Finite Group Databasebeta

← #13150150

Subgroups — #13150150

Order 1728.

712 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721815C211yesmaximal, characteristic, complemented
2C3×C4×#611C211yesmaximal, characteristic
3C9×#924C211yesmaximal, characteristic, complemented
4#11721257C212yesmaximal
5#11721246C212yesmaximal, complemented
6#11720969C211yesmaximal, characteristic, complemented
7#11720950C212yesmaximal, complemented
8#11720944C211yesmaximal, characteristic, complemented
9Dic6×Dic9C211yesmaximal, characteristic, complemented
10#11720907C211yesmaximal, characteristic, complemented
11#11720902C212yesmaximal, complemented
12#10594841—31nomaximal
13#10589743—31nomaximal
14C22×C3×C4×C9C2211yescharacteristic
15C2×C3×#611C2211yescharacteristic
16C2×C3×C4×D9C2211yescharacteristic
17C2×C9×Dic6C2211yescharacteristic
18C2×#6189C2212yes—
19#88551C2212yes—
20C4⋊(C3⋊Dic9)C2211yescharacteristic
21C4⋊(C3⋊Dic9)C2211yescharacteristic
22C4×C3⋊Dic9C2212yes—
23C9×#217C2212yes—
24C3×#3676C2211yescharacteristic
25C3×(C2×D9)⋊C4C2211yescharacteristic
26C9×C4⋊Dic3C2211yescharacteristic
27C9×Dic3⋊C4C2212yes—
28C3×Dic9⋊C4C2211yescharacteristic
29C3×C4×Dic9C2211yescharacteristic
30(C2×D9)⋊Dic3C2212yes—
31(C2×D9)⋊Dic3C2212yes—
32Dic3⋊Dic9C2212yes—
33#88442C2212yes—
34Dic9⋊Dic3C2212yes—
35Dic3×Dic9C2212yes—
36Dic3×Dic9C2212yes—
37#64135—31no—
38C3×C4×#67—31no—
39C3×#924D311yescharacteristic
40#63959—32no—
41#63956—32no—
42#63918—31no—
43#63862—32no—
44#63856—31no—
45Dic3×Dic6—31no—
46#63841—31no—
47#63838—32no—
48C9×C22⋊Q8—31no—
49#63508—31no—
50Q8×Dic9—31no—
51C4×#611D311yescharacteristic
52#63488—31no—
53#63488—61no—
54#63460—32no—
55#63454—31no—
56#63450—31no—
57#63449—32no—
58#63447—32no—
59#63441—32no—
60C23×C3×C9C2311yescharacteristic
61C22×C3×D9C2311yescharacteristic
62C2×C3×C4×C9—22no—
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3⋊Dic9C2312yes—
66C2×C3⋊Dic9C2312yes—
67C3×#611—22no—
68C2×C9×Dic3C2312yes—
69C2×C9×Dic3C2312yes—
70C2×C3×Dic9C2311yescharacteristic
71C2×C3×Dic9C2311yescharacteristic
72C2×C3×Dic9C2311yescharacteristic
73C3×C4×D9—22no—
74C9×Dic6—22no—
75#6189—24no—
76#5641—91no—
77#5542—91no—
78C22×C32×C4C2×D311yescharacteristic
79C2×C3×#67—31no—
80C2×C3×C4×D3—31no—
81C2×C3×Dic6C2×D311yescharacteristic
82C2×#627—32no—
83#3757—32no—
84C4⋊(C3⋊Dic3)—31no—
85C4⋊(C3⋊Dic3)—31no—
86C4×C3⋊Dic3—32no—
87C3×#217—31no—
88C3×#217C2×D312yes—
89C3×(C2×D3)⋊C4—31no—
90C3×C4⋊Dic3C2×D311yescharacteristic
91C3×Dic3⋊C4—31no—
92C3×Dic3⋊C4C2×D312yes—
93C3×C4×Dic3—31no—
94#3724—32no—
95Dic3⋊Dic3—32no—
96Dic3⋊Dic3—32no—
97(C2×D3)⋊Dic3—32no—
98(C2×D3)⋊Dic3—32no—
99Dic32—32no—
100Dic32—32no—
101C2×Q8×C9—31no—
102C22×C4×C9—21no—
103C22×C4×C9C2×D311yescharacteristic
104C2×#611C2×D311yescharacteristic
105C2×C4×D9C2×D311yescharacteristic
106C2×Dic18—32no—
107C9×C4⋊C4—32no—
108C9×C4⋊C4—31no—
109C9×C22⋊C4—32no—
110#3676—32no—
111#3676—62no—
112#3676C2×D311yescharacteristic
113(C2×D9)⋊C4—32no—
114(C2×D9)⋊C4—32no—
115(C2×D9)⋊C4C2×D311yescharacteristic
116C4⋊Dic9—31no—
117C4⋊Dic9—31no—
118C4⋊Dic9—32no—
119C4⋊Dic9—61no—
120C4⋊Dic9—61no—
121Dic9⋊C4—32no—
122Dic9⋊C4—32no—
123Dic9⋊C4C2×D311yescharacteristic
124C4×Dic9—32no—
125C4×Dic9—32no—
126C4×Dic9—32no—
127C4×Dic9—62no—
128C4×Dic9C2×D311yescharacteristic
129C22×C3×C9—21no—
130C22×C3×C9—21no—
131C22×C3×C9—21no—
132C22×C3×C9C2411yescharacteristic
133C2×C3×D9—21no—
134C2×C3×D9—21no—
135C2×C3×D9—21no—
136C3×C4×C9—21no—
137C3×C4×C9#4112yes—
138C3⋊Dic9—22no—
139C3⋊Dic9—22no—
140C9×Dic3—22no—
141C9×Dic3—22no—
142C3×Dic9—21no—
143C3×Dic9—21no—
144C3×Dic9#4112yes—
145C3×C22⋊Q8—31no—
146C3×C4×D4—91no—
147#946—91no—
148Q8×Dic3—91no—
149#936—92no—
150C4×#67—31no—
151#926—91no—
152#926—91no—
153#926—91no—
154#926—181no—
155#924—91no—
156#924D911yescharacteristic, complemented
157#898—92no—
158#892—91no—
159#888—91no—
160#887—92no—
161#887—92no—
162#885—92no—
163#879—92no—
164#879—92no—
165#879—92no—
166#869—91no—
167C4×Dic6—91no—
168C23×C32C22×D311yescharacteristic
169C22×C3×D3—31no—
170C2×C32×C4—22no—
171C2×C32×C4C22×D311yescharacteristic
172C2×C32×C4C22×D311yescharacteristic
173C2×C3⋊Dic3—32no—
174C2×C3⋊Dic3—32no—
175C3×#67—62no—
176C2×C3×Dic3—31no—
177C2×C3×Dic3—31no—
178C2×C3×Dic3—31no—
179C2×C3×Dic3C22×D312yes—
180C2×C3×Dic3C22×D312yes—
181C3×C4×D3—62no—
182C3×Dic6—22no—
183#627—64no—
184C23×C9—21no—
185C23×C9C22×D311yescharacteristic
186C22×D9C22×D311yescharacteristic
187Q8×C9—62no—
188C2×C4×C9—24no—
189C2×C4×C9—22no—
190C2×C4×C9—21no—
191C2×C4×C9—21no—
192C2×C4×C9—32no—
193C2×C4×C9—32no—
194C2×C4×C9C22×D311yescharacteristic
195C2×C4×C9C22×D311yescharacteristic
196#611—22no—
197C2×Dic9—32no—
198C2×Dic9—32no—
199C2×Dic9—62no—
200C2×Dic9—62no—
201C2×Dic9C22×D311yescharacteristic
202C2×Dic9C22×D311yescharacteristic
203C2×Dic9C22×D311yescharacteristic
204C4×D9—22no—
205Dic18—64no—
206#556—271noSylow
207C2×C3×C9—41no—
208C2×C3×C9C2×#4111yescharacteristic
209C2×C3×C9C2×#4111yescharacteristic
210C2×C3×C92−1+411yescharacteristic
211C3×D9—41no—
212C2×C3×Q8—31no—
213C2×C3×D4—91no—
214C22×C3×C4—21no—
215C22×C3×C4—91no—
216C22×C3×C4C2×D911yescharacteristic
217C22×C3×C4D3211yescharacteristic
218C2×#67—31no—
219C2×C4×D3—31no—
220C2×Dic6—92no—
221C2×Dic6—92no—
222C2×Dic6C2×D911yescharacteristic
223C3×C4⋊C4—32no—
224C3×C4⋊C4—31no—
225C3×C4⋊C4—91no—
226C3×C22⋊C4—32no—
227C3×C22⋊C4—91no—
228C3×C22⋊C4—91no—
229C3×C42—91no—
230#217—31no—
231#217—92no—
232#217—92no—
233#217—92no—
234#217—92no—
235#217—182no—
236#217C2×D912yes—
237(C2×D3)⋊C4—31no—
238(C2×D3)⋊C4—92no—
239(C2×D3)⋊C4—92no—
240C4⋊Dic3—92no—
241C4⋊Dic3—91no—
242C4⋊Dic3—91no—
243C4⋊Dic3—91no—
244C4⋊Dic3—91no—
245C4⋊Dic3—92no—
246C4⋊Dic3—181no—
247C4⋊Dic3—181no—
248C4⋊Dic3C2×D911yescharacteristic
249Dic3⋊C4—31no—
250Dic3⋊C4—92no—
251Dic3⋊C4—92no—
252Dic3⋊C4—92no—
253Dic3⋊C4—92no—
254Dic3⋊C4C2×D912yes—
255C4×Dic3—31no—
256C4×Dic3—92no—
257C4×Dic3—92no—
258C4×Dic3—92no—
259C4×Dic3—92no—
260C4×Dic3—92no—
261C4×Dic3—92no—
262C4×Dic3—182no—
263C22×C32—21no—
264C22×C32—21no—
265C22×C32—21no—
266C22×C32C23×D311yescharacteristic
267C2×C3×D3—61no—
268C2×C3×D3—61no—
269C2×C3×D3—61no—
270C32×C4—21no—
271C32×C4#23712yes—
272C3⋊Dic3—62no—
273C3⋊Dic3—62no—
274C3×Dic3—22no—
275C3×Dic3—22no—
276C3×Dic3—32no—
277C3×Dic3—61no—
278C3×Dic3—61no—
279C22×C9—22no—
280C22×C9—21no—
281C22×C9—21no—
282C22×C9—21no—
283C22×C9—21no—
284C22×C9—41no—
285C22×C9—41no—
286C22×C9C23×D311yescharacteristic
287C2×D9—21no—
288C2×D9—21no—
289C2×D9—21no—
290C4×C9—22no—
291C4×C9—21no—
292C4×C9—22no—
293C4×C9—62no—
294C4×C9—62no—
295C4×C9#23712yes—
296Dic9—21no—
297Dic9—21no—
298Dic9—62no—
299Dic9—62no—
300Dic9—122no—
301Dic9—122no—
302Dic9#23512yes—
303C4⋊Q8—271no—
304#126—272no—
305#126—272no—
306#124—272no—
307C22⋊Q8—31no—
308C22⋊Q8—271no—
309C4×Q8—271no—
310C4×Q8—272no—
311C4×D4—91no—
312#117—271no—
313#117—271no—
314C3×C9#55611yescharacteristic, complemented, Sylow
315C23×C3—21no—
316C23×C3—91no—
317C23×C3C22×D911yescharacteristic
318C23×C3C2×D3211yescharacteristic
319C22×D3—31no—
320C3×Q8—62no—
321C3×D4—182no—
322C2×C3×C4—24no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—32no—
328C2×C3×C4—32no—
329C2×C3×C4—91no—
330C2×C3×C4—91no—
331C2×C3×C4—91no—
332C2×C3×C4—182no—
333C2×C3×C4C22×D911yescharacteristic
334C2×C3×C4C22×D911yescharacteristic
335C2×C3×C4C2×D3211yescharacteristic
336C2×C3×C4C2×D3211yescharacteristic
337#67—62no—
338C2×Dic3—31no—
339C2×Dic3—31no—
340C2×Dic3—31no—
341C2×Dic3—92no—
342C2×Dic3—92no—
343C2×Dic3—92no—
344C2×Dic3—92no—
345C2×Dic3—182no—
346C2×Dic3—182no—
347C2×Dic3C22×D912yes—
348C2×Dic3C22×D912yes—
349C4×D3—62no—
350Dic6—22no—
351Dic6—184no—
352Dic6—184no—
353C2×C32—41no—
354C2×C32C2×#23711yescharacteristic
355C2×C32#100711yescharacteristic
356C2×C32D3×#4111yescharacteristic
357C3×D3—121no—
358C2×C9—21no—
359C2×C9—21no—
360C2×C9—21no—
361C2×C9—42no—
362C2×C9—41no—
363C2×C9C2×#23511yescharacteristic
364C2×C9C2×#23711yescharacteristic
365C2×C9#101011yescharacteristic
366D9—41no—
367C2×Q8—31no—
368C2×Q8—272no—
369C2×D4—91no—
370C22×C4—91no—
371C22×C4D3×D911yescharacteristic
372C4⋊C4—31no—
373C4⋊C4—32no—
374C4⋊C4—91no—
375C4⋊C4—272no—
376C4⋊C4—272no—
377C4⋊C4—272no—
378C4⋊C4—271no—
379C4⋊C4—271no—
380C22⋊C4—32no—
381C22⋊C4—91no—
382C22⋊C4—91no—
383C22⋊C4—272no—
384C22⋊C4—272no—
385C22⋊C4—272no—
386C42—91no—
387C42—272no—
388C42—272no—
389C42—272no—
390C22×C3—22no—
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—21no—
397C22×C3—21no—
398C22×C3—41no—
399C22×C3—41no—
400C22×C3—181no—
401C22×C3—181no—
402C22×C3—181no—
403C22×C3C23×D911yescharacteristic
404C22×C3C22×D3211yescharacteristic
405C2×D3—61no—
406C2×D3—61no—
407C2×D3—61no—
408C3×C4—22no—
409C3×C4—21no—
410C3×C4—21no—
411C3×C4—22no—
412C3×C4—62no—
413C3×C4—62no—
414C3×C4—92no—
415C3×C4—181no—
416C3×C4—181no—
417C3×C4#369912yes—
418C3×C4#380512yes—
419Dic3—22no—
420Dic3—22no—
421Dic3—32no—
422Dic3—61no—
423Dic3—61no—
424Dic3—182no—
425Dic3—182no—
426Dic3—182no—
427Dic3—182no—
428Dic3—362no—
429Dic3—362no—
430C32#564111yescharacteristic
431C9—21no—
432C9#554211yescharacteristic, complemented
433C23—91no—
434C23C2×D3×D911yescharacteristic
435Q8—62no—
436Q8—544no—
437D4—182no—
438C2×C4—22no—
439C2×C4—32no—
440C2×C4—32no—
441C2×C4—91no—
442C2×C4—91no—
443C2×C4—91no—
444C2×C4—182no—
445C2×C4—272no—
446C2×C4—272no—
447C2×C4C2×D3×D911yescharacteristic
448C2×C4C2×D3×D911yescharacteristic
449C2×C3—21no—
450C2×C3—21no—
451C2×C3—21no—
452C2×C3—42no—
453C2×C3—41no—
454C2×C3—41no—
455C2×C3—361no—
456C2×C3C2×#369911yescharacteristic
457C2×C3#6371311yescharacteristic
458C2×C3#41×D911yescharacteristic
459C2×C3#6429711yescharacteristic
460C2×C3D3×#23511yescharacteristic
461C2×C3C2×#380511yescharacteristic
462D3—121no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—181no—
467C22—181no—
468C22—181no—
469C22C22×D3×D911yescharacteristic
470C4—21no—
471C4—62no—
472C4—62no—
473C4—92no—
474C4—181no—
475C4—181no—
476C4—542no—
477C4—542no—
478C4#8866312yes—
479C3—21no—
480C3#1058974311yescharacteristic, complemented
481C3#1059484111yescharacteristic
482C2—41no—
483C2—361no—
484C2#1172245811yescharacteristic
485C2D9×#23511yescharacteristic
486C2C2×#8866311yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
prime radicalC2×C3×#611C22
hypercenterC22×C4D3×D9
hypocenterC3×C9#556
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. #11721815C2semidirect
3. C9×#924C2semidirect
5. #11721246C2semidirect
6. #11720969C2semidirect
7. #11720950C2semidirect
8. #11720944C2semidirect
9. Dic6×Dic9C2semidirect
10. #11720907C2semidirect
11. #11720902C2semidirect
156. #924D9semidirect
314. C3×C9#556semidirect
432. C9#5542semidirect
480. C3#10589743semidirect

Maximal subgroups

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

subgroupindex
#117218152
C3×C4×#6112
C9×#9242
#117212572
#117212462
#117209692
#117209502
#117209442
Dic6×Dic92
#117209072
#117209022
#105948413
#105897433