Finite Group Databasebeta

← #13148944

Subgroups — #13148944

Order 1728.

770 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4⋊C4×C3⋊D9C211yesmaximal, characteristic
2C9×#890C211yesmaximal, characteristic, complemented
3C3×#63452C211yesmaximal, characteristic, complemented
4#11721000C212yesmaximal
5#11721000C211yesmaximal, characteristic
6#11720980C212yesmaximal
7C4×(C3⋊D9)⋊C4C211yesmaximal, characteristic
8#11720935C211yesmaximal, characteristic, complemented
9#11720927C211yesmaximal, characteristic, complemented
10#11720919C212yesmaximal, complemented
11#11720918C212yesmaximal, complemented
12#10594482—31nomaximal
13D9×#125—31nomaximal
14C2×C4×C3⋊D9C2212yes—
15C2×C4×C3⋊D9C2211yescharacteristic
16C2×(C3⋊D9)⋊C4C2211yescharacteristic
17C2×(C3⋊D9)⋊C4C2211yescharacteristic
18C2×(C3⋊D9)⋊C4C2212yes—
19C3×C9×C4⋊C4C2211yescharacteristic
20C4⋊(C3⋊Dic9)C2211yescharacteristic
21#88534C2212yes—
22C9×C4⋊Dic3C2212yes—
23C9×C4⋊Dic3C2211yescharacteristic
24C9×Dic3⋊C4C2212yes—
25C4×C9×Dic3C2211yescharacteristic
26C3×C4⋊Dic9C2212yes—
27C3×C4⋊Dic9C2211yescharacteristic
28C3×Dic9⋊C4C2212yes—
29C3×C4×Dic9C2211yescharacteristic
30Dic3⋊Dic9C2212yes—
31#88442C2212yes—
32#88442C2212yes—
33#88442C2212yes—
34Dic9⋊Dic3C2212yes—
35Dic3×Dic9C2211yescharacteristic
36Dic3×Dic9C2211yescharacteristic
37C4⋊C4×C3⋊D3—31no—
38C3×#890—31no—
39C3×#890D311yescharacteristic
40#63900—31no—
41#63900—32no—
42#63883—32no—
43C4×#624—31no—
44#63869—31no—
45#63869—31no—
46#63847—32no—
47#63847—32no—
48C9×#125—31no—
49C4⋊C4×D9—31no—
50C4⋊C4×D9—32no—
51C4⋊C4×D9—31no—
52C4⋊C4×D9—32no—
53C4⋊C4×D9—61no—
54#63452—31no—
55#63452D311yescharacteristic, complemented
56#63451—32no—
57#63451—32no—
58C42×D9—31no—
59#63432—31no—
60C22×C3⋊D9C2311yescharacteristic
61C2×C3×C4×C9C2312yes—
62C2×C3×C4×C9C2311yescharacteristic
63C2×C3⋊Dic9C2312yes—
64C2×C3⋊Dic9C2311yescharacteristic
65C4×C3⋊D9—24no—
66C4×C3⋊D9Q814yes—
67C2×C9×Dic3C2311yescharacteristic
68C2×C9×Dic3C2311yescharacteristic
69C2×C9×Dic3C2312yes—
70C2×C3×Dic9C2311yescharacteristic
71C2×C3×Dic9C2311yescharacteristic
72C2×C3×Dic9C2312yes—
73(C3⋊D9)⋊C4—22no—
74(C3⋊D9)⋊C4—24no—
75(C3⋊D9)⋊C4—22no—
76D3×#125—91no—
77D3×#125—91no—
78C2×C4×C3⋊D3—31no—
79C2×C4×C3⋊D3—32no—
80C2×#624—31no—
81C2×#624—31no—
82C2×#624—32no—
83C32×C4⋊C4C2×D311yescharacteristic
84C4⋊(C3⋊Dic3)—31no—
85#3750—32no—
86C3×C4⋊Dic3—32no—
87C3×C4⋊Dic3—31no—
88C3×C4⋊Dic3C2×D312yes—
89C3×C4⋊Dic3C2×D311yescharacteristic
90C3×Dic3⋊C4—32no—
91C3×Dic3⋊C4C2×D312yes—
92C3×C4×Dic3—31no—
93C3×C4×Dic3C2×D311yescharacteristic
94#3724—32no—
95#3724—32no—
96#3724—32no—
97Dic3⋊Dic3—32no—
98Dic3⋊Dic3—32no—
99Dic32—31no—
100Dic32—31no—
101C2×C4×D9—31no—
102C2×C4×D9—32no—
103C2×C4×D9—31no—
104C2×C4×D9—31no—
105C2×C4×D9—32no—
106C2×C4×D9—61no—
107C2×C4×D9—62no—
108C9×C4⋊C4—21no—
109C9×C4⋊C4—32no—
110C9×C4⋊C4—32no—
111C9×C4⋊C4—31no—
112C9×C4⋊C4C2×D311yescharacteristic
113C42×C9—31no—
114C4⋊Dic9—31no—
115C4⋊Dic9—32no—
116C4⋊Dic9—61no—
117C4⋊Dic9C2×D311yescharacteristic
118C4⋊Dic9C2×D312yes—
119Dic9⋊C4—32no—
120Dic9⋊C4—32no—
121Dic9⋊C4—32no—
122Dic9⋊C4—32no—
123Dic9⋊C4—32no—
124Dic9⋊C4—62no—
125Dic9⋊C4C2×D312yes—
126C4×Dic9—31no—
127C4×Dic9—31no—
128C4×Dic9C2×D311yescharacteristic
129C22×C3×C9C2411yescharacteristic
130C2×C3⋊D9C2×Q812yes—
131C2×C3⋊D9#4112yes—
132C2×C3⋊D9#4112yes—
133C3×C4×C9—22no—
134C3×C4×C9C2×Q812yes—
135C3⋊Dic9—22no—
136C3⋊Dic9C2×Q812yes—
137C9×Dic3—21no—
138C9×Dic3—22no—
139C9×Dic3—21no—
140C3×Dic9—21no—
141C3×Dic9—22no—
142C3×Dic9—21no—
143C3×#125—31no—
144C3×#125—91no—
145D3×C4⋊C4—92no—
146D3×C4⋊C4—92no—
147D3×C4⋊C4—91no—
148D3×C4⋊C4—91no—
149D3×C4⋊C4—91no—
150D3×C4⋊C4—92no—
151D3×C4⋊C4—91no—
152D3×C4⋊C4—92no—
153D3×C4⋊C4—181no—
154#890—31no—
155#890—91no—
156#890—91no—
157#890D911yescharacteristic, complemented
158#889—92no—
159#889—92no—
160#889—92no—
161#889—92no—
162C42×D3—91no—
163C42×D3—91no—
164#870—91no—
165#870—91no—
166C22×C3⋊D3—31no—
167C2×C32×C4C22×D312yes—
168C2×C32×C4C22×D311yescharacteristic
169C2×C3⋊Dic3—31no—
170C2×C3⋊Dic3—32no—
171C4×C3⋊D3—34no—
172C4×C3⋊D3—64no—
173C2×C3×Dic3—31no—
174C2×C3×Dic3—31no—
175C2×C3×Dic3—32no—
176C2×C3×Dic3C22×D311yescharacteristic
177C2×C3×Dic3C22×D311yescharacteristic
178C2×C3×Dic3C22×D312yes—
179#624—62no—
180#624—64no—
181#624—62no—
182C22×D9—31no—
183C22×D9—61no—
184C2×C4×C9—21no—
185C2×C4×C9—22no—
186C2×C4×C9—31no—
187C2×C4×C9—31no—
188C2×C4×C9—32no—
189C2×C4×C9C22×D312yes—
190C2×C4×C9C22×D311yescharacteristic
191C2×Dic9—31no—
192C2×Dic9—32no—
193C2×Dic9—61no—
194C2×Dic9—62no—
195C2×Dic9C22×D311yescharacteristic
196C2×Dic9C22×D311yescharacteristic
197C2×Dic9C22×D312yes—
198C4×D9—34no—
199C4×D9—64no—
200C4×D9—62no—
201C4×D9—62no—
202C4×D9—64no—
203C4×D9—64no—
204C4×D9—124no—
205C2×#125—271noSylow
206C2×C3×C9C22×Q811yescharacteristic
207C2×C3×C9C2×#4111yescharacteristic
208C2×C3×C9C2×#4111yescharacteristic
209C3⋊D9#12514yescomplemented
210C2×C4×D3—91no—
211C2×C4×D3—91no—
212C2×C4×D3—91no—
213C2×C4×D3—91no—
214C2×C4×D3—92no—
215C2×C4×D3—92no—
216C2×C4×D3—92no—
217C2×C4×D3—91no—
218C2×C4×D3—91no—
219C2×C4×D3—92no—
220C2×C4×D3—181no—
221C2×C4×D3—182no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—32no—
224C3×C4⋊C4—32no—
225C3×C4⋊C4—31no—
226C3×C4⋊C4—92no—
227C3×C4⋊C4—92no—
228C3×C4⋊C4—91no—
229C3×C4⋊C4C2×D911yescharacteristic
230C3×C4⋊C4D3211yescharacteristic
231C3×C42—31no—
232C3×C42—91no—
233C4⋊Dic3—31no—
234C4⋊Dic3—32no—
235C4⋊Dic3—92no—
236C4⋊Dic3—91no—
237C4⋊Dic3—91no—
238C4⋊Dic3—92no—
239C4⋊Dic3—181no—
240C4⋊Dic3C2×D911yescharacteristic
241C4⋊Dic3C2×D912yes—
242Dic3⋊C4—32no—
243Dic3⋊C4—92no—
244Dic3⋊C4—92no—
245Dic3⋊C4—92no—
246Dic3⋊C4—92no—
247Dic3⋊C4—92no—
248Dic3⋊C4—92no—
249Dic3⋊C4—92no—
250Dic3⋊C4—92no—
251Dic3⋊C4—92no—
252Dic3⋊C4—92no—
253Dic3⋊C4—182no—
254Dic3⋊C4C2×D912yes—
255C4×Dic3—31no—
256C4×Dic3—91no—
257C4×Dic3—91no—
258C4×Dic3—91no—
259C4×Dic3—91no—
260C4×Dic3C2×D911yescharacteristic
261C22×C32C23×D311yescharacteristic
262C2×C3⋊D3—32no—
263C2×C3⋊D3—32no—
264C2×C3⋊D3—32no—
265C32×C4—22no—
266C32×C4D3×Q812yes—
267C3⋊Dic3—32no—
268C3⋊Dic3—62no—
269C3×Dic3—21no—
270C3×Dic3—21no—
271C3×Dic3—22no—
272C3×Dic3—61no—
273C3×Dic3—61no—
274C3×Dic3—62no—
275C22×C9—21no—
276C22×C9C23×D311yescharacteristic
277C2×D9—32no—
278C2×D9—32no—
279C2×D9—32no—
280C2×D9—62no—
281C2×D9—62no—
282C2×D9—62no—
283C4×C9—22no—
284C4×C9—22no—
285C4×C9—42no—
286C4×C9—61no—
287C4×C9—61no—
288C4×C9—62no—
289C4×C9D3×Q812yes—
290Dic9—21no—
291Dic9—21no—
292Dic9—22no—
293Dic9—32no—
294Dic9—62no—
295Dic9—62no—
296Dic9—122no—
297#125—31no—
298#125—91no—
299#125—271no—
300#125—271no—
301#125—272no—
302#125—272no—
303C2×C4⋊C4—272no—
304C2×C4⋊C4—272no—
305C2×C4⋊C4—271no—
306C2×C4⋊C4—271no—
307C2×C42—271no—
308C3×C9C2×#12511yescharacteristic, complemented, Sylow
309C22×D3—91no—
310C22×D3—91no—
311C22×D3—181no—
312C2×C3×C4—21no—
313C2×C3×C4—22no—
314C2×C3×C4—31no—
315C2×C3×C4—31no—
316C2×C3×C4—32no—
317C2×C3×C4—91no—
318C2×C3×C4—91no—
319C2×C3×C4—92no—
320C2×C3×C4C22×D912yes—
321C2×C3×C4C22×D911yescharacteristic
322C2×C3×C4C2×D3212yes—
323C2×C3×C4C2×D3211yescharacteristic
324C2×Dic3—31no—
325C2×Dic3—31no—
326C2×Dic3—32no—
327C2×Dic3—91no—
328C2×Dic3—91no—
329C2×Dic3—92no—
330C2×Dic3—92no—
331C2×Dic3—181no—
332C2×Dic3—182no—
333C2×Dic3C22×D911yescharacteristic
334C2×Dic3C22×D911yescharacteristic
335C2×Dic3C22×D912yes—
336C4×D3—94no—
337C4×D3—94no—
338C4×D3—184no—
339C4×D3—182no—
340C4×D3—182no—
341C4×D3—184no—
342C4×D3—182no—
343C4×D3—182no—
344C4×D3—184no—
345C4×D3—184no—
346C4×D3—184no—
347C4×D3—364no—
348C2×C32C2×D3×Q811yescharacteristic
349C2×C32D3×#4111yescharacteristic
350C2×C32D3×#4111yescharacteristic
351C3⋊D3—34no—
352C2×C9—21no—
353C2×C9—21no—
354C2×C9—21no—
355C2×C9C2×D3×Q811yescharacteristic
356C2×C9D3×#4111yescharacteristic
357C2×C9D3×#4111yescharacteristic
358D9—34no—
359D9—64no—
360C22×C4—271no—
361C22×C4—271no—
362C22×C4—272no—
363C22×C4—271no—
364C22×C4—272no—
365C4⋊C4—32no—
366C4⋊C4—32no—
367C4⋊C4—31no—
368C4⋊C4—92no—
369C4⋊C4—92no—
370C4⋊C4—91no—
371C4⋊C4—272no—
372C4⋊C4—272no—
373C4⋊C4—272no—
374C4⋊C4—272no—
375C4⋊C4—272no—
376C4⋊C4—272no—
377C4⋊C4—271no—
378C4⋊C4D3×D911yescharacteristic
379C42—31no—
380C42—91no—
381C42—271no—
382C42—271no—
383C22×C3—21no—
384C22×C3C23×D911yescharacteristic
385C22×C3C22×D3211yescharacteristic
386C2×D3—92no—
387C2×D3—92no—
388C2×D3—92no—
389C2×D3—92no—
390C2×D3—92no—
391C2×D3—92no—
392C2×D3—182no—
393C2×D3—182no—
394C2×D3—182no—
395C3×C4—22no—
396C3×C4—22no—
397C3×C4—22no—
398C3×C4—42no—
399C3×C4—61no—
400C3×C4—61no—
401C3×C4—62no—
402C3×C4—181no—
403C3×C4—181no—
404C3×C4—182no—
405C3×C4Q8×D912yes—
406C3×C4#379712yes—
407Dic3—21no—
408Dic3—21no—
409Dic3—22no—
410Dic3—61no—
411Dic3—61no—
412Dic3—62no—
413Dic3—92no—
414Dic3—92no—
415Dic3—182no—
416Dic3—182no—
417Dic3—182no—
418Dic3—362no—
419C32D3×#12511yescharacteristic
420C9—21no—
421C9D3×#12511yescharacteristic, complemented
422C23—271no—
423C2×C4—31no—
424C2×C4—31no—
425C2×C4—32no—
426C2×C4—91no—
427C2×C4—91no—
428C2×C4—92no—
429C2×C4—271no—
430C2×C4—274no—
431C2×C4—272no—
432C2×C4—544no—
433C2×C4—542no—
434C2×C4—542no—
435C2×C4—544no—
436C2×C4C2×D3×D912yes—
437C2×C4C2×D3×D911yescharacteristic
438C2×C3—21no—
439C2×C3—21no—
440C2×C3—21no—
441C2×C3C2×Q8×D911yescharacteristic
442C2×C3#41×D911yescharacteristic
443C2×C3#41×D911yescharacteristic
444C2×C3C2×#379711yescharacteristic
445C2×C3#6431211yescharacteristic
446C2×C3#6431911yescharacteristic
447D3—94no—
448D3—94no—
449D3—184no—
450C22—272no—
451C22—272no—
452C22—272no—
453C22C22×D3×D911yescharacteristic
454C4—22no—
455C4—61no—
456C4—61no—
457C4—62no—
458C4—181no—
459C4—181no—
460C4—182no—
461C4—272no—
462C4—542no—
463C4(C3⋊D9)⋊Q812yes—
464C3—21no—
465C3D9×#12511yescharacteristic, complemented
466C3#1059448211yescharacteristic
467C2—274no—
468C2C2×(C3⋊D9)⋊Q811yescharacteristic
469C2#1172249811yescharacteristic
470C2#1172251411yescharacteristic
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×C9C2×#125
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×#890C2semidirect
3. C3×#63452C2semidirect
8. #11720935C2semidirect
9. #11720927C2semidirect
10. #11720919C2semidirect
11. #11720918C2semidirect
55. #63452D3semidirect
157. #890D9semidirect
209. C3⋊D9#125semidirect
308. C3×C9C2×#125semidirect
421. C9D3×#125semidirect
465. C3D9×#125semidirect

Maximal subgroups

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

subgroupindex
C4⋊C4×C3⋊D92
C9×#8902
C3×#634522
#117210002
#117209802
C4×(C3⋊D9)⋊C42
#117209352
#117209272
#117209192
#117209182
#105944823
D9×#1253