Finite Group Databasebeta

← C2×#11723556 #13177707

Subgroups — C2×#11723556 #13177707

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×C32×#237C211yesmaximal, characteristic, complemented
2C2×#89021C211yesmaximal, characteristic, complemented
3C2×#88861C211yesmaximal, characteristic, complemented
4C2×C3×#3753C211yesmaximal, characteristic, complemented
5#11723556C218yesmaximal, direct product factor
6C2×#88799C211yesmaximal, characteristic, complemented
7C2×#88793C211yesmaximal, characteristic, complemented
8C2×D3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
9C2×#64159—31nomaximal
10C2×#63933—34nomaximal
11C2×C33×D4C2211yescharacteristic
12C2×#6308C2211yescharacteristic
13C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
14C2×C32×#67—21no—
15C22×C32×Dic3—21no—
16C32×#237—22no—
17C32×#237C2214yescomplemented
18C2×C32×C4×D3C2211yescharacteristic
19C2×C32×Dic6C2211yescharacteristic
20C2×#6290—21no—
21C2×Dic3×C3⋊D3—21no—
22#89021—22no—
23#89021C2214yescomplemented
24#88861C2214yescomplemented
25C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
26C3×#3753C2214yescomplemented
27C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
28#88799C2214yescomplemented
29#88793C2214yescomplemented
30D3×C3⋊(C3⋊C8)C2214yescomplemented
31C2×C32×#41—31no—
32C2×#3828—31no—
33C2×C3×#237D314yescomplemented
34C2×#3795—34no—
35#64159—38no—
36C2×#3756—31no—
37C2×#3755—31no—
38C2×#3754—31no—
39C2×#3754—34no—
40C2×#3754—64no—
41C2×#3753D311yescharacteristic, complemented
42C22×C3⋊(C3⋊C8)—31no—
43C2×C3×#213—34no—
44#63933—332no—
45C2×#3719—34no—
46C2×#3715—34no—
47C2×D3×C3⋊C8—34no—
48C23×C33—21no—
49C22×C3×C3⋊D3—21no—
50C22×C32×D3D411yescharacteristic
51C33×D4—21no—
52C33×D4C2312yes—
53C2×C33×C4C2311yescharacteristic
54C2×C3⋊(C3⋊Dic3)—21no—
55#6308—21no—
56#6308C2312yes—
57C3×C4⋊(C3⋊D3)—21no—
58C3×C4⋊(C3⋊D3)C2312yes—
59C32×#67—24no—
60C2×C32×Dic3—21no—
61C2×C32×Dic3—24no—
62C2×C32×Dic3D411yescharacteristic
63C32×C4×D3D412yes—
64C32×C4×D3C2312yes—
65C32×Dic6—21no—
66C32×Dic6C2312yes—
67#6290—24no—
68Dic3×C3⋊D3—24no—
69C3⋊(C3⋊(C3⋊C8))C2312yes—
70C3×C3⋊(C3⋊C8)C2312yes—
71C2×#950—94no—
72C2×#912—91no—
73C32×#41—34no—
74C32×#41—62no—
75C2×C32×Q8—31no—
76C2×C32×D4—24no—
77C2×C32×D4—61no—
78C2×C32×D4C2×D314yes—
79C2×C32×D4C2×D311yescharacteristic
80C22×C32×C4—31no—
81C22×C32×C4—61no—
82C2×#638—61no—
83#3828—34no—
84#3828—62no—
85C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
86C2×C4×C3⋊D3—61no—
87C2×#634—31no—
88C2×#634—34no—
89C2×#634—64no—
90C2×C3×#67—24no—
91C22×C3×Dic3—24no—
92C3×#237—28no—
93C3×#237C2×D3116yescomplemented
94C2×C3×D12—34no—
95C2×C3×C4×D3C2×D314yes—
96C2×C3×Dic6C2×D314yes—
97C2×#625—64no—
98C2×D3×Dic3—64no—
99#3795—316no—
100#3795—68no—
101#3756—34no—
102#3755—34no—
103#3754—34no—
104#3754—316no—
105#3754—616no—
106#3753C2×D314yescomplemented
107C2×C3⋊(C3⋊C8)—31no—
108C2×C3⋊(C3⋊C8)—34no—
109C2×C3⋊(C3⋊C8)—34no—
110C2×C3⋊(C3⋊C8)—64no—
111C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
112C3×#213—316no—
113C2×C3×C3⋊C8—34no—
114#3719—316no—
115#3715—316no—
116D3×C3⋊C8—316no—
117C22×C33—22no—
118C22×C33—42no—
119C22×C33C2×D411yescharacteristic
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—
127C32×Dic3—22no—
128C32×Dic3C2×D412yes—
129C2×C3×#41—34no—
130C2×#237—91no—
131C2×#237C3⋊D311yescharacteristic, complemented
132C2×#235—94no—
133C2×C3×D8—91no—
134#950—932no—
135C2×#216—94no—
136C2×#215—94no—
137C2×#214—94no—
138C2×#214—91no—
139C2×#214—91no—
140C2×#214—184no—
141C2×#213—34no—
142C22×C3⋊C8—94no—
143#912—98no—
144C2×Dic12—91no—
145C2×D3×C8—91no—
146C23×C32—21no—
147C23×C32—24no—
148C23×C32—31no—
149C23×C32—44no—
150C22×C3⋊D3—21no—
151C22×C3×D3—64no—
152C22×C3×D3#6714yes—
153C32×Q8—32no—
154C32×Q8—61no—
155C32×D4—21no—
156C32×D4—28no—
157C32×D4—24no—
158C32×D4—44no—
159C32×D4—64no—
160C32×D4C22×D318yes—
161C32×D4C22×D312yes—
162C2×C32×C4—24no—
163C2×C32×C4—31no—
164C2×C32×C4—32no—
165C2×C32×C4—32no—
166C2×C32×C4—64no—
167C2×C32×C4—61no—
168C2×C32×C4C22×D314yes—
169C2×C32×C4C22×D311yescharacteristic
170#638—64no—
171C2×C3⋊Dic3—61no—
172C2×C3⋊Dic3—64no—
173C2×C3⋊Dic3—124no—
174C4⋊(C3⋊D3)—21no—
175C4⋊(C3⋊D3)C22×D312yes—
176C4×C3⋊D3—64no—
177#634—32no—
178#634—38no—
179#634—68no—
180#634—61no—
181#634—64no—
182#634—124no—
183C3×#67—216no—
184C2×C3×Dic3—24no—
185C2×C3×Dic3—216no—
186C2×C3×Dic3#6714yes—
187C3×D12—38no—
188C3×D12—64no—
189C3×C4×D3#6718yes—
190C3×C4×D3C22×D318yes—
191C3×Dic6—24no—
192C3×Dic6C22×D318yes—
193#625—616no—
194D3×Dic3—616no—
195C3⋊(C3⋊C8)—32no—
196C3⋊(C3⋊C8)—38no—
197C3⋊(C3⋊C8)—68no—
198C3⋊(C3⋊C8)C22×D312yes—
199C3×C3⋊C8—38no—
200C2×#135—271noSylow
201C2×C33—42no—
202C2×C33#13512yescomplemented
203C2×C33C22×D411yescharacteristic
204C3×C3⋊D3—42no—
205C32×D3—22no—
206C3×#41—316no—
207C3×#41—68no—
208C2×C3×Q8—34no—
209C2×C3×D4—24no—
210C2×C3×D4—64no—
211C2×C3×D4—91no—
212C2×C3×D4D3214yes—
213C2×C3×D4C2×C3⋊D311yescharacteristic
214C22×C3×C4—34no—
215C22×C3×C4—64no—
216C2×#67—21no—
217C2×#67—184no—
218C2×#67—181no—
219C22×Dic3—21no—
220C22×Dic3—181no—
221#237—22no—
222#237—94no—
223#237—182no—
224#237C2×C3⋊D314yescomplemented
225#235—916no—
226#235—188no—
227C2×D12—34no—
228C2×C4×D3—184no—
229C2×C4×D3C2×C3⋊D311yescharacteristic
230C2×Dic6—94no—
231C2×Dic6—91no—
232C2×Dic6—184no—
233C2×Dic6C2×C3⋊D311yescharacteristic
234C3×D8—94no—
235C2×C3×C8—91no—
236#216—916no—
237#215—916no—
238#214—916no—
239#214—94no—
240#214—94no—
241#214—1816no—
242#213—316no—
243C2×C3⋊C8—34no—
244C2×C3⋊C8—916no—
245C2×C3⋊C8—94no—
246C2×C3⋊C8—91no—
247C2×C3⋊C8—184no—
248Dic12—94no—
249D3×C8—94no—
250C22×C32—22no—
251C22×C32—24no—
252C22×C32—28no—
253C22×C32—32no—
254C22×C32—48no—
255C22×C32—42no—
256C22×C32—48no—
257C22×C32—62no—
258C22×C32—88no—
259C22×C32D3×D411yescharacteristic
260C22×C32C2×#6714yes—
261C2×C3⋊D3—22no—
262C2×C3⋊D3—42no—
263C2×C3×D3—28no—
264C2×C3×D3—68no—
265C2×C3×D3—128no—
266C2×C3×D3C2×#6718yes—
267C32×C4—24no—
268C32×C4—24no—
269C32×C4—32no—
270C32×C4—62no—
271C32×C4D3×D411yescharacteristic
272C32×C4C2×#6714yes—
273C32×C4C23×D314yes—
274C32×C4C23×D311yescharacteristic
275C3⋊Dic3—62no—
276C3⋊Dic3—68no—
277C3⋊Dic3—128no—
278C3×Dic3—28no—
279C3×Dic3C2×#6718yes—
280C2×#41—31no—
281C2×#41—271no—
282#135—278no—
283C2×Q16—271no—
284C2×SD16—271no—
285C2×SD16—271no—
286C2×D8—91no—
287C22×C8—271no—
288C33C2×#13511yescharacteristic, complemented, Sylow
289C23×C3—24no—
290C23×C3—21no—
291C23×C3—34no—
292C23×C3—44no—
293C23×C3—181no—
294C22×D3—64no—
295C22×D3#63811yescharacteristic
296C3×Q8—38no—
297C3×Q8—64no—
298C3×D4—24no—
299C3×D4—21no—
300C3×D4—28no—
301C3×D4—44no—
302C3×D4—616no—
303C3×D4—92no—
304C3×D4—181no—
305C3×D4C2×D3218yes—
306C3×D4C22×C3⋊D312yes—
307C2×C3×C4—24no—
308C2×C3×C4—34no—
309C2×C3×C4—38no—
310C2×C3×C4—38no—
311C2×C3×C4—616no—
312C2×C3×C4—64no—
313C2×C3×C4C2×D3214yes—
314C2×C3×C4C22×C3⋊D311yescharacteristic
315#67—24no—
316#67—1816no—
317#67—184no—
318C2×Dic3—21no—
319C2×Dic3—24no—
320C2×Dic3—184no—
321C2×Dic3—181no—
322C2×Dic3—184no—
323C2×Dic3—364no—
324C2×Dic3#63811yescharacteristic
325D12—38no—
326D12—64no—
327C4×D3—1816no—
328C4×D3#63812yes—
329C4×D3C22×C3⋊D312yes—
330Dic6—21no—
331Dic6—98no—
332Dic6—92no—
333Dic6—188no—
334Dic6—184no—
335Dic6—181no—
336Dic6—364no—
337Dic6C22×C3⋊D312yes—
338C3×C8—92no—
339C3⋊C8—38no—
340C3⋊C8—98no—
341C3⋊C8—92no—
342C3⋊C8—188no—
343C2×C32—24no—
344C2×C32—28no—
345C2×C32—42no—
346C2×C32—48no—
347C2×C32—62no—
348C2×C32—88no—
349C2×C32#91212yescomplemented
350C2×C32#95018yescomplemented
351C2×C32C2×D3×D411yescharacteristic
352C2×C32C22×#6714yes—
353C3⋊D3—42no—
354C3×D3—28no—
355C3×D3—128no—
356#41—34no—
357#41—62no—
358#41—274no—
359#41—542no—
360C2×Q8—31no—
361C2×Q8—271no—
362C2×D4—61no—
363C2×D4—91no—
364C2×D4—541no—
365C2×D4D3×C3⋊D311yescharacteristic
366C22×C4—31no—
367C22×C4—61no—
368C22×C4—541no—
369Q16—274no—
370SD16—274no—
371SD16—274no—
372D8—94no—
373C2×C8—91no—
374C2×C8—274no—
375C2×C8—271no—
376C22×C3—28no—
377C22×C3—22no—
378C22×C3—24no—
379C22×C3—38no—
380C22×C3—48no—
381C22×C3—42no—
382C22×C3—48no—
383C22×C3—68no—
384C22×C3—88no—
385C22×C3—182no—
386C22×C3—362no—
387C22×C3D3×#6714yes—
388C22×C3C2×#63811yescharacteristic
389C2×D3—22no—
390C2×D3—68no—
391C2×D3—128no—
392C2×D3C2×#63812yes—
393C3×C4—24no—
394C3×C4—24no—
395C3×C4—38no—
396C3×C4—68no—
397C3×C4D3×#6714yes—
398C3×C4C2×#63811yescharacteristic
399C3×C4C22×D3214yes—
400C3×C4C23×C3⋊D311yescharacteristic
401Dic3—22no—
402Dic3—188no—
403Dic3—182no—
404Dic3—368no—
405Dic3C2×#63812yes—
406C32—24no—
407C32C2×#91211yescharacteristic, complemented
408C32C2×#95014yescomplemented
409C23—21no—
410C23—31no—
411C23—181no—
412Q8—32no—
413Q8—61no—
414Q8—272no—
415Q8—541no—
416D4—21no—
417D4—64no—
418D4—92no—
419D4—181no—
420D4—544no—
421D4C2×D3×C3⋊D312yes—
422C2×C4—31no—
423C2×C4—32no—
424C2×C4—32no—
425C2×C4—64no—
426C2×C4—61no—
427C2×C4—544no—
428C2×C4—541no—
429C2×C4C2×D3×C3⋊D311yescharacteristic
430C8—92no—
431C8—272no—
432C2×C3—28no—
433C2×C3—24no—
434C2×C3—48no—
435C2×C3—42no—
436C2×C3—68no—
437C2×C3—88no—
438C2×C3—362no—
439C2×C3#6393318yescomplemented
440C2×C3#6415912yescomplemented
441C2×C3C2×D3×#6714yes—
442C2×C3C22×#63811yescharacteristic
443D3—22no—
444D3—128no—
445C22—22no—
446C22—32no—
447C22—42no—
448C22—62no—
449C22—182no—
450C22—362no—
451C22D3×#63811yescharacteristic
452C4—32no—
453C4—62no—
454C4—542no—
455C4D3×#63811yescharacteristic
456C4C22×D3×C3⋊D311yescharacteristic
457C3—24no—
458C3C2×#6393314yescomplemented
459C3C2×#6415911yescharacteristic, complemented
460C2—42no—
461C2—62no—
462C2—362no—
463C2#1172355612yesdirect product factor
464C2C2×D3×#63811yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC33×C4C24
FrattiniC4C22×D3×C3⋊D3
FittingC2×C33×D4C22
socleC22×C33C2×D4
hypercenterC2×D4D3×C3⋊D3
hypocenterC33C2×#135
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C32×#237C2semidirect
2. C2×#89021C2semidirect
3. C2×#88861C2semidirect
4. C2×C3×#3753C2semidirect
5. #11723556C2direct (N × H)
6. C2×#88799C2semidirect
7. C2×#88793C2semidirect
8. C2×D3×C3⋊(C3⋊C8)C2semidirect
17. C32×#237C22semidirect
23. #89021C22semidirect
24. #88861C22semidirect
26. C3×#3753C22semidirect
28. #88799C22semidirect
29. #88793C22semidirect
30. D3×C3⋊(C3⋊C8)C22semidirect
33. C2×C3×#237D3semidirect
41. C2×#3753D3semidirect
93. C3×#237C2×D3semidirect
106. #3753C2×D3semidirect
131. C2×#237C3⋊D3semidirect
202. C2×C33#135semidirect
224. #237C2×C3⋊D3semidirect
288. C33C2×#135semidirect
349. C2×C32#912semidirect
350. C2×C32#950semidirect
407. C32C2×#912semidirect
408. C32C2×#950semidirect
439. C2×C3#63933semidirect
440. C2×C3#64159semidirect
458. C3C2×#63933semidirect
459. C3C2×#64159semidirect
463. C2#11723556direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×#2372
C2×#890212
C2×#888612
C2×C3×#37532
#117235562
C2×#887992
C2×#887932
C2×D3×C3⋊(C3⋊C8)2
C2×#641593
C2×#639333