 #13176672

Order 1728.

1900 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89017C211yesmaximal, characteristic, complemented
2C2×#89015C211yesmaximal, characteristic, complemented
3C2×C8×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
4C2×C3×#3744C211yesmaximal, characteristic, complemented
5C2×C32×#204C211yesmaximal, characteristic, complemented
6C2×#88798C211yesmaximal, characteristic, complemented
7C2×#88789C211yesmaximal, characteristic, complemented
8#11723342C218yesmaximal, direct product factor
9C2×#64125—31nomaximal
10C2×#63809—34nomaximal
11C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×#634C2211yescharacteristic
14C2×C32×D12C2211yescharacteristic
15C2×C32×Dic6C2211yescharacteristic
16C2×#6292—21no—
17C2×#6291—21no—
18C2×Dic3×C3⋊D3—21no—
19C2×D3×C3⋊Dic3—21no—
20#89017—22no—
21#89017C2214yescomplemented
22#89015—22no—
23#89015C2214yescomplemented
24C2×C33×C8C2211yescharacteristic
25C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
26C8×C3⋊(C3⋊D3)C2214yescomplemented
27C3×#3744C2214yescomplemented
28C32×#204C2214yescomplemented
29#88798C2214yescomplemented
30#88789C2214yescomplemented
31C2×#3832—31no—
32C2×#3830—31no—
33C2×#3796—34no—
34C2×#3796—34no—
35C2×C32×SD16—31no—
36C2×#3756—31no—
37C2×#3753—31no—
38#64125—38no—
39C2×#3744D311yescharacteristic, complemented
40C2×C8×C3⋊D3—31no—
41C2×C8×C3⋊D3—34no—
42C2×C8×C3⋊D3—64no—
43C2×C3×#204—34no—
44C2×C3×#204D314yescomplemented
45C2×#3718—34no—
46C2×#3713—34no—
47#63809—332no—
48C22×C3⋊(C3⋊D3)D411yescharacteristic
49C22×C3×C3⋊D3—21no—
50C22×C32×D3—21no—
51C2×C33×C4C2311yescharacteristic
52C2×C3⋊(C3⋊Dic3)D411yescharacteristic
53C4×C3⋊(C3⋊D3)D412yes—
54C4×C3⋊(C3⋊D3)C2312yes—
55C2×C3×C3⋊Dic3—21no—
56C3×C4⋊(C3⋊D3)—21no—
57C3×C4⋊(C3⋊D3)C2312yes—
58C3×#634—21no—
59C3×#634C2312yes—
60C2×C32×Dic3—21no—
61C32×D12—21no—
62C32×D12C2312yes—
63C32×Dic6—21no—
64C32×Dic6C2312yes—
65#6292—24no—
66#6291—24no—
67Dic3×C3⋊D3—24no—
68D3×C3⋊Dic3—24no—
69C33×C8C2312yes—
70C3⋊(C3⋊(C3⋊C8))C2312yes—
71C2×#916—94no—
72C2×#916—91no—
73C2×C32×Q8—31no—
74C2×C32×D4—31no—
75C2×#638—61no—
76C22×C3⋊Dic3—61no—
77#3832—34no—
78#3832—62no—
79#3830—34no—
80#3830—62no—
81C2×C4⋊(C3⋊D3)—61no—
82C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
83C2×C4×C3⋊D3—31no—
84C2×C4×C3⋊D3—34no—
85C2×C4×C3⋊D3—61no—
86C2×C4×C3⋊D3—64no—
87C2×#634C2×D311yescharacteristic
88C2×C3×D12—34no—
89C2×C3×D12C2×D314yes—
90C2×C3×Dic6—34no—
91C2×C3×Dic6C2×D314yes—
92C2×#626—64no—
93C2×#626—64no—
94C2×D3×Dic3—64no—
95C2×D3×Dic3—64no—
96#3796—316no—
97#3796—316no—
98#3796—68no—
99#3796—68no—
100C32×SD16—34no—
101C2×C32×C8—24no—
102C2×C32×C8C2×D314yes—
103C2×C32×C8C2×D311yescharacteristic
104#3756—34no—
105#3753—34no—
106C2×C3⋊(C3⋊C8)—31no—
107C2×C3⋊(C3⋊C8)—34no—
108C2×C3⋊(C3⋊C8)—64no—
109#3744C2×D314yescomplemented
110C8×C3⋊D3—34no—
111C8×C3⋊D3—316no—
112C8×C3⋊D3—616no—
113C3×#204—316no—
114C3×#204C2×D3116yescomplemented
115#3718—316no—
116#3713—316no—
117C22×C33C2×D411yescharacteristic
118C2×C3⋊(C3⋊D3)—22no—
119C2×C3⋊(C3⋊D3)C2×D412yes—
120C2×C3×C3⋊D3—22no—
121C2×C3×C3⋊D3—42no—
122C2×C32×D3—22no—
123C2×C32×D3—42no—
124C33×C4C2×D411yescharacteristic
125C33×C4C2411yescharacteristic
126C3⋊(C3⋊Dic3)C2×D412yes—
127C3×C3⋊Dic3—22no—
128C32×Dic3—22no—
129C2×#239—94no—
130C2×#239—91no—
131C2×#237—94no—
132C2×#237—91no—
133C2×C3×SD16—34no—
134C2×C3×SD16—91no—
135C2×#216—94no—
136C2×#216—91no—
137C2×#213—94no—
138C2×#213—91no—
139#916—932no—
140#916—98no—
141C2×#204—34no—
142C2×#204C3⋊D311yescharacteristic, complemented
143C2×D3×C8—94no—
144C2×D3×C8—91no—
145C2×D3×C8—184no—
146C23×C32—61no—
147C22×C3⋊D3—21no—
148C22×C3⋊D3—31no—
149C22×C3⋊D3—34no—
150C22×C3⋊D3—64no—
151C22×C3×D3—24no—
152C22×C3×D3—64no—
153C32×Q8—32no—
154C32×Q8—61no—
155C32×D4—32no—
156C32×D4—61no—
157C2×C32×C4—24no—
158C2×C32×C4—61no—
159C2×C32×C4C22×D314yes—
160C2×C32×C4C22×D311yescharacteristic
161#638—64no—
162C2×C3⋊Dic3—21no—
163C2×C3⋊Dic3—31no—
164C2×C3⋊Dic3—34no—
165C2×C3⋊Dic3—64no—
166C2×C3⋊Dic3—64no—
167C4⋊(C3⋊D3)—21no—
168C4⋊(C3⋊D3)—64no—
169C4⋊(C3⋊D3)C22×D312yes—
170C4×C3⋊D3—32no—
171C4×C3⋊D3—32no—
172C4×C3⋊D3—38no—
173C4×C3⋊D3—38no—
174C4×C3⋊D3—68no—
175C4×C3⋊D3—68no—
176C4×C3⋊D3—64no—
177#634—21no—
178#634C22×D312yes—
179C2×C3×Dic3—24no—
180C2×C3×Dic3—64no—
181C3×D12—24no—
182C3×D12—38no—
183C3×D12—64no—
184C3×D12C22×D318yes—
185C3×Dic6—24no—
186C3×Dic6—38no—
187C3×Dic6—64no—
188C3×Dic6C22×D318yes—
189#626—616no—
190#626—616no—
191D3×Dic3—616no—
192D3×Dic3—616no—
193C32×C8—28no—
194C32×C8C22×D318yes—
195C32×C8C22×D312yes—
196C3⋊(C3⋊C8)—32no—
197C3⋊(C3⋊C8)—38no—
198C3⋊(C3⋊C8)—68no—
199C2×#135—271noSylow
200C2×C33#13512yescomplemented
201C2×C33C22×D411yescharacteristic
202C3⋊(C3⋊D3)—22no—
203C3×C3⋊D3—42no—
204C32×D3—42no—
205C2×C3×Q8—34no—
206C2×C3×Q8—91no—
207C2×C3×D4—34no—
208C2×C3×D4—91no—
209C2×#67—181no—
210C2×#67—184no—
211C22×Dic3—181no—
212C22×Dic3—184no—
213#239—916no—
214#239—94no—
215#239—182no—
216#239—188no—
217#237—916no—
218#237—94no—
219#237—182no—
220#237—188no—
221C2×D12—34no—
222C2×D12—184no—
223C2×D12—181no—
224C2×D12C2×C3⋊D311yescharacteristic
225C2×C4×D3—94no—
226C2×C4×D3—91no—
227C2×C4×D3—184no—
228C2×C4×D3—184no—
229C2×C4×D3—181no—
230C2×Dic6—34no—
231C2×Dic6C2×C3⋊D311yescharacteristic
232C3×SD16—316no—
233C3×SD16—94no—
234C2×C3×C8—24no—
235C2×C3×C8D3214yes—
236C2×C3×C8C2×C3⋊D311yescharacteristic
237#216—916no—
238#216—94no—
239#213—916no—
240#213—94no—
241C2×C3⋊C8—94no—
242C2×C3⋊C8—91no—
243C2×C3⋊C8—184no—
244#204—316no—
245#204C2×C3⋊D314yescomplemented
246D3×C8—916no—
247D3×C8—94no—
248D3×C8—1816no—
249C22×C32—24no—
250C22×C32—62no—
251C22×C32—122no—
252C22×C32D3×D414yes—
253C22×C32D3×D411yescharacteristic
254C2×C3⋊D3—22no—
255C2×C3⋊D3—32no—
256C2×C3⋊D3—38no—
257C2×C3⋊D3—42no—
258C2×C3⋊D3—68no—
259C2×C3⋊D3—616no—
260C2×C3⋊D3—68no—
261C2×C3⋊D3—62no—
262C2×C3×D3—28no—
263C2×C3×D3—48no—
264C2×C3×D3—68no—
265C2×C3×D3—128no—
266C32×C4—24no—
267C32×C4—24no—
268C32×C4—62no—
269C32×C4D3×D414yes—
270C32×C4D3×D411yescharacteristic
271C32×C4C23×D314yes—
272C32×C4C23×D311yescharacteristic
273C3⋊Dic3—22no—
274C3⋊Dic3—32no—
275C3⋊Dic3—38no—
276C3⋊Dic3—68no—
277C3×Dic3—28no—
278C3×Dic3—68no—
279C2×#41—271no—
280C2×#41—271no—
281#135—278no—
282C2×Q16—271no—
283C2×SD16—31no—
284C2×SD16—91no—
285C2×D8—271no—
286C22×C8—271no—
287C33C2×#13511yescharacteristic, complemented, Sylow
288C23×C3—64no—
289C23×C3—181no—
290C22×D3—21no—
291C22×D3—64no—
292C22×D3—94no—
293C22×D3—91no—
294C22×D3—184no—
295C3×Q8—38no—
296C3×Q8—64no—
297C3×Q8—92no—
298C3×Q8—181no—
299C3×D4—38no—
300C3×D4—64no—
301C3×D4—92no—
302C3×D4—181no—
303C2×C3×C4—24no—
304C2×C3×C4—64no—
305C2×C3×C4—181no—
306C2×C3×C4C2×D3214yes—
307C2×C3×C4C22×C3⋊D311yescharacteristic
308#67—184no—
309#67—1816no—
310C2×Dic3—21no—
311C2×Dic3—64no—
312C2×Dic3—94no—
313C2×Dic3—91no—
314C2×Dic3—184no—
315C2×Dic3—184no—
316C2×Dic3—1816no—
317D12—21no—
318D12—38no—
319D12—64no—
320D12—1816no—
321D12—184no—
322D12C22×C3⋊D312yes—
323C4×D3—98no—
324C4×D3—92no—
325C4×D3—98no—
326C4×D3—92no—
327C4×D3—188no—
328C4×D3—188no—
329C4×D3—1816no—
330C4×D3—184no—
331Dic6—21no—
332Dic6—38no—
333Dic6—64no—
334Dic6C22×C3⋊D312yes—
335C3×C8—28no—
336C3×C8C2×D3218yes—
337C3×C8C22×C3⋊D312yes—
338C3⋊C8—98no—
339C3⋊C8—92no—
340C3⋊C8—188no—
341C2×C32—24no—
342C2×C32—28no—
343C2×C32—122no—
344C2×C32#91618yescomplemented
345C2×C32#91612yescomplemented
346C2×C32C2×D3×D414yes—
347C2×C32C2×D3×D411yescharacteristic
348C3⋊D3—42no—
349C3⋊D3—616no—
350C3⋊D3—68no—
351C3⋊D3—62no—
352C3×D3—48no—
353C3×D3—128no—
354#41—274no—
355#41—274no—
356#41—542no—
357#41—542no—
358C2×Q8—31no—
359C2×Q8—91no—
360C2×D4—31no—
361C2×D4—91no—
362C2×D4—541no—
363C2×D4—541no—
364C22×C4—271no—
365C22×C4—541no—
366C22×C4—541no—
367Q16—274no—
368SD16—34no—
369SD16—94no—
370D8—274no—
371C2×C8—271no—
372C2×C8—274no—
373C2×C8D3×C3⋊D311yescharacteristic
374C22×C3—24no—
375C22×C3—68no—
376C22×C3—128no—
377C22×C3—182no—
378C22×C3—362no—
379C22×C3#380214yes—
380C22×C3D4×C3⋊D311yescharacteristic
381C2×D3—22no—
382C2×D3—42no—
383C2×D3—68no—
384C2×D3—98no—
385C2×D3—92no—
386C2×D3—128no—
387C2×D3—188no—
388C2×D3—1816no—
389C2×D3—188no—
390C2×D3—182no—
391C3×C4—24no—
392C3×C4—24no—
393C3×C4—68no—
394C3×C4—182no—
395C3×C4#380214yes—
396C3×C4D4×C3⋊D311yescharacteristic
397C3×C4C22×D3214yes—
398C3×C4C23×C3⋊D311yescharacteristic
399Dic3—22no—
400Dic3—68no—
401Dic3—98no—
402Dic3—92no—
403Dic3—188no—
404C32—24no—
405C32C2×#91614yescomplemented
406C32C2×#91611yescharacteristic, complemented
407C23—61no—
408C23—181no—
409C23—271no—
410Q8—32no—
411Q8—61no—
412Q8—92no—
413Q8—181no—
414D4—32no—
415D4—61no—
416D4—92no—
417D4—181no—
418D4—544no—
419D4—544no—
420C2×C4—61no—
421C2×C4—181no—
422C2×C4—271no—
423C2×C4—272no—
424C2×C4—272no—
425C2×C4—544no—
426C2×C4—544no—
427C2×C4C2×D3×C3⋊D311yescharacteristic
428C8—272no—
429C8C2×D3×C3⋊D312yes—
430C2×C3—24no—
431C2×C3—28no—
432C2×C3—128no—
433C2×C3—362no—
434C2×C3#6380918yescomplemented
435C2×C3#6412512yescomplemented
436C2×C3C2×#380214yes—
437C2×C3C2×D4×C3⋊D311yescharacteristic
438D3—42no—
439D3—128no—
440D3—1816no—
441D3—182no—
442D3—188no—
443C22—62no—
444C22—122no—
445C22—182no—
446C22—272no—
447C22—362no—
448C22—542no—
449C22#8902611yescharacteristic
450C4—62no—
451C4—182no—
452C4—272no—
453C4#8902611yescharacteristic
454C4C22×D3×C3⋊D311yescharacteristic
455C3—24no—
456C3C2×#6380914yescomplemented
457C3C2×#6412511yescharacteristic, complemented
458C2—122no—
459C2—362no—
460C2—542no—
461C2#1172334212yesdirect product factor
462C2C2×#8902611yescharacteristic
subgroupG/H
centerC22#89026
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×#89017C2semidirect
2. C2×#89015C2semidirect
3. C2×C8×C3⋊(C3⋊D3)C2semidirect
4. C2×C3×#3744C2semidirect
5. C2×C32×#204C2semidirect
6. C2×#88798C2semidirect
7. C2×#88789C2semidirect
8. #11723342C2direct (N × H)
21. #89017C22semidirect
23. #89015C22semidirect
26. C8×C3⋊(C3⋊D3)C22semidirect
27. C3×#3744C22semidirect
28. C32×#204C22semidirect
29. #88798C22semidirect
30. #88789C22semidirect
39. C2×#3744D3semidirect
44. C2×C3×#204D3semidirect
109. #3744C2×D3semidirect
114. C3×#204C2×D3semidirect
142. C2×#204C3⋊D3semidirect
200. C2×C33#135semidirect
245. #204C2×C3⋊D3semidirect
287. C33C2×#135semidirect
344. C2×C32#916semidirect
345. C2×C32#916semidirect
405. C32C2×#916semidirect
406. C32C2×#916semidirect
434. C2×C3#63809semidirect
435. C2×C3#64125semidirect
456. C3C2×#63809semidirect
457. C3C2×#64125semidirect
461. C2#11723342direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#890172
C2×#890152
C2×C8×C3⋊(C3⋊D3)2
C2×C3×#37442
C2×C32×#2042
C2×#887982
C2×#887892
#117233422
C2×#641253
C2×#638093