Finite Group Databasebeta

← #13148850

Subgroups — #13148850

Order 1728.

740 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721753C211yesmaximal, characteristic
2C9×#887C211yesmaximal, characteristic, complemented
3C3×#63450C211yesmaximal, characteristic, complemented
4#11720985C212yesmaximal
5C4×(C3⋊D9)⋊C4C211yesmaximal, characteristic
6#11720929C211yesmaximal, characteristic
7#11720924C211yesmaximal, characteristic, complemented
8#11720915C212yesmaximal, complemented
9#11720912C211yesmaximal, characteristic, complemented
10Dic3×Dic18C212yesmaximal, complemented
11#11720898C212yesmaximal
12#10594397—31nomaximal
13#10589680—31nomaximal
14C2×C4×C3⋊D9C2211yescharacteristic
15C2×#6226C2211yescharacteristic
16C2×C9×Dic6C2211yescharacteristic
17C2×C3×Dic18C2212yes—
18C2×(C3⋊D9)⋊C4C2211yescharacteristic
19C2×(C3⋊D9)⋊C4C2211yescharacteristic
20C3×C9×C4⋊C4C2211yescharacteristic
21#88536C2212yes—
22C4⋊(C3⋊Dic9)C2212yes—
23C9×Dic3⋊C4C2212yes—
24C4×C9×Dic3C2211yescharacteristic
25C4×C9×Dic3C2212yes—
26C3×C4⋊Dic9C2211yescharacteristic
27C3×Dic9⋊C4C2212yes—
28C3×C4×Dic9C2211yescharacteristic
29#88445C2212yes—
30#88442C2212yes—
31Dic9⋊Dic3C2212yes—
32Dic9⋊Dic3C2212yes—
33Dic9⋊Dic3C2212yes—
34Dic3×Dic9C2211yescharacteristic
35Dic3×Dic9C2211yescharacteristic
36Dic3×Dic9C2212yes—
37#64106—31no—
38C3×#888—31no—
39C3×#887D311yescharacteristic
40#63888—32no—
41C4×#624—31no—
42#63871—31no—
43#63866—31no—
44#63845—32no—
45#63844—31no—
46Dic3×Dic6—32no—
47#63836—32no—
48C4×Q8×C9—31no—
49#63508—31no—
50#63506—31no—
51#63459—31no—
52#63459—61no—
53#63458—32no—
54#63451—32no—
55#63450D311yescharacteristic, complemented
56#63434—32no—
57C42×D9—31no—
58C4×Dic18—32no—
59C4×Dic18—31no—
60C22×C3⋊D9C2311yescharacteristic
61C2×C3×C4×C9C2312yes—
62C2×C3×C4×C9C2311yescharacteristic
63C2×C3⋊Dic9C2311yescharacteristic
64C2×C3⋊Dic9C2312yes—
65C4×C3⋊D9—22no—
66#6226—22no—
67C2×C9×Dic3C2311yescharacteristic
68C2×C9×Dic3C2312yes—
69C2×C9×Dic3C2311yescharacteristic
70C2×C3×Dic9C2312yes—
71C2×C3×Dic9C2311yescharacteristic
72C2×C3×Dic9C2311yescharacteristic
73C9×Dic6—22no—
74C3×Dic18—24no—
75(C3⋊D9)⋊C4—22no—
76(C3⋊D9)⋊C4Q814yescomplemented
77#5728—91no—
78#5578—91no—
79C2×C4×C3⋊D3—31no—
80C2×#634—31no—
81C2×C3×Dic6—32no—
82C2×C3×Dic6C2×D311yescharacteristic
83C2×#624—31no—
84C2×#624—31no—
85C32×C4⋊C4C2×D311yescharacteristic
86#3752—32no—
87C4⋊(C3⋊Dic3)—32no—
88C3×C4⋊Dic3—31no—
89C3×Dic3⋊C4—32no—
90C3×Dic3⋊C4C2×D312yes—
91C3×C4×Dic3—31no—
92C3×C4×Dic3C2×D312yes—
93C3×C4×Dic3C2×D311yescharacteristic
94#3724—32no—
95Dic3⋊Dic3—32no—
96Dic3⋊Dic3—32no—
97Dic3⋊Dic3—32no—
98#3722—32no—
99Dic32—32no—
100Dic32—31no—
101Dic32—31no—
102C2×Q8×C9—31no—
103C2×C4×D9—31no—
104C2×C4×D9—31no—
105C2×C4×D9—31no—
106C2×C4×D9—61no—
107C2×Dic18—31no—
108C2×Dic18—61no—
109C2×Dic18C2×D312yes—
110C9×C4⋊C4—21no—
111C9×C4⋊C4—32no—
112C9×C4⋊C4C2×D311yescharacteristic
113C42×C9—31no—
114C42×C9—32no—
115(C2×D9)⋊C4—32no—
116(C2×D9)⋊C4—32no—
117(C2×D9)⋊C4—62no—
118C4⋊Dic9—32no—
119C4⋊Dic9—62no—
120C4⋊Dic9C2×D311yescharacteristic
121Dic9⋊C4—32no—
122Dic9⋊C4—32no—
123Dic9⋊C4—32no—
124Dic9⋊C4—32no—
125Dic9⋊C4C2×D312yes—
126C4×Dic9—31no—
127C4×Dic9—32no—
128C4×Dic9—31no—
129C4×Dic9C2×D311yescharacteristic
130C22×C3×C9C2411yescharacteristic
131C2×C3⋊D9—21no—
132C2×C3⋊D9—21no—
133C2×C3⋊D9C2×Q812yes—
134C3×C4×C9—22no—
135C3×C4×C9#4112yes—
136C3⋊Dic9—22no—
137C3⋊Dic9—21no—
138C9×Dic3—21no—
139C9×Dic3—22no—
140C9×Dic3C2×Q812yes—
141C3×Dic9—22no—
142C3×Dic9C2×Q812yes—
143C3×Dic9#4112yes—
144C3×C4⋊Q8—91no—
145C3×C4×Q8—31no—
146#946—91no—
147#946—92no—
148Q8×Dic3—92no—
149#944—91no—
150#897—91no—
151#897—91no—
152#897—91no—
153#897—181no—
154#896—92no—
155#892—92no—
156#890—92no—
157#889—92no—
158#888—31no—
159#887—91no—
160#887D911yescharacteristic, complemented
161#872—92no—
162C42×D3—91no—
163C42×D3—91no—
164#869—91no—
165C4×Dic6—91no—
166C4×Dic6—92no—
167C22×C3⋊D3—31no—
168C2×C32×C4C22×D312yes—
169C2×C32×C4C22×D311yescharacteristic
170C2×C3⋊Dic3—31no—
171C2×C3⋊Dic3—32no—
172C4×C3⋊D3—62no—
173#634—62no—
174C2×C3×Dic3—31no—
175C2×C3×Dic3—31no—
176C2×C3×Dic3—32no—
177C2×C3×Dic3C22×D311yescharacteristic
178C2×C3×Dic3C22×D312yes—
179C2×C3×Dic3C22×D311yescharacteristic
180C3×Dic6—22no—
181C3×Dic6—64no—
182#624—34no—
183#624—62no—
184C22×D9—31no—
185C22×D9—61no—
186Q8×C9—62no—
187C2×C4×C9—21no—
188C2×C4×C9—22no—
189C2×C4×C9—31no—
190C2×C4×C9—31no—
191C2×C4×C9—32no—
192C2×C4×C9C22×D312yes—
193C2×C4×C9C22×D311yescharacteristic
194C2×Dic9—31no—
195C2×Dic9—32no—
196C2×Dic9—61no—
197C2×Dic9—62no—
198C2×Dic9C22×D312yes—
199C2×Dic9C22×D311yescharacteristic
200C2×Dic9C22×D311yescharacteristic
201C4×D9—34no—
202C4×D9—62no—
203C4×D9—62no—
204C4×D9—122no—
205Dic18—24no—
206Dic18—62no—
207Dic18—122no—
208#533—271noSylow
209C2×C3×C9C22×Q811yescharacteristic
210C2×C3×C9C2×#4111yescharacteristic
211C2×C3×C9C2×#4111yescharacteristic
212C3⋊D9—22no—
213C2×C3×Q8—31no—
214C2×C3×Q8—92no—
215C2×C4×D3—91no—
216C2×C4×D3—91no—
217C2×C4×D3—91no—
218C2×C4×D3—91no—
219C2×C4×D3—91no—
220C2×C4×D3—91no—
221C2×C4×D3—181no—
222C2×Dic6—32no—
223C2×Dic6—91no—
224C2×Dic6—91no—
225C2×Dic6—181no—
226C2×Dic6C2×D911yescharacteristic
227C3×C4⋊C4—21no—
228C3×C4⋊C4—32no—
229C3×C4⋊C4—91no—
230C3×C4⋊C4—92no—
231C3×C4⋊C4C2×D911yescharacteristic
232C3×C4⋊C4D3211yescharacteristic
233C3×C42—31no—
234C3×C42—32no—
235C3×C42—91no—
236(C2×D3)⋊C4—92no—
237(C2×D3)⋊C4—92no—
238(C2×D3)⋊C4—92no—
239(C2×D3)⋊C4—92no—
240(C2×D3)⋊C4—182no—
241C4⋊Dic3—31no—
242C4⋊Dic3—92no—
243C4⋊Dic3—92no—
244C4⋊Dic3—92no—
245C4⋊Dic3—92no—
246C4⋊Dic3—92no—
247C4⋊Dic3—182no—
248Dic3⋊C4—32no—
249Dic3⋊C4—92no—
250Dic3⋊C4—92no—
251Dic3⋊C4—92no—
252Dic3⋊C4—92no—
253Dic3⋊C4—92no—
254Dic3⋊C4C2×D912yes—
255C4×Dic3—31no—
256C4×Dic3—91no—
257C4×Dic3—92no—
258C4×Dic3—91no—
259C4×Dic3—91no—
260C4×Dic3—91no—
261C4×Dic3—92no—
262C4×Dic3C2×D911yescharacteristic
263C4×Dic3C2×D912yes—
264C22×C32C23×D311yescharacteristic
265C2×C3⋊D3—32no—
266C2×C3⋊D3—61no—
267C2×C3⋊D3—61no—
268C32×C4—22no—
269C32×C4#23512yes—
270C3⋊Dic3—62no—
271C3⋊Dic3—61no—
272C3×Dic3—22no—
273C3×Dic3—21no—
274C3×Dic3—32no—
275C3×Dic3—32no—
276C3×Dic3—62no—
277C3×Dic3D3×Q812yes—
278C22×C9—21no—
279C22×C9C23×D311yescharacteristic
280C2×D9—32no—
281C2×D9—62no—
282C2×D9—61no—
283C2×D9—61no—
284C2×D9—121no—
285C2×D9—121no—
286C4×C9—22no—
287C4×C9—22no—
288C4×C9—32no—
289C4×C9—42no—
290C4×C9—61no—
291C4×C9—62no—
292C4×C9#23912yes—
293Dic9—22no—
294Dic9—62no—
295Dic9—61no—
296Dic9—122no—
297Dic9—121no—
298Dic9#23712yes—
299Dic9D3×Q812yes—
300C4⋊Q8—91no—
301C4⋊Q8—271no—
302#125—272no—
303C22⋊Q8—272no—
304C22⋊Q8—271no—
305C22⋊Q8—271no—
306C4×Q8—31no—
307C4×Q8—271no—
308C4×Q8—272no—
309#117—272no—
310C2×C42—271no—
311C3×C9#53311yescharacteristic, complemented, Sylow
312C22×D3—91no—
313C22×D3—91no—
314C22×D3—181no—
315C3×Q8—62no—
316C3×Q8—184no—
317C2×C3×C4—21no—
318C2×C3×C4—22no—
319C2×C3×C4—31no—
320C2×C3×C4—31no—
321C2×C3×C4—32no—
322C2×C3×C4—91no—
323C2×C3×C4—91no—
324C2×C3×C4—92no—
325C2×C3×C4C22×D912yes—
326C2×C3×C4C22×D911yescharacteristic
327C2×C3×C4C2×D3212yes—
328C2×C3×C4C2×D3211yescharacteristic
329C2×Dic3—31no—
330C2×Dic3—31no—
331C2×Dic3—32no—
332C2×Dic3—91no—
333C2×Dic3—92no—
334C2×Dic3—91no—
335C2×Dic3—92no—
336C2×Dic3—181no—
337C2×Dic3—182no—
338C2×Dic3C22×D911yescharacteristic
339C2×Dic3C22×D912yes—
340C2×Dic3C22×D911yescharacteristic
341C4×D3—94no—
342C4×D3—94no—
343C4×D3—182no—
344C4×D3—182no—
345C4×D3—182no—
346C4×D3—182no—
347C4×D3—362no—
348Dic6—22no—
349Dic6—64no—
350Dic6—182no—
351Dic6—182no—
352Dic6—362no—
353C2×C32C2×#23511yescharacteristic
354C2×C32C2×D3×Q811yescharacteristic
355C2×C32D3×#4111yescharacteristic
356C3⋊D3—62no—
357C2×C9—21no—
358C2×C9—21no—
359C2×C9—21no—
360C2×C9C2×#23711yescharacteristic
361C2×C9C2×D3×Q811yescharacteristic
362C2×C9C2×#23911yescharacteristic
363D9—62no—
364D9—122no—
365C2×Q8—31no—
366C2×Q8—92no—
367C2×Q8—271no—
368C22×C4—271no—
369C22×C4—271no—
370C22×C4—271no—
371C4⋊C4—32no—
372C4⋊C4—92no—
373C4⋊C4—91no—
374C4⋊C4—272no—
375C4⋊C4—272no—
376C4⋊C4—272no—
377C4⋊C4—272no—
378C4⋊C4—272no—
379C4⋊C4D3×D911yescharacteristic
380C22⋊C4—272no—
381C22⋊C4—272no—
382C42—31no—
383C42—32no—
384C42—91no—
385C42—271no—
386C42—271no—
387C42—272no—
388C22×C3—21no—
389C22×C3C23×D911yescharacteristic
390C22×C3C22×D3211yescharacteristic
391C2×D3—92no—
392C2×D3—92no—
393C2×D3—182no—
394C2×D3—181no—
395C2×D3—181no—
396C2×D3—181no—
397C2×D3—181no—
398C2×D3—361no—
399C2×D3—361no—
400C3×C4—22no—
401C3×C4—22no—
402C3×C4—22no—
403C3×C4—32no—
404C3×C4—42no—
405C3×C4—62no—
406C3×C4—61no—
407C3×C4—92no—
408C3×C4—92no—
409C3×C4—182no—
410C3×C4#369712yes—
411C3×C4#379912yes—
412Dic3—22no—
413Dic3—21no—
414Dic3—32no—
415Dic3—32no—
416Dic3—62no—
417Dic3—182no—
418Dic3—181no—
419Dic3—182no—
420Dic3—181no—
421Dic3—362no—
422Dic3—361no—
423Dic3Q8×D912yes—
424C32#557811yescharacteristic
425C9—21no—
426C9#572811yescharacteristic, complemented
427C23—271no—
428Q8—62no—
429Q8—184no—
430Q8—542no—
431C2×C4—31no—
432C2×C4—31no—
433C2×C4—32no—
434C2×C4—91no—
435C2×C4—91no—
436C2×C4—92no—
437C2×C4—274no—
438C2×C4—271no—
439C2×C4—272no—
440C2×C4—542no—
441C2×C4—542no—
442C2×C4C2×D3×D912yes—
443C2×C4C2×D3×D911yescharacteristic
444C2×C3—21no—
445C2×C3—21no—
446C2×C3—21no—
447C2×C3C2×#369711yescharacteristic
448C2×C3C2×Q8×D911yescharacteristic
449C2×C3#41×D911yescharacteristic
450C2×C3C2×#379911yescharacteristic
451C2×C3D3×#23711yescharacteristic
452C2×C3D32×Q811yescharacteristic
453D3—182no—
454D3—182no—
455D3—362no—
456C22—272no—
457C22—541no—
458C22—541no—
459C22C22×D3×D911yescharacteristic
460C4—22no—
461C4—32no—
462C4—61no—
463C4—62no—
464C4—92no—
465C4—92no—
466C4—182no—
467C4—541no—
468C4—542no—
469C4#8864212yes—
470C3—21no—
471C3#1058968011yescharacteristic, complemented
472C3#1059439711yescharacteristic
473C2—542no—
474C2C2×#8864211yescharacteristic
475C2D9×#23711yescharacteristic
476C2D3×Q8×D911yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
prime radicalC22×C3⋊D9C23
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#533
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C9×#887C2semidirect
3. C3×#63450C2semidirect
7. #11720924C2semidirect
8. #11720915C2semidirect
9. #11720912C2semidirect
10. Dic3×Dic18C2semidirect
55. #63450D3semidirect
76. (C3⋊D9)⋊C4Q8semidirect
160. #887D9semidirect
311. C3×C9#533semidirect
426. C9#5728semidirect
471. C3#10589680semidirect

Maximal subgroups

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

subgroupindex
#117217532
C9×#8872
C3×#634502
#117209852
C4×(C3⋊D9)⋊C42
#117209292
#117209242
#117209152
#117209122
Dic3×Dic182
#117208982
#105943973
#105896803