Finite Group Databasebeta

← C2×#11723550 #13177661

Subgroups — C2×#11723550 #13177661

Order 1728.

1820 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#237C211yesmaximal, characteristic
2C2×D3×#634C211yesmaximal, characteristic, complemented
3C2×#88861C211yesmaximal, characteristic, complemented
4C2×C3×#3754C211yesmaximal, characteristic, complemented
5#11723550C218yesmaximal, direct product factor
6C2×#88799C211yesmaximal, characteristic, complemented
7C2×#88798C211yesmaximal, characteristic, complemented
8C2×#88786C211yesmaximal, characteristic, complemented
9C2×#64160—31nomaximal
10C2×#63929—34nomaximal
11C2×C33×D4C2211yescharacteristic
12C2×#6308C2211yescharacteristic
13C2×C3×#634C2211yescharacteristic
14C2×C32×#67—21no—
15C22×C32×Dic3—21no—
16C32×#237—22no—
17C32×#237C2214yes—
18C2×C32×C4×D3C2211yescharacteristic
19C2×C32×Dic6C2211yescharacteristic
20C2×#6293—21no—
21C2×D3×C3⋊Dic3—21no—
22D3×#634—22no—
23D3×#634C2214yescomplemented
24#88861C2214yescomplemented
25C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
26C3×#3754C2214yescomplemented
27C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
28#88799C2214yescomplemented
29#88798C2214yescomplemented
30#88786C2214yescomplemented
31C2×C32×#41—31no—
32C22×#634—31no—
33C2×C3×#237D314yes—
34C2×D3×Dic6—34no—
35#64160—38no—
36C2×#3756—31no—
37C2×#3756—31no—
38C2×#3754—31no—
39C2×#3754—34no—
40C2×#3754—64no—
41C2×#3754D311yescharacteristic, complemented
42C2×#3748—31no—
43C2×C3×#214—34no—
44#63929—332no—
45C2×#3719—34no—
46C2×#3718—34no—
47C2×#3711—34no—
48C23×C33—21no—
49C22×C32×D3D411yescharacteristic
50C33×D4—21no—
51C33×D4C2312yes—
52C2×C33×C4C2311yescharacteristic
53C2×C3⋊(C3⋊Dic3)—21no—
54#6308—21no—
55#6308C2312yes—
56C2×C3×C3⋊Dic3—21no—
57C3×#634—21no—
58C3×#634C2312yes—
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#6293—24no—
68D3×C3⋊Dic3—24no—
69C3⋊(C3⋊(C3⋊C8))C2312yes—
70C3×C3⋊(C3⋊C8)C2312yes—
71C2×#951—94no—
72C2×#915—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—
82C22×C3⋊Dic3—61no—
83C2×#634—31no—
84C2×#634—34no—
85C2×#634—34no—
86C2×#634—62no—
87C2×#634—61no—
88C2×#634—64no—
89C2×#634C2×D311yescharacteristic
90C2×C3×#67—24no—
91C22×C3×Dic3—24no—
92C3×#237—28no—
93C3×#237C2×D3116yes—
94C2×C3×C4×D3C2×D314yes—
95C2×C3×Dic6—34no—
96C2×C3×Dic6C2×D314yes—
97C2×#627—64no—
98C2×D3×Dic3—64no—
99D3×Dic6—316no—
100D3×Dic6—68no—
101#3756—34no—
102#3756—34no—
103#3754—34no—
104#3754—316no—
105#3754—616no—
106#3754C2×D314yescomplemented
107#3748—34no—
108C2×C3⋊(C3⋊C8)—31no—
109C2×C3⋊(C3⋊C8)—34no—
110C2×C3⋊(C3⋊C8)—64no—
111C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
112C3×#214—316no—
113C2×C3×C3⋊C8—34no—
114#3719—316no—
115#3718—316no—
116#3711—316no—
117C22×C33—22no—
118C22×C33—42no—
119C22×C33C2×D411yescharacteristic
120C2×C32×D3—22no—
121C2×C32×D3C2×D412yes—
122C33×C4C2×D411yescharacteristic
123C33×C4C2411yescharacteristic
124C3⋊(C3⋊Dic3)—22no—
125C3×C3⋊Dic3—22no—
126C32×Dic3—22no—
127C32×Dic3C2×D412yes—
128C2×C3×#41—34no—
129C2×D3×Q8—91no—
130C2×#237C3⋊D311yescharacteristic
131C22×Dic6—94no—
132C2×C3×SD16—91no—
133#951—932no—
134C2×#216—94no—
135C2×#216—94no—
136C2×#216—91no—
137C2×#214—34no—
138C2×#214—94no—
139C2×#214—91no—
140C2×#214—184no—
141C2×#208—94no—
142#915—98no—
143C2×Dic12—91no—
144C2×ΓL(1,𝔽25)—91no—
145C23×C32—21no—
146C23×C32—24no—
147C23×C32—31no—
148C23×C32—44no—
149C22×C3×D3#6714yes—
150C32×Q8—32no—
151C32×Q8—61no—
152C32×D4—21no—
153C32×D4—28no—
154C32×D4—24no—
155C32×D4—44no—
156C32×D4—64no—
157C32×D4C22×D318yes—
158C32×D4C22×D312yes—
159C2×C32×C4—24no—
160C2×C32×C4—31no—
161C2×C32×C4—32no—
162C2×C32×C4—32no—
163C2×C32×C4—64no—
164C2×C32×C4—61no—
165C2×C32×C4C22×D314yes—
166C2×C32×C4C22×D311yescharacteristic
167C2×C3⋊Dic3—21no—
168C2×C3⋊Dic3—64no—
169C2×C3⋊Dic3—61no—
170C2×C3⋊Dic3—64no—
171C2×C3⋊Dic3—124no—
172#634—21no—
173#634—32no—
174#634—38no—
175#634—68no—
176#634—64no—
177#634—61no—
178#634—64no—
179#634—124no—
180#634C22×D312yes—
181C3×#67—216no—
182C2×C3×Dic3—24no—
183C2×C3×Dic3—216no—
184C2×C3×Dic3—64no—
185C2×C3×Dic3#6714yes—
186C3×C4×D3#6718yes—
187C3×C4×D3C22×D318yes—
188C3×Dic6—24no—
189C3×Dic6—38no—
190C3×Dic6—64no—
191C3×Dic6C22×D318yes—
192#627—616no—
193D3×Dic3—616no—
194C3⋊(C3⋊C8)—32no—
195C3⋊(C3⋊C8)—38no—
196C3⋊(C3⋊C8)—68no—
197C3⋊(C3⋊C8)C22×D312yes—
198C3×C3⋊C8—38no—
199C2×#137—271noSylow
200C2×C33—42no—
201C2×C33#13712yescomplemented
202C2×C33C22×D411yescharacteristic
203C32×D3—22no—
204C3×#41—316no—
205C3×#41—68no—
206C2×C3×Q8—34no—
207C2×C3×Q8—91no—
208C2×C3×D4—24no—
209C2×C3×D4—64no—
210C2×C3×D4D3214yes—
211C2×C3×D4C2×C3⋊D311yescharacteristic
212C22×C3×C4—34no—
213C22×C3×C4—64no—
214C2×#67—21no—
215C22×Dic3—21no—
216C22×Dic3—184no—
217D3×Q8—94no—
218D3×Q8—182no—
219#237—22no—
220#237C2×C3⋊D314yes—
221C2×C4×D3—181no—
222C2×C4×D3C2×C3⋊D311yescharacteristic
223C2×Dic6—34no—
224C2×Dic6—916no—
225C2×Dic6—94no—
226C2×Dic6—91no—
227C2×Dic6—184no—
228C2×Dic6—188no—
229C2×Dic6—184no—
230C2×Dic6—181no—
231C2×Dic6C2×C3⋊D311yescharacteristic
232C3×SD16—94no—
233C2×C3×C8—91no—
234#216—916no—
235#216—916no—
236#216—94no—
237#214—316no—
238#214—916no—
239#214—94no—
240#214—1816no—
241#208—916no—
242C2×C3⋊C8—34no—
243C2×C3⋊C8—94no—
244C2×C3⋊C8—91no—
245C2×C3⋊C8—184no—
246Dic12—94no—
247ΓL(1,𝔽25)—94no—
248C22×C32—22no—
249C22×C32—24no—
250C22×C32—28no—
251C22×C32—32no—
252C22×C32—48no—
253C22×C32—42no—
254C22×C32—48no—
255C22×C32—62no—
256C22×C32—88no—
257C22×C32D3×D411yescharacteristic
258C22×C32C2×#6714yes—
259C2×C3×D3—28no—
260C2×C3×D3C2×#6718yes—
261C32×C4—24no—
262C32×C4—24no—
263C32×C4—32no—
264C32×C4—62no—
265C32×C4D3×D411yescharacteristic
266C32×C4C2×#6714yes—
267C32×C4C23×D314yes—
268C32×C4C23×D311yescharacteristic
269C3⋊Dic3—22no—
270C3⋊Dic3—62no—
271C3⋊Dic3—68no—
272C3⋊Dic3—128no—
273C3×Dic3—28no—
274C3×Dic3—68no—
275C3×Dic3C2×#6718yes—
276C2×#41—31no—
277C22×Q8—271no—
278#137—278no—
279C2×Q16—271no—
280C2×Q16—271no—
281C2×SD16—91no—
282C2×SD16—271no—
283C2×M16—271no—
284C33C2×#13711yescharacteristic, complemented, Sylow
285C23×C3—24no—
286C23×C3—21no—
287C23×C3—34no—
288C23×C3—44no—
289C22×D3#63811yescharacteristic
290C3×Q8—38no—
291C3×Q8—64no—
292C3×Q8—92no—
293C3×Q8—181no—
294C3×D4—24no—
295C3×D4—21no—
296C3×D4—28no—
297C3×D4—44no—
298C3×D4—616no—
299C3×D4C2×D3218yes—
300C3×D4C22×C3⋊D312yes—
301C2×C3×C4—24no—
302C2×C3×C4—34no—
303C2×C3×C4—38no—
304C2×C3×C4—38no—
305C2×C3×C4—616no—
306C2×C3×C4—64no—
307C2×C3×C4—181no—
308C2×C3×C4C2×D3214yes—
309C2×C3×C4C22×C3⋊D311yescharacteristic
310#67—24no—
311C2×Dic3—21no—
312C2×Dic3—24no—
313C2×Dic3—64no—
314C2×Dic3—1816no—
315C2×Dic3—184no—
316C2×Dic3—181no—
317C2×Dic3—364no—
318C2×Dic3#63811yescharacteristic
319C4×D3—184no—
320C4×D3#63812yes—
321C4×D3C22×C3⋊D312yes—
322Dic6—21no—
323Dic6—38no—
324Dic6—64no—
325Dic6—98no—
326Dic6—92no—
327Dic6—188no—
328Dic6—1816no—
329Dic6—184no—
330Dic6—181no—
331Dic6—184no—
332Dic6—364no—
333Dic6C22×C3⋊D312yes—
334C3×C8—92no—
335C3⋊C8—38no—
336C3⋊C8—98no—
337C3⋊C8—92no—
338C3⋊C8—188no—
339C2×C32—24no—
340C2×C32—28no—
341C2×C32—42no—
342C2×C32—48no—
343C2×C32—62no—
344C2×C32—88no—
345C2×C32#91512yescomplemented
346C2×C32#95118yescomplemented
347C2×C32C2×D3×D411yescharacteristic
348C2×C32C22×#6714yes—
349C3×D3—28no—
350#41—34no—
351#41—62no—
352C2×Q8—31no—
353C2×Q8—91no—
354C2×Q8—274no—
355C2×Q8—271no—
356C2×Q8—541no—
357C2×Q8—542no—
358C2×D4—61no—
359C2×D4D3×C3⋊D311yescharacteristic
360C22×C4—31no—
361C22×C4—61no—
362C22×C4—541no—
363Q16—274no—
364Q16—274no—
365SD16—94no—
366SD16—274no—
367M16—274no—
368C2×C8—91no—
369C2×C8—271no—
370C22×C3—28no—
371C22×C3—22no—
372C22×C3—24no—
373C22×C3—38no—
374C22×C3—48no—
375C22×C3—42no—
376C22×C3—48no—
377C22×C3—68no—
378C22×C3—88no—
379C22×C3D3×#6714yes—
380C22×C3C2×#63811yescharacteristic
381C2×D3—22no—
382C2×D3C2×#63812yes—
383C3×C4—24no—
384C3×C4—24no—
385C3×C4—38no—
386C3×C4—68no—
387C3×C4—182no—
388C3×C4D3×#6714yes—
389C3×C4C2×#63811yescharacteristic
390C3×C4C22×D3214yes—
391C3×C4C23×C3⋊D311yescharacteristic
392Dic3—22no—
393Dic3—68no—
394Dic3—188no—
395Dic3—182no—
396Dic3—368no—
397Dic3C2×#63812yes—
398C32—24no—
399C32C2×#91511yescharacteristic, complemented
400C32C2×#95114yescomplemented
401C23—21no—
402C23—31no—
403Q8—32no—
404Q8—61no—
405Q8—92no—
406Q8—181no—
407Q8—272no—
408Q8—544no—
409Q8—541no—
410D4—21no—
411D4—64no—
412D4C2×D3×C3⋊D312yes—
413C2×C4—31no—
414C2×C4—32no—
415C2×C4—32no—
416C2×C4—64no—
417C2×C4—61no—
418C2×C4—181no—
419C2×C4—544no—
420C2×C4—541no—
421C2×C4C2×D3×C3⋊D311yescharacteristic
422C8—92no—
423C8—272no—
424C2×C3—28no—
425C2×C3—24no—
426C2×C3—48no—
427C2×C3—42no—
428C2×C3—68no—
429C2×C3—88no—
430C2×C3#6392918yescomplemented
431C2×C3#6416012yescomplemented
432C2×C3C2×D3×#6714yes—
433C2×C3C22×#63811yescharacteristic
434D3—22no—
435C22—22no—
436C22—32no—
437C22—42no—
438C22—62no—
439C22D3×#63811yescharacteristic
440C4—32no—
441C4—62no—
442C4—182no—
443C4—542no—
444C4D3×#63811yescharacteristic
445C4C22×D3×C3⋊D311yescharacteristic
446C3—24no—
447C3C2×#6392914yescomplemented
448C3C2×#6416011yescharacteristic, complemented
449C2—42no—
450C2—62no—
451C2#1172355012yesdirect product factor
452C2C2×D3×#63811yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC33×C4C24
FrattiniC4C22×D3×C3⋊D3
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radicalC2×C32×#237C2
hypercenterC2×D4D3×C3⋊D3
hypocenterC33C2×#137
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
2. C2×D3×#634C2semidirect
3. C2×#88861C2semidirect
4. C2×C3×#3754C2semidirect
5. #11723550C2direct (N × H)
6. C2×#88799C2semidirect
7. C2×#88798C2semidirect
8. C2×#88786C2semidirect
23. D3×#634C22semidirect
24. #88861C22semidirect
26. C3×#3754C22semidirect
28. #88799C22semidirect
29. #88798C22semidirect
30. #88786C22semidirect
41. C2×#3754D3semidirect
106. #3754C2×D3semidirect
201. C2×C33#137semidirect
284. C33C2×#137semidirect
345. C2×C32#915semidirect
346. C2×C32#951semidirect
399. C32C2×#915semidirect
400. C32C2×#951semidirect
430. C2×C3#63929semidirect
431. C2×C3#64160semidirect
447. C3C2×#63929semidirect
448. C3C2×#64160semidirect
451. C2#11723550direct (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×D3×#6342
C2×#888612
C2×C3×#37542
#117235502
C2×#887992
C2×#887982
C2×#887862
C2×#641603
C2×#639293