Finite Group Databasebeta

← #13156839

Subgroups — #13156839

Order 1728.

1026 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88815C211yesmaximal, characteristic
2C2×Dic3⋊(C3⋊Dic3)C212yesmaximal
3C2×Dic3×C3⋊Dic3C211yesmaximal, characteristic
4C2×C3×#3724C211yesmaximal, characteristic
5C3×#63579C212yesmaximal
6#10591328—31nomaximal
7#10591268—32nomaximal
8C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
9C22×C3×C3⋊Dic3C2212yes—
10C22×C3×C3⋊Dic3C2211yescharacteristic
11C22×C32×Dic3C2212yes—
12C22×C32×Dic3C2211yescharacteristic
13#88815—22no—
14Dic3⋊(C3⋊Dic3)—24no—
15Dic3×C3⋊Dic3—22no—
16C3×#3724C414yescomplemented
17C2×C4⋊(C3⋊Dic3)—32no—
18C2×C4×C3⋊Dic3—31no—
19C2×C3×Dic3⋊C4—32no—
20C2×#3724—31no—
21C2×#3724D311yescharacteristic
22C2×Dic3⋊Dic3—32no—
23C2×Dic3⋊Dic3—32no—
24C2×Dic3⋊Dic3—32no—
25C2×Dic3⋊Dic3—62no—
26C2×Dic32—32no—
27C2×Dic32—61no—
28C3×C3⋊#95—32no—
29C3×C3⋊#95—32no—
30#63579D312yes—
31C23×C33C2311yescharacteristic
32C2×C3⋊(C3⋊Dic3)—21no—
33C2×C3⋊(C3⋊Dic3)—21no—
34C2×C3⋊(C3⋊Dic3)D412yes—
35C2×C3×C3⋊Dic3—22no—
36C2×C3×C3⋊Dic3—22no—
37C2×C3×C3⋊Dic3—22no—
38C2×C3×C3⋊Dic3C2×C412yes—
39C2×C3×C3⋊Dic3D412yes—
40C2×C3×C3⋊Dic3Q812yes—
41C2×C32×Dic3—21no—
42C2×C32×Dic3—22no—
43C2×C32×Dic3—22no—
44C2×C32×Dic3—21no—
45C2×C32×Dic3C2×C414yes—
46C2×C32×Dic3Q812yes—
47#4984—91no—
48#4647—92no—
49C22×C32×C4—32no—
50C22×C32×C4—31no—
51C22×C3⋊Dic3—31no—
52C22×C3⋊Dic3—32no—
53C22×C3⋊Dic3—61no—
54C22×C3⋊Dic3—62no—
55C22×C3⋊Dic3—121no—
56C22×C3⋊Dic3C2×D312yes—
57C22×C3⋊Dic3C2×D311yescharacteristic
58C22×C3×Dic3—21no—
59C22×C3×Dic3—22no—
60C22×C3×Dic3—32no—
61C22×C3×Dic3—32no—
62C22×C3×Dic3—32no—
63C22×C3×Dic3—61no—
64C22×C3×Dic3—62no—
65C22×C3×Dic3C2×D312yes—
66C22×C3×Dic3C2×D312yes—
67C22×C3×Dic3C2×D312yes—
68C4⋊(C3⋊Dic3)—64no—
69C4×C3⋊Dic3—62no—
70C3×Dic3⋊C4—38no—
71#3724—62no—
72#3724Dic314yescomplemented
73Dic3⋊Dic3—64no—
74Dic3⋊Dic3—64no—
75Dic3⋊Dic3—68no—
76Dic3⋊Dic3—64no—
77Dic32—64no—
78Dic32—64no—
79C22×C33C22×C411yescharacteristic
80C22×C33C2×D411yescharacteristic
81C22×C33C2×D411yescharacteristic
82C22×C33C2×Q811yescharacteristic
83C22×C33C2×Q811yescharacteristic
84C22×C33#4111yescharacteristic
85C22×C33#4111yescharacteristic
86C3⋊(C3⋊Dic3)—22no—
87C3×C3⋊Dic3—42no—
88C3×C3⋊Dic3C4⋊C414yescomplemented
89C32×Dic3—24no—
90C32×Dic3—22no—
91C2×C3×C4⋊C4—91no—
92C2×C4⋊Dic3—91no—
93C2×C4⋊Dic3—92no—
94C2×C4⋊Dic3—92no—
95C2×C4⋊Dic3—182no—
96C2×Dic3⋊C4—32no—
97C2×Dic3⋊C4—92no—
98C2×Dic3⋊C4—92no—
99C2×C4×Dic3—92no—
100C2×C4×Dic3—91no—
101C2×C4×Dic3—181no—
102C3×#95—92no—
103C3⋊#95—32no—
104C3⋊#95—32no—
105C23×C32—22no—
106C23×C32—21no—
107C23×C32—41no—
108C23×C32C22×D312yes—
109C23×C32C22×D311yescharacteristic
110C2×C32×C4—34no—
111C2×C32×C4—32no—
112C2×C32×C4—61no—
113C2×C32×C4—62no—
114C2×C32×C4—62no—
115C2×C32×C4—61no—
116C2×C3⋊Dic3—22no—
117C2×C3⋊Dic3—22no—
118C2×C3⋊Dic3—22no—
119C2×C3⋊Dic3—32no—
120C2×C3⋊Dic3—34no—
121C2×C3⋊Dic3—62no—
122C2×C3⋊Dic3—64no—
123C2×C3⋊Dic3—61no—
124C2×C3⋊Dic3—61no—
125C2×C3⋊Dic3—62no—
126C2×C3⋊Dic3—62no—
127C2×C3⋊Dic3—62no—
128C2×C3⋊Dic3—62no—
129C2×C3⋊Dic3—122no—
130C2×C3⋊Dic3—122no—
131C2×C3⋊Dic3—122no—
132C2×C3⋊Dic3—122no—
133C2×C3⋊Dic3—122no—
134C2×C3⋊Dic3Dic612yes—
135C2×C3⋊Dic3D1212yes—
136C2×C3⋊Dic3C2×Dic312yes—
137C2×C3×Dic3—22no—
138C2×C3×Dic3—24no—
139C2×C3×Dic3—22no—
140C2×C3×Dic3—24no—
141C2×C3×Dic3—22no—
142C2×C3×Dic3—22no—
143C2×C3×Dic3—22no—
144C2×C3×Dic3—22no—
145C2×C3×Dic3—22no—
146C2×C3×Dic3—22no—
147C2×C3×Dic3—24no—
148C2×C3×Dic3—22no—
149C2×C3×Dic3—34no—
150C2×C3×Dic3—34no—
151C2×C3×Dic3—34no—
152C2×C3×Dic3—62no—
153C2×C3×Dic3—62no—
154C2×C3×Dic3—62no—
155C2×C3×Dic3—64no—
156C2×C3×Dic3—62no—
157C2×C3×Dic3—62no—
158C2×C3×Dic3—62no—
159C2×C3×Dic3—62no—
160C2×C3×Dic3—62no—
161C2×C3×Dic3—62no—
162C2×C3×Dic3—122no—
163C2×C3×Dic3—122no—
164C2×C3×Dic3Dic614yes—
165C2×C3×Dic3C4×D314yes—
166C2×C3×Dic3C2×Dic314yes—
167#389—271noSylow
168C2×C33C2×C4⋊C411yescharacteristic
169C2×C33C4×D411yescharacteristic
170C2×C33C4×Q811yescharacteristic
171C2×C33#12111yescharacteristic
172C2×C33C22⋊Q811yescharacteristic
173C2×C33#12511yescharacteristic
174C2×C33C4⋊Q811yescharacteristic
175C22×C3×C4—32no—
176C22×C3×C4—32no—
177C22×C3×C4—32no—
178C22×C3×C4—61no—
179C22×C3×C4—62no—
180C22×C3×C4—91no—
181C22×C3×C4—92no—
182C22×Dic3—32no—
183C22×Dic3—32no—
184C22×Dic3—32no—
185C22×Dic3—61no—
186C22×Dic3—62no—
187C22×Dic3—92no—
188C22×Dic3—91no—
189C22×Dic3—181no—
190C22×Dic3—182no—
191C22×Dic3—361no—
192C22×Dic3D3212yes—
193C22×Dic3D3211yescharacteristic
194C3×C4⋊C4—94no—
195C4⋊Dic3—182no—
196C4⋊Dic3—184no—
197C4⋊Dic3—184no—
198C4⋊Dic3—188no—
199Dic3⋊C4—38no—
200Dic3⋊C4—184no—
201Dic3⋊C4—184no—
202C4×Dic3—182no—
203C4×Dic3—184no—
204C4×Dic3—184no—
205C22×C32—22no—
206C22×C32—22no—
207C22×C32—21no—
208C22×C32—21no—
209C22×C32—22no—
210C22×C32—21no—
211C22×C32—22no—
212C22×C32—21no—
213C22×C32—22no—
214C22×C32—22no—
215C22×C32—21no—
216C22×C32—21no—
217C22×C32—22no—
218C22×C32—21no—
219C22×C32—41no—
220C22×C32—41no—
221C22×C32—41no—
222C22×C32—41no—
223C22×C32—41no—
224C22×C32—41no—
225C22×C32—41no—
226C22×C32C2×Dic611yescharacteristic
227C22×C32C2×Dic612yes—
228C22×C32C2×C4×D312yes—
229C22×C32C2×D1211yescharacteristic
230C22×C32#23512yes—
231C22×C32D3×D412yes—
232C22×C32D3×D412yes—
233C22×C32D3×D411yescharacteristic
234C22×C32#23712yes—
235C22×C32#23711yescharacteristic
236C22×C32D3×Q811yescharacteristic
237C22×C32D3×Q812yes—
238C22×C32#23911yescharacteristic
239C22×C32C22×Dic311yescharacteristic
240C32×C4—64no—
241C32×C4—62no—
242C3⋊Dic3—42no—
243C3⋊Dic3—62no—
244C3⋊Dic3—64no—
245C3⋊Dic3—64no—
246C3⋊Dic3—124no—
247C3⋊Dic3—124no—
248C3⋊Dic3C4⋊Dic314yescomplemented
249C3×Dic3—28no—
250C3×Dic3—24no—
251C3×Dic3—24no—
252C3×Dic3—24no—
253C3×Dic3—24no—
254C3×Dic3—38no—
255C3×Dic3—64no—
256C3×Dic3—124no—
257C3×Dic3—122no—
258C3×Dic3—122no—
259C2×C4⋊C4—91no—
260C2×C4⋊C4—272no—
261C2×C4⋊C4—271no—
262C2×C42—271no—
263#95—92no—
264C33#38911yescharacteristic, complemented, Sylow
265C23×C3—21no—
266C23×C3—22no—
267C23×C3—41no—
268C23×C3C2×D3212yes—
269C23×C3C2×D3211yescharacteristic
270C2×C3×C4—34no—
271C2×C3×C4—34no—
272C2×C3×C4—34no—
273C2×C3×C4—64no—
274C2×C3×C4—62no—
275C2×C3×C4—62no—
276C2×C3×C4—64no—
277C2×C3×C4—62no—
278C2×C3×C4—64no—
279C2×C3×C4—62no—
280C2×C3×C4—62no—
281C2×C3×C4—62no—
282C2×C3×C4—62no—
283C2×C3×C4—62no—
284C2×C3×C4—62no—
285C2×C3×C4—92no—
286C2×C3×C4—92no—
287C2×C3×C4—92no—
288C2×C3×C4—182no—
289C2×C3×C4—182no—
290C2×C3×C4—182no—
291C2×Dic3—22no—
292C2×Dic3—21no—
293C2×Dic3—21no—
294C2×Dic3—22no—
295C2×Dic3—34no—
296C2×Dic3—34no—
297C2×Dic3—34no—
298C2×Dic3—62no—
299C2×Dic3—62no—
300C2×Dic3—62no—
301C2×Dic3—64no—
302C2×Dic3—62no—
303C2×Dic3—62no—
304C2×Dic3—62no—
305C2×Dic3—62no—
306C2×Dic3—62no—
307C2×Dic3—62no—
308C2×Dic3—92no—
309C2×Dic3—94no—
310C2×Dic3—122no—
311C2×Dic3—122no—
312C2×Dic3—182no—
313C2×Dic3—184no—
314C2×Dic3—182no—
315C2×Dic3—182no—
316C2×Dic3—181no—
317C2×Dic3—182no—
318C2×Dic3—182no—
319C2×Dic3—181no—
320C2×Dic3—362no—
321C2×Dic3—362no—
322C2×Dic3—362no—
323C2×Dic3—362no—
324C2×Dic3—362no—
325C2×Dic3D3×Dic314yes—
326C2×Dic3#62712yes—
327C2×C32—22no—
328C2×C32—22no—
329C2×C32—22no—
330C2×C32—21no—
331C2×C32—21no—
332C2×C32—22no—
333C2×C32—22no—
334C2×C32—21no—
335C2×C32—22no—
336C2×C32—21no—
337C2×C32—21no—
338C2×C32—21no—
339C2×C32—22no—
340C2×C32—21no—
341C2×C32—41no—
342C2×C32—41no—
343C2×C32—41no—
344C2×C32—41no—
345C2×C32—41no—
346C2×C32—41no—
347C2×C32—41no—
348C2×C32C4×Dic612yes—
349C2×C32#87812yes—
350C2×C32#67⋊C412yes—
351C2×C32#88412yes—
352C2×C32#88811yescharacteristic
353C2×C32#88812yes—
354C2×C32#88912yes—
355C2×C32#89011yescharacteristic
356C2×C32D3×C4⋊C412yes—
357C2×C32#89511yescharacteristic
358C2×C32#89711yescharacteristic
359C2×C32C2×C4⋊Dic311yescharacteristic
360C2×C32D4×Dic311yescharacteristic
361C2×C32Q8×Dic311yescharacteristic
362C22×C4—31no—
363C22×C4—32no—
364C22×C4—91no—
365C22×C4—92no—
366C22×C4—271no—
367C4⋊C4—94no—
368C4⋊C4—544no—
369C4⋊C4—542no—
370C42—542no—
371C22×C3—21no—
372C22×C3—22no—
373C22×C3—22no—
374C22×C3—21no—
375C22×C3—21no—
376C22×C3—22no—
377C22×C3—21no—
378C22×C3—22no—
379C22×C3—21no—
380C22×C3—22no—
381C22×C3—22no—
382C22×C3—21no—
383C22×C3—22no—
384C22×C3—21no—
385C22×C3—41no—
386C22×C3—41no—
387C22×C3—41no—
388C22×C3—41no—
389C22×C3—41no—
390C22×C3—41no—
391C22×C3—41no—
392C22×C3D3×Dic612yes—
393C22×C3D3×Dic612yes—
394C22×C3#379711yescharacteristic
395C22×C3#379811yescharacteristic
396C22×C3#379912yes—
397C22×C3C4×D3211yescharacteristic
398C22×C3D3×D1212yes—
399C22×C3#380211yescharacteristic
400C22×C3C2×D3×Dic312yes—
401C22×C3#380512yes—
402C22×C3#380511yescharacteristic
403C22×C3C2×#62711yescharacteristic
404C22×C3#381112yes—
405C22×C3#381111yescharacteristic
406C3×C4—68no—
407C3×C4—64no—
408C3×C4—64no—
409C3×C4—64no—
410C3×C4—64no—
411C3×C4—94no—
412C3×C4—362no—
413Dic3—24no—
414Dic3—22no—
415Dic3—38no—
416Dic3—64no—
417Dic3—124no—
418Dic3—122no—
419Dic3—122no—
420Dic3—184no—
421Dic3—184no—
422Dic3—182no—
423Dic3—364no—
424Dic3—364no—
425C32—22no—
426C32—21no—
427C32—41no—
428C32#464712yescomplemented
429C32#498411yescharacteristic, complemented
430C23D3311yescharacteristic
431C2×C4—34no—
432C2×C4—32no—
433C2×C4—62no—
434C2×C4—61no—
435C2×C4—62no—
436C2×C4—61no—
437C2×C4—92no—
438C2×C4—92no—
439C2×C4—92no—
440C2×C4—182no—
441C2×C4—182no—
442C2×C4—182no—
443C2×C4—272no—
444C2×C4—541no—
445C2×C4—541no—
446C2×C3—22no—
447C2×C3—21no—
448C2×C3—22no—
449C2×C3—21no—
450C2×C3—21no—
451C2×C3—22no—
452C2×C3—22no—
453C2×C3—21no—
454C2×C3—22no—
455C2×C3—21no—
456C2×C3—22no—
457C2×C3—21no—
458C2×C3—21no—
459C2×C3—22no—
460C2×C3—41no—
461C2×C3—41no—
462C2×C3—41no—
463C2×C3—41no—
464C2×C3—41no—
465C2×C3—41no—
466C2×C3—41no—
467C2×C3#6384012yes—
468C2×C3Dic3×Dic612yes—
469C2×C3#6384512yes—
470C2×C3#6387011yescharacteristic
471C2×C3#6388311yescharacteristic
472C2×C3D3×C4⋊Dic312yes—
473C2×C3#6389312yes—
474C2×C3#6390211yescharacteristic
475C2×C3#6391012yes—
476C2×C3#6391211yescharacteristic
477C2×C3C4×#62711yescharacteristic
478C2×C3#6391911yescharacteristic
479C2×C3#67⋊Dic312yes—
480C2×C3#6398211yescharacteristic
481C22D32×Dic311yescharacteristic
482C22#8895411yescharacteristic
483C22#8895511yescharacteristic
484C22D3×#62711yescharacteristic
485C22#8895711yescharacteristic
486C22#8896311yescharacteristic
487C22#8896511yescharacteristic
488C4—64no—
489C4—62no—
490C4—94no—
491C4—362no—
492C4—542no—
493C3—22no—
494C3—21no—
495C3—41no—
496C3#1059126812yescomplemented
497C3#1059132811yescharacteristic, complemented
498C2#1172290811yescharacteristic
499C2#1172292511yescharacteristic
500C2Dic3×#62711yescharacteristic
501C2#1172296811yescharacteristic
502C2#1172298111yescharacteristic
503C2#1172301811yescharacteristic
504C2#1172303711yescharacteristic
subgroupG/H
center, Frattini, hypercenterC23D33
commutatorC22×C33C22×C4
Fitting, socle, prime radical, cosocleC23×C33C23
hypocenterC33#389
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
16. C3×#3724C4semidirect
72. #3724Dic3semidirect
88. C3×C3⋊Dic3C4⋊C4semidirect
248. C3⋊Dic3C4⋊Dic3semidirect
264. C33#389semidirect
428. C32#4647semidirect
429. C32#4984semidirect
496. C3#10591268semidirect
497. C3#10591328semidirect

Maximal subgroups

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

subgroupindex
C2×#888152
C2×Dic3⋊(C3⋊Dic3)2
C2×Dic3×C3⋊Dic32
C2×C3×#37242
C3×#635792
#105913283
#105912683