Finite Group Databasebeta

← #13149017

Subgroups — #13149017

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#11721748C211yesmaximal, characteristic, complemented
2C3×#63454C211yesmaximal, characteristic, complemented
3C9×#887C211yesmaximal, characteristic, complemented
4C4×#6192C211yesmaximal, characteristic
5#11720970C211yesmaximal, characteristic, complemented
6C4×#6189C212yesmaximal, complemented
7#11720950C212yesmaximal, complemented
8#11720944C211yesmaximal, characteristic, complemented
9#11720940C212yesmaximal
10#11720934C212yesmaximal, complemented
11#11720929C211yesmaximal, characteristic, complemented
12#10594565—31nomaximal
13#10589686—31nomaximal
14C2×C4×C3⋊D9C2211yescharacteristic
15C2×C3×C4×D9C2211yescharacteristic
16C2×C9×Dic6C2211yescharacteristic
17C2×#6192C2211yescharacteristic
18C2×#6189C2212yes—
19C3×C9×C4⋊C4C2211yescharacteristic
20#88536C2212yes—
21C4⋊(C3⋊Dic9)C2211yescharacteristic
22C4×C3⋊Dic9C2212yes—
23C3×(C2×D9)⋊C4C2212yes—
24C9×Dic3⋊C4C2212yes—
25C4×C9×Dic3C2212yes—
26C4×C9×Dic3C2211yescharacteristic
27C3×C4⋊Dic9C2211yescharacteristic
28C3×C4×Dic9C2212yes—
29#88445C2211yescharacteristic
30#88445C2212yes—
31(C2×D9)⋊Dic3C2211yescharacteristic
32(C2×D9)⋊Dic3C2212yes—
33Dic3⋊Dic9C2211yescharacteristic
34Dic3⋊Dic9C2212yes—
35#88442C2212yes—
36Dic9⋊Dic3C2212yes—
37#64101—31no—
38C3×#892—31no—
39C3×#887D311yescharacteristic
40#63918—31no—
41C4×#627—32no—
42C4×#626—31no—
43#63871—31no—
44#63868—32no—
45#63862—32no—
46#63856—31no—
47#63852—32no—
48C4×Q8×C9—31no—
49#63508—31no—
50#63508—31no—
51#63460—32no—
52#63460—32no—
53#63454—31no—
54#63454—61no—
55#63454D311yescharacteristic, complemented
56#63437—32no—
57C4×D36—31no—
58#63431—31no—
59C4×Dic18—32no—
60C22×C3⋊D9C2311yescharacteristic
61C22×C3×D9C2311yescharacteristic
62C2×C3×C4×C9C2312yes—
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3⋊Dic9C2312yes—
65C2×C3⋊Dic9C2311yescharacteristic
66C4×C3⋊D9—22no—
67C2×C9×Dic3C2312yes—
68C2×C9×Dic3C2311yescharacteristic
69C2×C9×Dic3C2311yescharacteristic
70C2×C3×Dic9C2312yes—
71C2×C3×Dic9C2311yescharacteristic
72C3×C4×D9—22no—
73C9×Dic6—22no—
74#6192—22no—
75#6189—24no—
76#5584—91no—
77#5533—91no—
78C2×C4×C3⋊D3—31no—
79C2×C3×C4×D3—31no—
80C2×C3×Dic6C2×D311yescharacteristic
81C2×#627—32no—
82C2×#626—31no—
83C32×C4⋊C4C2×D311yescharacteristic
84#3752—32no—
85C4⋊(C3⋊Dic3)—31no—
86C4×C3⋊Dic3—32no—
87C3×(C2×D3)⋊C4—32no—
88C3×C4⋊Dic3—31no—
89C3×Dic3⋊C4C2×D312yes—
90C3×C4×Dic3—32no—
91C3×C4×Dic3C2×D312yes—
92C3×C4×Dic3C2×D311yescharacteristic
93#3724—32no—
94Dic3⋊Dic3—32no—
95Dic3⋊Dic3—32no—
96Dic3⋊Dic3—31no—
97#3722—32no—
98#3722—31no—
99(C2×D3)⋊Dic3—32no—
100(C2×D3)⋊Dic3—31no—
101C2×Q8×C9—31no—
102C2×D36—31no—
103C2×C4×D9—31no—
104C2×C4×D9—61no—
105C2×C4×D9C2×D311yescharacteristic
106C2×Dic18—32no—
107C9×C4⋊C4—21no—
108C9×C4⋊C4—32no—
109C9×C4⋊C4C2×D311yescharacteristic
110C42×C9—32no—
111C42×C9—31no—
112(C2×D9)⋊C4—32no—
113(C2×D9)⋊C4—31no—
114(C2×D9)⋊C4—32no—
115(C2×D9)⋊C4—32no—
116(C2×D9)⋊C4—31no—
117(C2×D9)⋊C4—62no—
118(C2×D9)⋊C4C2×D312yes—
119C4⋊Dic9—31no—
120C4⋊Dic9—31no—
121C4⋊Dic9—32no—
122C4⋊Dic9—61no—
123C4⋊Dic9C2×D311yescharacteristic
124Dic9⋊C4—32no—
125Dic9⋊C4—32no—
126C4×Dic9—32no—
127C4×Dic9—62no—
128C4×Dic9C2×D312yes—
129C22×C3×C9C2411yescharacteristic
130C2×C3⋊D9—21no—
131C2×C3⋊D9—21no—
132C2×C3⋊D9—21no—
133C2×C3×D9—21no—
134C2×C3×D9—21no—
135C2×C3×D9—21no—
136C3×C4×C9—22no—
137C3×C4×C9#4112yes—
138C3⋊Dic9—21no—
139C3⋊Dic9—22no—
140C9×Dic3—22no—
141C9×Dic3—21no—
142C9×Dic3#4112yes—
143C3×Dic9—22no—
144C3×Dic9—21no—
145C3×C4×Q8—31no—
146C3×#117—91no—
147#946—91no—
148#946—91no—
149C4×#67—91no—
150#924—91no—
151#898—92no—
152#898—92no—
153#897—91no—
154#892—31no—
155#892—91no—
156#892—91no—
157#892—181no—
158#888—91no—
159#887D911yescharacteristic, complemented
160#885—92no—
161#879—92no—
162#876—92no—
163#875—92no—
164C4×D12—91no—
165#869—91no—
166C4×Dic6—92no—
167C4×Dic6—92no—
168C22×C3⋊D3—31no—
169C22×C3×D3—31no—
170C2×C32×C4C22×D312yes—
171C2×C32×C4C22×D311yescharacteristic
172C2×C3⋊Dic3—31no—
173C2×C3⋊Dic3—32no—
174C4×C3⋊D3—62no—
175C2×C3×Dic3—32no—
176C2×C3×Dic3—31no—
177C2×C3×Dic3C22×D311yescharacteristic
178C2×C3×Dic3C22×D312yes—
179C2×C3×Dic3C22×D311yescharacteristic
180C3×C4×D3—62no—
181C3×Dic6—22no—
182#627—64no—
183#626—62no—
184C22×D9—31no—
185C22×D9—61no—
186C22×D9C22×D311yescharacteristic
187Q8×C9—62no—
188C2×C4×C9—21no—
189C2×C4×C9—22no—
190C2×C4×C9—31no—
191C2×C4×C9—32no—
192C2×C4×C9—31no—
193C2×C4×C9C22×D312yes—
194C2×C4×C9C22×D311yescharacteristic
195C2×Dic9—31no—
196C2×Dic9—32no—
197C2×Dic9—61no—
198C2×Dic9—62no—
199C2×Dic9C22×D311yescharacteristic
200C2×Dic9C22×D312yes—
201D36—62no—
202C4×D9—22no—
203C4×D9—64no—
204C4×D9—62no—
205Dic18—64no—
206#556—271noSylow
207C2×C3×C9C2×#4111yescharacteristic
208C2×C3×C9C2×#4111yescharacteristic
209C2×C3×C92−1+411yescharacteristic
210C3⋊D9—41no—
211C3×D9—41no—
212C2×C3×Q8—31no—
213C22×C3×C4—91no—
214C2×#67—91no—
215C2×D12—91no—
216C2×C4×D3—31no—
217C2×C4×D3—91no—
218C2×C4×D3—91no—
219C2×C4×D3—181no—
220C2×Dic6—92no—
221C2×Dic6—92no—
222C2×Dic6C2×D911yescharacteristic
223C3×C4⋊C4—21no—
224C3×C4⋊C4—32no—
225C3×C4⋊C4—91no—
226C3×C4⋊C4C2×D911yescharacteristic
227C3×C4⋊C4D3211yescharacteristic
228C3×C22⋊C4—92no—
229C3×C42—32no—
230C3×C42—31no—
231C3×C42—92no—
232#217—91no—
233#217—92no—
234(C2×D3)⋊C4—32no—
235(C2×D3)⋊C4—91no—
236(C2×D3)⋊C4—92no—
237(C2×D3)⋊C4—91no—
238(C2×D3)⋊C4—92no—
239(C2×D3)⋊C4—92no—
240(C2×D3)⋊C4—92no—
241(C2×D3)⋊C4—91no—
242(C2×D3)⋊C4—92no—
243(C2×D3)⋊C4—182no—
244C4⋊Dic3—31no—
245C4⋊Dic3—91no—
246C4⋊Dic3—92no—
247C4⋊Dic3—91no—
248C4⋊Dic3—91no—
249C4⋊Dic3—92no—
250C4⋊Dic3—181no—
251Dic3⋊C4—92no—
252Dic3⋊C4—92no—
253Dic3⋊C4—91no—
254Dic3⋊C4—92no—
255Dic3⋊C4—92no—
256Dic3⋊C4C2×D912yes—
257C4×Dic3—32no—
258C4×Dic3—92no—
259C4×Dic3—92no—
260C4×Dic3—182no—
261C4×Dic3C2×D912yes—
262C4×Dic3C2×D911yescharacteristic
263C22×C32C23×D311yescharacteristic
264C2×C3⋊D3—61no—
265C2×C3⋊D3—61no—
266C2×C3⋊D3—61no—
267C2×C3×D3—61no—
268C2×C3×D3—61no—
269C2×C3×D3—61no—
270C32×C4—22no—
271C32×C4#23512yes—
272C3⋊Dic3—61no—
273C3⋊Dic3—62no—
274C3×Dic3—22no—
275C3×Dic3—21no—
276C3×Dic3—61no—
277C3×Dic3—62no—
278C3×Dic3#23912yes—
279C22×C9—21no—
280C22×C9C23×D311yescharacteristic
281C2×D9—21no—
282C2×D9—21no—
283C2×D9—21no—
284C2×D9—62no—
285C2×D9—61no—
286C2×D9—61no—
287C2×D9—61no—
288C2×D9—121no—
289C2×D9—121no—
290C4×C9—22no—
291C4×C9—22no—
292C4×C9—32no—
293C4×C9—42no—
294C4×C9—62no—
295C4×C9—61no—
296C4×C9#23512yes—
297Dic9—21no—
298Dic9—22no—
299Dic9—62no—
300Dic9—61no—
301Dic9—62no—
302Dic9—122no—
303C4⋊Q8—271no—
304#126—272no—
305#126—272no—
306#124—272no—
307C22⋊Q8—271no—
308C22⋊Q8—271no—
309C4×Q8—31no—
310C4×Q8—272no—
311C4×D4—271no—
312#117—91no—
313#117—271no—
314C3×C9#55611yescharacteristic, complemented, Sylow
315C23×C3—91no—
316C22×D3—31no—
317C22×D3—91no—
318C22×D3—91no—
319C22×D3—181no—
320C3×Q8—62no—
321C2×C3×C4—21no—
322C2×C3×C4—22no—
323C2×C3×C4—31no—
324C2×C3×C4—32no—
325C2×C3×C4—31no—
326C2×C3×C4—91no—
327C2×C3×C4—92no—
328C2×C3×C4—182no—
329C2×C3×C4C22×D912yes—
330C2×C3×C4C22×D911yescharacteristic
331C2×C3×C4C2×D3212yes—
332C2×C3×C4C2×D3211yescharacteristic
333#67—182no—
334C2×Dic3—31no—
335C2×Dic3—32no—
336C2×Dic3—91no—
337C2×Dic3—92no—
338C2×Dic3—91no—
339C2×Dic3—92no—
340C2×Dic3—181no—
341C2×Dic3—182no—
342C2×Dic3C22×D911yescharacteristic
343C2×Dic3C22×D912yes—
344C2×Dic3C22×D911yescharacteristic
345D12—182no—
346C4×D3—62no—
347C4×D3—184no—
348C4×D3—182no—
349C4×D3—182no—
350Dic6—22no—
351Dic6—184no—
352Dic6—184no—
353C2×C32C2×#23511yescharacteristic
354C2×C32C2×#23911yescharacteristic
355C2×C32#101011yescharacteristic
356C3⋊D3—121no—
357C3×D3—121no—
358C2×C9—21no—
359C2×C9—21no—
360C2×C9—21no—
361C2×C9C2×#23511yescharacteristic
362C2×C9D3×#4111yescharacteristic
363C2×C9#101011yescharacteristic
364D9—41no—
365D9—122no—
366D9—121no—
367C2×Q8—31no—
368C2×Q8—272no—
369C2×D4—271no—
370C22×C4—91no—
371C22×C4—271no—
372C4⋊C4—32no—
373C4⋊C4—91no—
374C4⋊C4—271no—
375C4⋊C4—272no—
376C4⋊C4—272no—
377C4⋊C4—272no—
378C4⋊C4—271no—
379C4⋊C4D3×D911yescharacteristic
380C22⋊C4—92no—
381C22⋊C4—271no—
382C22⋊C4—272no—
383C22⋊C4—271no—
384C22⋊C4—272no—
385C22⋊C4—272no—
386C42—31no—
387C42—32no—
388C42—92no—
389C42—272no—
390C22×C3—21no—
391C22×C3—181no—
392C22×C3—181no—
393C22×C3—181no—
394C22×C3C23×D911yescharacteristic
395C22×C3C22×D3211yescharacteristic
396C2×D3—61no—
397C2×D3—61no—
398C2×D3—61no—
399C2×D3—182no—
400C2×D3—181no—
401C2×D3—181no—
402C2×D3—181no—
403C2×D3—181no—
404C2×D3—181no—
405C2×D3—181no—
406C2×D3—361no—
407C2×D3—361no—
408C3×C4—22no—
409C3×C4—22no—
410C3×C4—22no—
411C3×C4—32no—
412C3×C4—42no—
413C3×C4—62no—
414C3×C4—61no—
415C3×C4—181no—
416C3×C4—182no—
417C3×C4#369712yes—
418C3×C4#379812yes—
419Dic3—22no—
420Dic3—21no—
421Dic3—61no—
422Dic3—62no—
423Dic3—182no—
424Dic3—181no—
425Dic3—182no—
426Dic3—181no—
427Dic3—182no—
428Dic3—362no—
429Dic3#370112yes—
430C32#558411yescharacteristic
431C9—21no—
432C9#553311yescharacteristic, complemented
433C23—91no—
434C23—271no—
435Q8—62no—
436Q8—544no—
437D4—542no—
438C2×C4—31no—
439C2×C4—31no—
440C2×C4—32no—
441C2×C4—91no—
442C2×C4—92no—
443C2×C4—182no—
444C2×C4—271no—
445C2×C4—272no—
446C2×C4—542no—
447C2×C4C2×D3×D912yes—
448C2×C4C2×D3×D911yescharacteristic
449C2×C3—21no—
450C2×C3—21no—
451C2×C3—21no—
452C2×C3—361no—
453C2×C3C2×#369711yescharacteristic
454C2×C3C2×#370111yescharacteristic
455C2×C3#6371611yescharacteristic
456C2×C3C2×#379811yescharacteristic
457C2×C3#6430911yescharacteristic
458C2×C3D3×#23911yescharacteristic
459D3—121no—
460D3—362no—
461D3—361no—
462D3—361no—
463C22—181no—
464C22—181no—
465C22—181no—
466C22—541no—
467C22—541no—
468C22—541no—
469C22C22×D3×D911yescharacteristic
470C4—22no—
471C4—32no—
472C4—62no—
473C4—61no—
474C4—182no—
475C4—181no—
476C4—542no—
477C4—541no—
478C4#8864012yes—
479C3—21no—
480C3#1058968611yescharacteristic, complemented
481C3#1059456511yescharacteristic
482C2—361no—
483C2—1081no—
484C2C2×#8864011yescharacteristic
485C2#1172249411yescharacteristic
486C2D3×#370111yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
prime radicalC2×#6192C22
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#556
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. #11721748C2semidirect
2. C3×#63454C2semidirect
3. C9×#887C2semidirect
5. #11720970C2semidirect
6. C4×#6189C2semidirect
7. #11720950C2semidirect
8. #11720944C2semidirect
10. #11720934C2semidirect
11. #11720929C2semidirect
55. #63454D3semidirect
159. #887D9semidirect
314. C3×C9#556semidirect
432. C9#5533semidirect
480. C3#10589686semidirect

Maximal subgroups

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

subgroupindex
#117217482
C3×#634542
C9×#8872
C4×#61922
#117209702
C4×#61892
#117209502
#117209442
#117209402
#117209342
#117209292
#105945653
#105896863