Finite Group Databasebeta

← #13148791

Subgroups — #13148791

Order 1728.

778 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#11721749C211yesmaximal, characteristic
2C9×#890C211yesmaximal, characteristic, complemented
3C3×#63452C211yesmaximal, characteristic, complemented
4#11721000C212yesmaximal, complemented
5#11720995C212yesmaximal, complemented
6#11720994C212yesmaximal, complemented
7#11720993C211yesmaximal, characteristic
8#11720992C211yesmaximal, characteristic
9#11720971C211yesmaximal, characteristic, complemented
10C4×#6189C211yesmaximal, characteristic, complemented
11#11720923C212yesmaximal, complemented
12#10594484—31nomaximal
13#10589792—31nomaximal
14C2×C4⋊(C3⋊D9)C2211yescharacteristic
15C2×C4×C3⋊D9C2212yes—
16C2×(C3⋊D9)⋊C4C2212yes—
17C2×(C3⋊D9)⋊C4C2212yes—
18C2×#6189C2211yescharacteristic
19C2×#6189C2212yes—
20C3×C9×C4⋊C4C2211yescharacteristic
21#88536C2212yes—
22C4×C3⋊Dic9C2211yescharacteristic
23C9×C4⋊Dic3C2211yescharacteristic
24C9×C4⋊Dic3C2212yes—
25C9×Dic3⋊C4C2212yes—
26C4×C9×Dic3C2211yescharacteristic
27C3×C4⋊Dic9C2211yescharacteristic
28C3×C4⋊Dic9C2212yes—
29C3×Dic9⋊C4C2212yes—
30C3×C4×Dic9C2211yescharacteristic
31#88445C2212yes—
32#88445C2212yes—
33Dic3⋊Dic9C2211yescharacteristic
34#88442C2211yescharacteristic
35#88442C2212yes—
36Dic9⋊Dic3C2211yescharacteristic
37#64102—31no—
38C3×#890—31no—
39C3×#890D311yescharacteristic
40#63919—31no—
41C4×#627—31no—
42#63900—32no—
43#63895—32no—
44#63895—32no—
45#63894—31no—
46#63894—31no—
47#63865—32no—
48C9×#125—31no—
49#63459—32no—
50#63458—32no—
51#63458—32no—
52#63455—31no—
53#63455—31no—
54#63455—61no—
55C4⋊C4×D9—32no—
56#63452D311yescharacteristic, complemented
57#63450—31no—
58C4×D36—31no—
59C4×Dic18—31no—
60C22×C3⋊D9C2312yes—
61C2×C3×C4×C9C2312yes—
62C2×C3×C4×C9C2311yescharacteristic
63C2×C3⋊Dic9C2311yescharacteristic
64C2×C3⋊Dic9C2311yescharacteristic
65C4⋊(C3⋊D9)Q814yescomplemented
66C4×C3⋊D9—24no—
67C2×C9×Dic3C2312yes—
68C2×C9×Dic3C2311yescharacteristic
69C2×C9×Dic3C2311yescharacteristic
70C2×C3×Dic9C2312yes—
71C2×C3×Dic9C2311yescharacteristic
72C2×C3×Dic9C2311yescharacteristic
73(C3⋊D9)⋊C4—42no—
74(C3⋊D9)⋊C4—42no—
75#6189—22no—
76#6189—42no—
77#5690—91no—
78#5690—91no—
79C2×C4⋊(C3⋊D3)—31no—
80C2×C4×C3⋊D3—32no—
81C2×#627—31no—
82C2×#627—32no—
83C2×#624—32no—
84C2×#624—32no—
85C32×C4⋊C4C2×D311yescharacteristic
86#3752—32no—
87C4×C3⋊Dic3—31no—
88C3×C4⋊Dic3—31no—
89C3×C4⋊Dic3—32no—
90C3×C4⋊Dic3C2×D311yescharacteristic
91C3×C4⋊Dic3C2×D312yes—
92C3×Dic3⋊C4—32no—
93C3×Dic3⋊C4C2×D312yes—
94C3×C4×Dic3—31no—
95C3×C4×Dic3C2×D311yescharacteristic
96#3724—32no—
97#3724—31no—
98Dic3⋊Dic3—31no—
99Dic3⋊Dic3—31no—
100#3722—32no—
101#3722—32no—
102C2×D36—31no—
103C2×D36—61no—
104C2×C4×D9—32no—
105C2×C4×D9—32no—
106C2×C4×D9—32no—
107C2×C4×D9—62no—
108C2×Dic18—31no—
109C2×Dic18—32no—
110C9×C4⋊C4—21no—
111C9×C4⋊C4—32no—
112C9×C4⋊C4—31no—
113C9×C4⋊C4—32no—
114C9×C4⋊C4C2×D311yescharacteristic
115C42×C9—31no—
116(C2×D9)⋊C4—32no—
117(C2×D9)⋊C4—32no—
118(C2×D9)⋊C4—32no—
119(C2×D9)⋊C4—62no—
120C4⋊Dic9—31no—
121C4⋊Dic9C2×D311yescharacteristic
122C4⋊Dic9C2×D312yes—
123Dic9⋊C4—31no—
124Dic9⋊C4—31no—
125Dic9⋊C4—32no—
126Dic9⋊C4C2×D312yes—
127C4×Dic9—31no—
128C4×Dic9—61no—
129C4×Dic9C2×D311yescharacteristic
130C22×C3×C9C2411yescharacteristic
131C2×C3⋊D9—22no—
132C2×C3⋊D9—22no—
133C2×C3⋊D9C2×Q814yes—
134C3×C4×C9—22no—
135C3×C4×C9C2×Q812yes—
136C3⋊Dic9—21no—
137C3⋊Dic9#4112yes—
138C9×Dic3—21no—
139C9×Dic3—22no—
140C9×Dic3—21no—
141C3×Dic9—21no—
142C3×Dic9—22no—
143C3×Dic9—21no—
144C3×#125—31no—
145C3×#125—91no—
146#897—92no—
147#897—92no—
148#896—92no—
149#896—92no—
150#896—92no—
151#896—92no—
152#893—91no—
153#893—91no—
154#893—91no—
155#893—91no—
156#893—181no—
157D3×C4⋊C4—92no—
158D3×C4⋊C4—92no—
159#890—31no—
160#890D911yescharacteristic, complemented
161#888—91no—
162#888—91no—
163C4×D12—91no—
164C4×D12—91no—
165C4×Dic6—91no—
166C4×Dic6—91no—
167C22×C3⋊D3—32no—
168C2×C32×C4C22×D312yes—
169C2×C32×C4C22×D311yescharacteristic
170C2×C3⋊Dic3—31no—
171C2×C3⋊Dic3—31no—
172C4⋊(C3⋊D3)—34no—
173C4×C3⋊D3—64no—
174C2×C3×Dic3—31no—
175C2×C3×Dic3—32no—
176C2×C3×Dic3—31no—
177C2×C3×Dic3C22×D311yescharacteristic
178C2×C3×Dic3C22×D312yes—
179C2×C3×Dic3C22×D311yescharacteristic
180#627—62no—
181#627—122no—
182#624—122no—
183#624—122no—
184C22×D9—32no—
185C22×D9—62no—
186C2×C4×C9—21no—
187C2×C4×C9—22no—
188C2×C4×C9—31no—
189C2×C4×C9—32no—
190C2×C4×C9—31no—
191C2×C4×C9C22×D312yes—
192C2×C4×C9C22×D311yescharacteristic
193C2×Dic9—31no—
194C2×Dic9—31no—
195C2×Dic9—61no—
196C2×Dic9—61no—
197C2×Dic9C22×D311yescharacteristic
198C2×Dic9C22×D311yescharacteristic
199C2×Dic9C22×D312yes—
200D36—34no—
201D36—64no—
202C4×D9—64no—
203C4×D9—124no—
204C4×D9—122no—
205C4×D9—122no—
206Dic18—62no—
207Dic18—122no—
208#554—271noSylow
209C2×C3×C9C22×Q811yescharacteristic
210C2×C3×C9C2×#4111yescharacteristic
211C2×C3×C9H(5,𝔽2)11yescharacteristic
212C3⋊D9—24no—
213C2×D12—91no—
214C2×D12—91no—
215C2×D12—181no—
216C2×C4×D3—92no—
217C2×C4×D3—92no—
218C2×C4×D3—92no—
219C2×C4×D3—92no—
220C2×C4×D3—92no—
221C2×C4×D3—92no—
222C2×C4×D3—182no—
223C2×Dic6—91no—
224C2×Dic6—91no—
225C2×Dic6—92no—
226C2×Dic6—92no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—31no—
229C3×C4⋊C4—32no—
230C3×C4⋊C4—32no—
231C3×C4⋊C4—91no—
232C3×C4⋊C4—92no—
233C3×C4⋊C4—92no—
234C3×C4⋊C4C2×D911yescharacteristic
235C3×C4⋊C4D3211yescharacteristic
236C3×C42—31no—
237C3×C42—91no—
238(C2×D3)⋊C4—92no—
239(C2×D3)⋊C4—92no—
240(C2×D3)⋊C4—92no—
241(C2×D3)⋊C4—92no—
242(C2×D3)⋊C4—92no—
243(C2×D3)⋊C4—92no—
244(C2×D3)⋊C4—182no—
245C4⋊Dic3—31no—
246C4⋊Dic3—32no—
247C4⋊Dic3—91no—
248C4⋊Dic3—91no—
249C4⋊Dic3C2×D911yescharacteristic
250C4⋊Dic3C2×D912yes—
251Dic3⋊C4—32no—
252Dic3⋊C4—91no—
253Dic3⋊C4—91no—
254Dic3⋊C4—92no—
255Dic3⋊C4—91no—
256Dic3⋊C4—91no—
257Dic3⋊C4—92no—
258Dic3⋊C4C2×D912yes—
259C4×Dic3—31no—
260C4×Dic3—91no—
261C4×Dic3—91no—
262C4×Dic3—181no—
263C4×Dic3C2×D911yescharacteristic
264C22×C32C23×D311yescharacteristic
265C2×C3⋊D3—34no—
266C2×C3⋊D3—62no—
267C2×C3⋊D3—62no—
268C32×C4—22no—
269C32×C4D3×Q812yes—
270C3⋊Dic3—32no—
271C3⋊Dic3—61no—
272C3×Dic3—21no—
273C3×Dic3—21no—
274C3×Dic3—22no—
275C3×Dic3—61no—
276C3×Dic3—61no—
277C3×Dic3—62no—
278C22×C9—21no—
279C22×C9C23×D311yescharacteristic
280C2×D9—34no—
281C2×D9—64no—
282C2×D9—62no—
283C2×D9—62no—
284C2×D9—122no—
285C2×D9—122no—
286C4×C9—22no—
287C4×C9—22no—
288C4×C9—42no—
289C4×C9—61no—
290C4×C9—62no—
291C4×C9—61no—
292C4×C9D3×Q812yes—
293Dic9—21no—
294Dic9—22no—
295Dic9—21no—
296Dic9—32no—
297Dic9—62no—
298Dic9—61no—
299Dic9—121no—
300C4⋊Q8—271no—
301#125—31no—
302#125—91no—
303C22⋊Q8—272no—
304C22⋊Q8—272no—
305C22⋊Q8—272no—
306C4×Q8—271no—
307C4×D4—271no—
308C4×D4—271no—
309C4×D4—271no—
310C2×C4⋊C4—272no—
311C3×C9#55411yescharacteristic, complemented, Sylow
312C22×D3—92no—
313C22×D3—92no—
314C22×D3—182no—
315C2×C3×C4—21no—
316C2×C3×C4—22no—
317C2×C3×C4—31no—
318C2×C3×C4—32no—
319C2×C3×C4—31no—
320C2×C3×C4—91no—
321C2×C3×C4—92no—
322C2×C3×C4—91no—
323C2×C3×C4C22×D912yes—
324C2×C3×C4C22×D911yescharacteristic
325C2×C3×C4C2×D3212yes—
326C2×C3×C4C2×D3211yescharacteristic
327C2×Dic3—31no—
328C2×Dic3—31no—
329C2×Dic3—32no—
330C2×Dic3—91no—
331C2×Dic3—91no—
332C2×Dic3—91no—
333C2×Dic3—91no—
334C2×Dic3—181no—
335C2×Dic3—181no—
336C2×Dic3C22×D911yescharacteristic
337C2×Dic3C22×D911yescharacteristic
338C2×Dic3C22×D912yes—
339D12—94no—
340D12—94no—
341D12—184no—
342C4×D3—184no—
343C4×D3—184no—
344C4×D3—364no—
345C4×D3—362no—
346C4×D3—362no—
347C4×D3—362no—
348C4×D3—362no—
349Dic6—182no—
350Dic6—182no—
351Dic6—362no—
352Dic6—362no—
353C2×C32C2×D3×Q811yescharacteristic
354C2×C32D3×#4111yescharacteristic
355C2×C32#100911yescharacteristic
356C3⋊D3—64no—
357C2×C9—21no—
358C2×C9—21no—
359C2×C9—21no—
360C2×C9C2×D3×Q811yescharacteristic
361C2×C9D3×#4111yescharacteristic
362C2×C9#100911yescharacteristic
363D9—64no—
364D9—124no—
365C2×Q8—271no—
366C2×Q8—272no—
367C2×D4—271no—
368C22×C4—272no—
369C22×C4—272no—
370C22×C4—272no—
371C4⋊C4—32no—
372C4⋊C4—31no—
373C4⋊C4—32no—
374C4⋊C4—92no—
375C4⋊C4—91no—
376C4⋊C4—92no—
377C4⋊C4—271no—
378C4⋊C4—272no—
379C4⋊C4—271no—
380C4⋊C4—271no—
381C4⋊C4D3×D911yescharacteristic
382C22⋊C4—272no—
383C22⋊C4—272no—
384C22⋊C4—272no—
385C42—31no—
386C42—91no—
387C42—271no—
388C22×C3—21no—
389C22×C3C23×D911yescharacteristic
390C22×C3C22×D3211yescharacteristic
391C2×D3—94no—
392C2×D3—94no—
393C2×D3—184no—
394C2×D3—182no—
395C2×D3—182no—
396C2×D3—182no—
397C2×D3—182no—
398C2×D3—362no—
399C2×D3—362no—
400C3×C4—22no—
401C3×C4—22no—
402C3×C4—22no—
403C3×C4—42no—
404C3×C4—61no—
405C3×C4—62no—
406C3×C4—61no—
407C3×C4—181no—
408C3×C4—182no—
409C3×C4—181no—
410C3×C4Q8×D912yes—
411C3×C4#379712yes—
412Dic3—21no—
413Dic3—22no—
414Dic3—21no—
415Dic3—61no—
416Dic3—62no—
417Dic3—61no—
418Dic3—92no—
419Dic3—92no—
420Dic3—182no—
421Dic3—181no—
422Dic3—181no—
423Dic3—361no—
424C32#569011yescharacteristic
425C9—21no—
426C9#569011yescharacteristic, complemented
427C23—272no—
428Q8—542no—
429Q8—1082no—
430D4—274no—
431C2×C4—31no—
432C2×C4—31no—
433C2×C4—32no—
434C2×C4—91no—
435C2×C4—91no—
436C2×C4—92no—
437C2×C4—271no—
438C2×C4—271no—
439C2×C4—544no—
440C2×C4—1082no—
441C2×C4—1082no—
442C2×C4C2×D3×D912yes—
443C2×C4C2×D3×D911yescharacteristic
444C2×C3—21no—
445C2×C3—21no—
446C2×C3—21no—
447C2×C3C2×Q8×D911yescharacteristic
448C2×C3#41×D911yescharacteristic
449C2×C3#6371511yescharacteristic
450C2×C3C2×#379711yescharacteristic
451C2×C3#6431211yescharacteristic
452C2×C3#6432011yescharacteristic
453D3—184no—
454D3—184no—
455D3—364no—
456C22—274no—
457C22—542no—
458C22—542no—
459C22C22×D3×D911yescharacteristic
460C4—22no—
461C4—61no—
462C4—61no—
463C4—62no—
464C4—181no—
465C4—181no—
466C4—182no—
467C4—272no—
468C4—541no—
469C4(C3⋊D9)⋊Q812yes—
470C3—21no—
471C3#1058979211yescharacteristic, complemented
472C3#1059448411yescharacteristic
473C2—544no—
474C2C2×(C3⋊D9)⋊Q811yescharacteristic
475C2#1172249811yescharacteristic
476C2#1172251511yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
prime radicalC2×C4⋊(C3⋊D9)C22
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#554
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
4. #11721000C2semidirect
5. #11720995C2semidirect
6. #11720994C2semidirect
9. #11720971C2semidirect
10. C4×#6189C2semidirect
11. #11720923C2semidirect
56. #63452D3semidirect
65. C4⋊(C3⋊D9)Q8semidirect
160. #890D9semidirect
311. C3×C9#554semidirect
426. C9#5690semidirect
471. C3#10589792semidirect

Maximal subgroups

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

subgroupindex
#117217492
C9×#8902
C3×#634522
#117210002
#117209952
#117209942
#117209932
#117209922
#117209712
C4×#61892
#117209232
#105944843
#105897923