Finite Group Databasebeta

← #13149016

Subgroups — #13149016

Order 1728.

802 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721749C211yesmaximal, characteristic, complemented
2C3×#63455C211yesmaximal, characteristic, complemented
3C9×#887C211yesmaximal, characteristic, complemented
4#11721006C211yesmaximal, characteristic
5C4×#6192C212yesmaximal, complemented
6C4×#6189C211yesmaximal, characteristic, complemented
7#11720953C211yesmaximal, characteristic, complemented
8#11720951C211yesmaximal, characteristic, complemented
9#11720941C212yesmaximal, complemented
10#11720940C212yesmaximal, complemented
11#11720938C212yesmaximal, complemented
12#10594566—31nomaximal
13#10589687—31nomaximal
14C2×C4⋊(C3⋊D9)C2211yescharacteristic
15C2×C4×C3⋊D9C2212yes—
16C2×C3×D36C2211yescharacteristic
17C2×C3×C4×D9C2212yes—
18C2×C9×Dic6C2211yescharacteristic
19C2×#6192C2212yes—
20C2×#6192C2212yes—
21C2×#6189C2211yescharacteristic
22C3×C9×C4⋊C4C2211yescharacteristic
23#88536C2212yes—
24C4×C3⋊Dic9C2211yescharacteristic
25C3×(C2×D9)⋊C4C2212yes—
26C9×Dic3⋊C4C2212yes—
27C4×C9×Dic3C2211yescharacteristic
28C4×C9×Dic3C2212yes—
29C3×C4×Dic9C2211yescharacteristic
30#88445C2212yes—
31#88445C2212yes—
32(C2×D9)⋊Dic3C2212yes—
33(C2×D9)⋊Dic3C2212yes—
34Dic3⋊Dic9C2211yescharacteristic
35#88442C2211yescharacteristic
36Dic9⋊Dic3C2211yescharacteristic
37#64102—31no—
38C3×#893—31no—
39C3×#887D311yescharacteristic
40C4×#627—31no—
41#63911—31no—
42C4×#626—32no—
43#63864—31no—
44#63863—31no—
45#63853—32no—
46#63852—32no—
47#63850—32no—
48C4×Q8×C9—31no—
49#63509—31no—
50#63509—31no—
51#63456—32no—
52#63456—32no—
53#63455—31no—
54#63455—61no—
55#63455D311yescharacteristic, complemented
56#63437—32no—
57#63436—31no—
58C4×D36—32no—
59C4×Dic18—31no—
60C22×C3⋊D9C2312yes—
61C22×C3×D9C2312yes—
62C2×C3×C4×C9C2312yes—
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3⋊Dic9C2311yescharacteristic
65C2×C3⋊Dic9C2311yescharacteristic
66C4⋊(C3⋊D9)—22no—
67C4×C3⋊D9—42no—
68C2×C9×Dic3C2311yescharacteristic
69C2×C9×Dic3C2312yes—
70C2×C9×Dic3C2311yescharacteristic
71C2×C3×Dic9C2311yescharacteristic
72C2×C3×Dic9C2311yescharacteristic
73C3×D36—22no—
74C3×C4×D9—42no—
75C9×Dic6—22no—
76#6192—24no—
77#6192—42no—
78#6189—22no—
79#5585—91no—
80#5559—91no—
81C2×C4⋊(C3⋊D3)—31no—
82C2×C4×C3⋊D3—32no—
83C2×C3×D12—31no—
84C2×C3×C4×D3—32no—
85C2×C3×Dic6C2×D311yescharacteristic
86C2×#627—31no—
87C2×#626—32no—
88C2×#626—32no—
89C32×C4⋊C4C2×D311yescharacteristic
90#3752—32no—
91C4×C3⋊Dic3—31no—
92C3×(C2×D3)⋊C4—32no—
93C3×Dic3⋊C4C2×D312yes—
94C3×C4×Dic3—31no—
95C3×C4×Dic3C2×D312yes—
96C3×C4×Dic3C2×D311yescharacteristic
97#3724—31no—
98Dic3⋊Dic3—31no—
99Dic3⋊Dic3—31no—
100#3722—32no—
101#3722—32no—
102(C2×D3)⋊Dic3—32no—
103(C2×D3)⋊Dic3—32no—
104C2×Q8×C9—31no—
105C2×D36—32no—
106C2×D36—31no—
107C2×D36—32no—
108C2×D36—61no—
109C2×D36C2×D311yescharacteristic
110C2×C4×D9—32no—
111C2×C4×D9—62no—
112C2×C4×D9C2×D312yes—
113C2×Dic18—31no—
114C9×C4⋊C4—21no—
115C9×C4⋊C4—32no—
116C9×C4⋊C4C2×D311yescharacteristic
117C42×C9—31no—
118C42×C9—32no—
119(C2×D9)⋊C4—32no—
120(C2×D9)⋊C4—32no—
121(C2×D9)⋊C4—32no—
122(C2×D9)⋊C4—32no—
123(C2×D9)⋊C4—32no—
124(C2×D9)⋊C4—62no—
125(C2×D9)⋊C4C2×D312yes—
126C4⋊Dic9—31no—
127Dic9⋊C4—31no—
128Dic9⋊C4—31no—
129C4×Dic9—31no—
130C4×Dic9—61no—
131C4×Dic9C2×D311yescharacteristic
132C22×C3×C9C2411yescharacteristic
133C2×C3⋊D9—22no—
134C2×C3⋊D9—22no—
135C2×C3⋊D9—22no—
136C2×C3×D9—22no—
137C2×C3×D9—22no—
138C2×C3×D9—22no—
139C3×C4×C9—22no—
140C3×C4×C9#4112yes—
141C3⋊Dic9—21no—
142C3⋊Dic9—21no—
143C9×Dic3—22no—
144C9×Dic3—21no—
145C9×Dic3#4112yes—
146C3×Dic9—21no—
147C3×Dic9—21no—
148C3×C4×Q8—31no—
149C3×C4×D4—91no—
150#947—91no—
151#947—91no—
152#940—91no—
153#936—91no—
154#929—92no—
155C4×#67—92no—
156#894—92no—
157#894—92no—
158#893—31no—
159#893—91no—
160#893—91no—
161#893—181no—
162#887D911yescharacteristic, complemented
163#885—92no—
164#883—92no—
165#875—92no—
166#875—91no—
167#874—91no—
168C4×D12—92no—
169C4×Dic6—91no—
170C4×Dic6—91no—
171C22×C3⋊D3—32no—
172C22×C3×D3—32no—
173C2×C32×C4C22×D312yes—
174C2×C32×C4C22×D311yescharacteristic
175C2×C3⋊Dic3—31no—
176C2×C3⋊Dic3—31no—
177C4⋊(C3⋊D3)—62no—
178C4×C3⋊D3—122no—
179C2×C3×Dic3—31no—
180C2×C3×Dic3—31no—
181C2×C3×Dic3C22×D312yes—
182C2×C3×Dic3C22×D311yescharacteristic
183C2×C3×Dic3C22×D311yescharacteristic
184C3×D12—62no—
185C3×C4×D3—122no—
186C3×Dic6—22no—
187#627—62no—
188#626—64no—
189#626—122no—
190C22×D9—32no—
191C22×D9—62no—
192C22×D9C22×D312yes—
193Q8×C9—62no—
194C2×C4×C9—21no—
195C2×C4×C9—22no—
196C2×C4×C9—31no—
197C2×C4×C9—31no—
198C2×C4×C9—32no—
199C2×C4×C9C22×D312yes—
200C2×C4×C9C22×D311yescharacteristic
201C2×Dic9—31no—
202C2×Dic9—31no—
203C2×Dic9—61no—
204C2×Dic9—61no—
205C2×Dic9C22×D311yescharacteristic
206C2×Dic9C22×D311yescharacteristic
207D36—22no—
208D36—64no—
209D36—64no—
210D36—62no—
211D36—122no—
212C4×D9—42no—
213C4×D9—122no—
214C4×D9—124no—
215Dic18—62no—
216#559—271noSylow
217C2×C3×C9C2×#4111yescharacteristic
218C2×C3×C9C2×#4111yescharacteristic
219C2×C3×C9H(5,𝔽2)11yescharacteristic
220C3⋊D9—42no—
221C3×D9—42no—
222C2×C3×Q8—31no—
223C2×C3×D4—91no—
224C22×C3×C4—92no—
225C2×#67—92no—
226C2×#67—92no—
227C2×D12—31no—
228C2×D12—92no—
229C2×D12—91no—
230C2×D12—91no—
231C2×D12—92no—
232C2×D12—181no—
233C2×C4×D3—32no—
234C2×C4×D3—92no—
235C2×C4×D3—92no—
236C2×C4×D3—182no—
237C2×Dic6—91no—
238C2×Dic6—91no—
239C2×Dic6C2×D911yescharacteristic
240C3×C4⋊C4—21no—
241C3×C4⋊C4—32no—
242C3×C4⋊C4C2×D911yescharacteristic
243C3×C4⋊C4D3211yescharacteristic
244C3×C22⋊C4—92no—
245C3×C42—31no—
246C3×C42—32no—
247C3×C42—91no—
248#217—92no—
249#217—92no—
250(C2×D3)⋊C4—32no—
251(C2×D3)⋊C4—92no—
252(C2×D3)⋊C4—92no—
253(C2×D3)⋊C4—92no—
254(C2×D3)⋊C4—92no—
255(C2×D3)⋊C4—92no—
256(C2×D3)⋊C4—92no—
257(C2×D3)⋊C4—92no—
258(C2×D3)⋊C4—92no—
259(C2×D3)⋊C4—182no—
260C4⋊Dic3—91no—
261C4⋊Dic3—91no—
262Dic3⋊C4—91no—
263Dic3⋊C4—91no—
264Dic3⋊C4—91no—
265Dic3⋊C4—91no—
266Dic3⋊C4C2×D912yes—
267C4×Dic3—31no—
268C4×Dic3—91no—
269C4×Dic3—91no—
270C4×Dic3—181no—
271C4×Dic3C2×D911yescharacteristic
272C4×Dic3C2×D912yes—
273C22×C32C23×D311yescharacteristic
274C2×C3⋊D3—62no—
275C2×C3⋊D3—62no—
276C2×C3⋊D3—62no—
277C2×C3×D3—62no—
278C2×C3×D3—62no—
279C2×C3×D3—62no—
280C32×C4—22no—
281C32×C4#23512yes—
282C3⋊Dic3—61no—
283C3⋊Dic3—61no—
284C3×Dic3—22no—
285C3×Dic3—21no—
286C3×Dic3—61no—
287C3×Dic3—61no—
288C3×Dic3#23912yes—
289C22×C9—21no—
290C22×C9C23×D311yescharacteristic
291C2×D9—22no—
292C2×D9—22no—
293C2×D9—22no—
294C2×D9—64no—
295C2×D9—62no—
296C2×D9—62no—
297C2×D9—62no—
298C2×D9—122no—
299C2×D9—122no—
300C4×C9—22no—
301C4×C9—22no—
302C4×C9—32no—
303C4×C9—42no—
304C4×C9—62no—
305C4×C9—61no—
306C4×C9#23512yes—
307Dic9—21no—
308Dic9—21no—
309Dic9—62no—
310Dic9—61no—
311Dic9—61no—
312Dic9—121no—
313#127—271no—
314#124—272no—
315#124—271no—
316#124—271no—
317#123—272no—
318#123—272no—
319C4×Q8—31no—
320C4×Q8—271no—
321C4×D4—91no—
322C4×D4—272no—
323C4×D4—271no—
324C3×C9#55911yescharacteristic, complemented, Sylow
325C23×C3—92no—
326C22×D3—32no—
327C22×D3—92no—
328C22×D3—92no—
329C22×D3—182no—
330C3×Q8—62no—
331C3×D4—182no—
332C2×C3×C4—21no—
333C2×C3×C4—22no—
334C2×C3×C4—31no—
335C2×C3×C4—31no—
336C2×C3×C4—32no—
337C2×C3×C4—91no—
338C2×C3×C4—91no—
339C2×C3×C4—362no—
340C2×C3×C4C22×D912yes—
341C2×C3×C4C22×D911yescharacteristic
342C2×C3×C4C2×D3212yes—
343C2×C3×C4C2×D3211yescharacteristic
344#67—184no—
345#67—362no—
346C2×Dic3—31no—
347C2×Dic3—31no—
348C2×Dic3—91no—
349C2×Dic3—91no—
350C2×Dic3—91no—
351C2×Dic3—91no—
352C2×Dic3—181no—
353C2×Dic3—181no—
354C2×Dic3C22×D911yescharacteristic
355C2×Dic3C22×D912yes—
356C2×Dic3C22×D911yescharacteristic
357D12—62no—
358D12—184no—
359D12—184no—
360D12—182no—
361D12—182no—
362D12—362no—
363C4×D3—122no—
364C4×D3—362no—
365C4×D3—362no—
366C4×D3—364no—
367Dic6—22no—
368Dic6—182no—
369Dic6—182no—
370C2×C32C2×#23511yescharacteristic
371C2×C32C2×#23911yescharacteristic
372C2×C32#100911yescharacteristic
373C3⋊D3—122no—
374C3×D3—122no—
375C2×C9—21no—
376C2×C9—21no—
377C2×C9—21no—
378C2×C9C2×#23511yescharacteristic
379C2×C9#100411yescharacteristic
380C2×C9D3×#4111yescharacteristic
381D9—42no—
382D9—124no—
383D9—122no—
384C2×Q8—31no—
385C2×Q8—271no—
386C2×D4—91no—
387C2×D4—272no—
388C2×D4—272no—
389C2×D4—271no—
390C22×C4—92no—
391C22×C4—272no—
392C4⋊C4—32no—
393C4⋊C4—271no—
394C4⋊C4—271no—
395C4⋊C4—271no—
396C4⋊C4D3×D911yescharacteristic
397C22⋊C4—92no—
398C22⋊C4—272no—
399C22⋊C4—272no—
400C22⋊C4—272no—
401C22⋊C4—272no—
402C22⋊C4—272no—
403C42—32no—
404C42—31no—
405C42—91no—
406C42—271no—
407C22×C3—21no—
408C22×C3—182no—
409C22×C3—182no—
410C22×C3—182no—
411C22×C3C23×D911yescharacteristic
412C22×C3C22×D3211yescharacteristic
413C2×D3—62no—
414C2×D3—62no—
415C2×D3—62no—
416C2×D3—184no—
417C2×D3—182no—
418C2×D3—182no—
419C2×D3—182no—
420C2×D3—182no—
421C2×D3—182no—
422C2×D3—182no—
423C2×D3—362no—
424C2×D3—362no—
425C3×C4—22no—
426C3×C4—22no—
427C3×C4—22no—
428C3×C4—32no—
429C3×C4—42no—
430C3×C4—62no—
431C3×C4—61no—
432C3×C4—181no—
433C3×C4—181no—
434C3×C4#369712yes—
435C3×C4#379812yes—
436Dic3—22no—
437Dic3—21no—
438Dic3—61no—
439Dic3—61no—
440Dic3—182no—
441Dic3—181no—
442Dic3—181no—
443Dic3—181no—
444Dic3—181no—
445Dic3—361no—
446Dic3#370112yes—
447C32#558511yescharacteristic
448C9—21no—
449C9#555911yescharacteristic, complemented
450C23—92no—
451C23—272no—
452Q8—62no—
453Q8—542no—
454D4—182no—
455D4—544no—
456D4—542no—
457D4—1082no—
458C2×C4—31no—
459C2×C4—32no—
460C2×C4—31no—
461C2×C4—91no—
462C2×C4—91no—
463C2×C4—271no—
464C2×C4—271no—
465C2×C4—362no—
466C2×C4—1082no—
467C2×C4C2×D3×D912yes—
468C2×C4C2×D3×D911yescharacteristic
469C2×C3—21no—
470C2×C3—21no—
471C2×C3—21no—
472C2×C3—362no—
473C2×C3C2×#369711yescharacteristic
474C2×C3C2×#370111yescharacteristic
475C2×C3#6371511yescharacteristic
476C2×C3C2×#379811yescharacteristic
477C2×C3#6431411yescharacteristic
478C2×C3D3×#23911yescharacteristic
479D3—122no—
480D3—364no—
481D3—362no—
482D3—362no—
483C22—182no—
484C22—182no—
485C22—182no—
486C22—542no—
487C22—542no—
488C22—542no—
489C22C22×D3×D911yescharacteristic
490C4—22no—
491C4—32no—
492C4—61no—
493C4—62no—
494C4—181no—
495C4—181no—
496C4—541no—
497C4—541no—
498C4#8864012yes—
499C3—21no—
500C3#1058968711yescharacteristic, complemented
501C3#1059456611yescharacteristic
502C2—362no—
503C2—1082no—
504C2C2×#8864011yescharacteristic
505C2#1172250111yescharacteristic
506C2D3×#370111yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
prime radical#11721006C2
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#559
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #11721749C2semidirect
2. C3×#63455C2semidirect
3. C9×#887C2semidirect
5. C4×#6192C2semidirect
6. C4×#6189C2semidirect
7. #11720953C2semidirect
8. #11720951C2semidirect
9. #11720941C2semidirect
10. #11720940C2semidirect
11. #11720938C2semidirect
55. #63455D3semidirect
162. #887D9semidirect
324. C3×C9#559semidirect
449. C9#5559semidirect
500. C3#10589687semidirect

Maximal subgroups

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

subgroupindex
#117217492
C3×#634552
C9×#8872
#117210062
C4×#61922
C4×#61892
#117209532
#117209512
#117209412
#117209402
#117209382
#105945663
#105896873