Finite Group Databasebeta

← #13150261

Subgroups — #13150261

Order 1728.

796 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721819C211yesmaximal, characteristic
2C9×#924C211yesmaximal, characteristic, complemented
3C2×C3×C4×Dic9C211yesmaximal, characteristic, complemented
4#11721255C212yesmaximal, complemented
5#11721241C212yesmaximal, complemented
6#11720992C211yesmaximal, characteristic, complemented
7#11720953C211yesmaximal, characteristic, complemented
8#11720935C211yesmaximal, characteristic, complemented
9#11720934C212yesmaximal, complemented
10Dic6×Dic9C211yesmaximal, characteristic, complemented
11#11720899C212yesmaximal, complemented
12#10594955—31nomaximal
13#10589726—31nomaximal
14C22×C3×C4×C9C2211yescharacteristic
15C2×#6233C2212yes—
16C2×C4⋊(C3⋊D9)C2211yescharacteristic
17C22×C3×Dic9C2211yescharacteristic
18C22×C3×Dic9C2211yescharacteristic
19C2×C9×Dic6C2211yescharacteristic
20C2×(C3⋊D9)⋊C4C2212yes—
21#88536C2212yes—
22C4⋊(C3⋊Dic9)C2211yescharacteristic
23C9×#217C2212yes—
24C9×C4⋊Dic3C2211yescharacteristic
25C9×Dic3⋊C4C2212yes—
26C3×C4×Dic9C2211yescharacteristic
27C3×C4×Dic9C2211yescharacteristic
28C3×C4×Dic9C2211yescharacteristic
29C3×C4×Dic9C2211yescharacteristic
30#88445C2212yes—
31#88445C2212yes—
32#88445C2212yes—
33Dic3⋊Dic9C2212yes—
34Dic3⋊Dic9C2212yes—
35#88442C2212yes—
36Dic3×Dic9C2212yes—
37#64139—31no—
38C3×#924D311yescharacteristic
39C2×C3×C4×Dic3—31no—
40#63966—32no—
41#63952—32no—
42#63894—31no—
43#63869—31no—
44#63868—32no—
45#63864—31no—
46Dic3×Dic6—31no—
47#63836—32no—
48C9×C22⋊Q8—31no—
49#63509—31no—
50Q8×Dic9—31no—
51#63492—31no—
52#63492—61no—
53C2×C4×Dic9D311yescharacteristic, complemented
54#63460—32no—
55#63455—31no—
56#63454—32no—
57#63452—31no—
58#63448—32no—
59#611⋊C4—32no—
60C23×C3×C9C2311yescharacteristic
61C22×C3⋊D9C2312yes—
62C2×C3×C4×C9—22no—
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3×C4×C9C2311yescharacteristic
65#6233—24no—
66C2×C3⋊Dic9C2312yes—
67C4⋊(C3⋊D9)—22no—
68C2×C9×Dic3C2312yes—
69C2×C9×Dic3C2312yes—
70C2×C3×Dic9—22no—
71C2×C3×Dic9—22no—
72C2×C3×Dic9C2311yescharacteristic
73C2×C3×Dic9C2311yescharacteristic
74C2×C3×Dic9C2311yescharacteristic
75C2×C3×Dic9C2311yescharacteristic
76C9×Dic6—22no—
77(C3⋊D9)⋊C4—24no—
78#5624—91no—
79#5479—91no—
80C22×C32×C4C2×D311yescharacteristic
81C2×#638—32no—
82C2×C4⋊(C3⋊D3)—31no—
83C22×C3×Dic3—31no—
84C22×C3×Dic3—31no—
85C2×C3×Dic6C2×D311yescharacteristic
86C2×#624—32no—
87#3752—32no—
88C4⋊(C3⋊Dic3)—31no—
89C3×#217C2×D312yes—
90C3×C4⋊Dic3C2×D311yescharacteristic
91C3×Dic3⋊C4C2×D312yes—
92C3×C4×Dic3—31no—
93C3×C4×Dic3—31no—
94C3×C4×Dic3—31no—
95C3×C4×Dic3—31no—
96#3724—32no—
97Dic3⋊Dic3—32no—
98Dic3⋊Dic3—32no—
99#3722—32no—
100#3722—32no—
101#3722—32no—
102Dic32—32no—
103C2×Q8×C9—31no—
104C22×C4×C9—21no—
105C22×C4×C9C2×D311yescharacteristic
106C2×#611—32no—
107C2×#611—62no—
108C22×Dic9C2×D311yescharacteristic
109C22×Dic9C2×D311yescharacteristic
110C2×D36—31no—
111C2×D36—61no—
112C2×C4×D9—32no—
113C9×C4⋊C4—32no—
114C9×C4⋊C4—31no—
115C9×C22⋊C4—32no—
116(C2×D9)⋊C4—32no—
117(C2×D9)⋊C4—32no—
118(C2×D9)⋊C4—32no—
119(C2×D9)⋊C4—32no—
120(C2×D9)⋊C4—62no—
121C4⋊Dic9—31no—
122C4⋊Dic9—32no—
123C4⋊Dic9—32no—
124C4⋊Dic9—61no—
125Dic9⋊C4—32no—
126C4×Dic9—32no—
127C4×Dic9C2×D311yescharacteristic
128C4×Dic9C2×D311yescharacteristic
129C4×Dic9C2×D311yescharacteristic
130C4×Dic9C2×D311yescharacteristic
131C22×C3×C9—21no—
132C22×C3×C9—21no—
133C22×C3×C9#4112yes—
134C22×C3×C9C2411yescharacteristic
135C2×C3⋊D9—22no—
136C2×C3⋊D9—22no—
137C2×C3⋊D9—22no—
138C3×C4×C9—21no—
139C3×C4×C9#4112yes—
140C3⋊Dic9—22no—
141C9×Dic3—22no—
142C9×Dic3—22no—
143C3×Dic9—21no—
144C3×Dic9—21no—
145C3×Dic9—21no—
146C3×Dic9#4112yes—
147C3×C22⋊Q8—31no—
148C2×C3×C42—91no—
149#947—91no—
150Q8×Dic3—91no—
151#930—91no—
152#930—91no—
153#930—181no—
154#929—92no—
155C4×#67—92no—
156#924D911yescharacteristic, complemented
157C2×C4×Dic3—31no—
158#898—92no—
159#893—91no—
160#892—92no—
161#890—91no—
162#886—92no—
163#67⋊C4—92no—
164#876—92no—
165#875—91no—
166C4×D12—91no—
167#872—92no—
168#870—91no—
169C4×Dic6—91no—
170C23×C32C22×D311yescharacteristic
171C22×C3⋊D3—32no—
172C2×C32×C4—22no—
173C2×C32×C4C22×D311yescharacteristic
174C2×C32×C4C22×D311yescharacteristic
175#638—64no—
176C2×C3⋊Dic3—32no—
177C4⋊(C3⋊D3)—62no—
178C2×C3×Dic3—31no—
179C2×C3×Dic3—31no—
180C2×C3×Dic3—31no—
181C2×C3×Dic3—31no—
182C2×C3×Dic3—62no—
183C2×C3×Dic3—62no—
184C2×C3×Dic3C22×D312yes—
185C2×C3×Dic3C22×D312yes—
186C3×Dic6—22no—
187#624—64no—
188C23×C9—21no—
189C23×C9C22×D311yescharacteristic
190C22×D9—32no—
191C22×D9—62no—
192Q8×C9—62no—
193C2×C4×C9—22no—
194C2×C4×C9—21no—
195C2×C4×C9—21no—
196C2×C4×C9—32no—
197C2×C4×C9—32no—
198C2×C4×C9—42no—
199C2×C4×C9C22×D311yescharacteristic
200C2×C4×C9C22×D311yescharacteristic
201#611—64no—
202#611—124no—
203C2×Dic9—22no—
204C2×Dic9—22no—
205C2×Dic9—32no—
206C2×Dic9—62no—
207C2×Dic9C22×D311yescharacteristic
208C2×Dic9C22×D311yescharacteristic
209C2×Dic9C22×D311yescharacteristic
210C2×Dic9C22×D311yescharacteristic
211D36—62no—
212D36—122no—
213C4×D9—64no—
214#529—271noSylow
215C2×C3×C9—22no—
216C2×C3×C9C2×#4111yescharacteristic
217C2×C3×C9C2×#4111yescharacteristic
218C2×C3×C9C2×#4111yescharacteristic
219C3⋊D9—42no—
220C2×C3×Q8—31no—
221C22×C3×C4—21no—
222C22×C3×C4—91no—
223C22×C3×C4—91no—
224C22×C3×C4C2×D911yescharacteristic
225C22×C3×C4D3211yescharacteristic
226C2×#67—92no—
227C2×#67—92no—
228C2×#67—182no—
229C22×Dic3—31no—
230C22×Dic3—31no—
231C2×D12—91no—
232C2×D12—91no—
233C2×D12—181no—
234C2×C4×D3—92no—
235C2×C4×D3—92no—
236C2×Dic6C2×D911yescharacteristic
237C3×C4⋊C4—31no—
238C3×C4⋊C4—32no—
239C3×C22⋊C4—32no—
240C3×C42—91no—
241C3×C42—91no—
242C3×C42—91no—
243C3×C42—91no—
244#217C2×D912yes—
245(C2×D3)⋊C4—92no—
246(C2×D3)⋊C4—92no—
247(C2×D3)⋊C4—92no—
248(C2×D3)⋊C4—92no—
249(C2×D3)⋊C4—92no—
250(C2×D3)⋊C4—92no—
251(C2×D3)⋊C4—92no—
252(C2×D3)⋊C4—92no—
253(C2×D3)⋊C4—182no—
254C4⋊Dic3—92no—
255C4⋊Dic3—92no—
256C4⋊Dic3—91no—
257C4⋊Dic3—91no—
258C4⋊Dic3—181no—
259C4⋊Dic3C2×D911yescharacteristic
260Dic3⋊C4—92no—
261Dic3⋊C4—92no—
262Dic3⋊C4—92no—
263Dic3⋊C4—92no—
264Dic3⋊C4C2×D912yes—
265C4×Dic3—31no—
266C4×Dic3—31no—
267C4×Dic3—31no—
268C4×Dic3—31no—
269C4×Dic3—92no—
270C4×Dic3—92no—
271C22×C32—21no—
272C22×C32—21no—
273C22×C32#23912yes—
274C22×C32C23×D311yescharacteristic
275C2×C3⋊D3—62no—
276C2×C3⋊D3—62no—
277C2×C3⋊D3—62no—
278C32×C4—21no—
279C32×C4#23712yes—
280C3⋊Dic3—62no—
281C3×Dic3—22no—
282C3×Dic3—22no—
283C3×Dic3—32no—
284C3×Dic3—61no—
285C3×Dic3—61no—
286C3×Dic3—61no—
287C22×C9—21no—
288C22×C9—21no—
289C22×C9—21no—
290C22×C9—22no—
291C22×C9—41no—
292C22×C9—41no—
293C22×C9#23512yes—
294C22×C9C23×D311yescharacteristic
295C2×D9—62no—
296C2×D9—62no—
297C2×D9—62no—
298C2×D9—122no—
299C2×D9—122no—
300C2×D9—122no—
301C4×C9—21no—
302C4×C9—22no—
303C4×C9—41no—
304C4×C9—62no—
305C4×C9—62no—
306C4×C9#23512yes—
307Dic9—21no—
308Dic9—21no—
309Dic9—21no—
310Dic9—62no—
311Dic9—122no—
312Dic9#23512yes—
313#126—272no—
314#125—271no—
315#124—271no—
316#123—272no—
317C22⋊Q8—31no—
318#121—271no—
319C4×Q8—271no—
320C4×D4—271no—
321C4×D4—272no—
322#117—272no—
323C2×C42—91no—
324C3×C9#52911yescharacteristic, complemented, Sylow
325C23×C3—21no—
326C23×C3C22×D911yescharacteristic
327C23×C3C2×D3211yescharacteristic
328C22×D3—92no—
329C22×D3—92no—
330C22×D3—182no—
331C3×Q8—62no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—32no—
337C2×C3×C4—32no—
338C2×C3×C4—42no—
339C2×C3×C4—91no—
340C2×C3×C4—91no—
341C2×C3×C4—91no—
342C2×C3×C4—91no—
343C2×C3×C4—182no—
344C2×C3×C4—182no—
345C2×C3×C4C22×D911yescharacteristic
346C2×C3×C4C22×D911yescharacteristic
347C2×C3×C4C2×D3211yescharacteristic
348C2×C3×C4C2×D3211yescharacteristic
349#67—184no—
350#67—184no—
351#67—364no—
352C2×Dic3—31no—
353C2×Dic3—31no—
354C2×Dic3—31no—
355C2×Dic3—31no—
356C2×Dic3—62no—
357C2×Dic3—62no—
358C2×Dic3—92no—
359C2×Dic3—92no—
360C2×Dic3—182no—
361C2×Dic3C22×D912yes—
362C2×Dic3C22×D912yes—
363D12—182no—
364D12—182no—
365D12—362no—
366C4×D3—184no—
367C4×D3—184no—
368Dic6—22no—
369C2×C32—22no—
370C2×C32C2×#23711yescharacteristic
371C2×C32C2×#23911yescharacteristic
372C2×C32D3×#4111yescharacteristic
373C3⋊D3—122no—
374C2×C9—22no—
375C2×C9—21no—
376C2×C9—21no—
377C2×C9—21no—
378C2×C9—42no—
379C2×C9C2×#23511yescharacteristic
380C2×C9C2×#23511yescharacteristic
381C2×C9C2×#23511yescharacteristic
382D9—122no—
383D9—242no—
384C2×Q8—31no—
385C2×D4—272no—
386C2×D4—271no—
387C22×C4—91no—
388C22×C4—91no—
389C22×C4—272no—
390C22×C4D3×D911yescharacteristic
391C4⋊C4—32no—
392C4⋊C4—31no—
393C4⋊C4—272no—
394C4⋊C4—272no—
395C4⋊C4—272no—
396C4⋊C4—271no—
397C22⋊C4—32no—
398C22⋊C4—272no—
399C22⋊C4—272no—
400C22⋊C4—272no—
401C22⋊C4—272no—
402C42—91no—
403C42—91no—
404C42—91no—
405C42—91no—
406C42—272no—
407C22×C3—21no—
408C22×C3—21no—
409C22×C3—21no—
410C22×C3—21no—
411C22×C3—22no—
412C22×C3—21no—
413C22×C3—41no—
414C22×C3—41no—
415C22×C3#370112yes—
416C22×C3C23×D911yescharacteristic
417C22×C3#379912yes—
418C22×C3C22×D3211yescharacteristic
419C2×D3—182no—
420C2×D3—182no—
421C2×D3—182no—
422C2×D3—182no—
423C2×D3—182no—
424C2×D3—182no—
425C2×D3—362no—
426C2×D3—362no—
427C2×D3—362no—
428C3×C4—21no—
429C3×C4—21no—
430C3×C4—22no—
431C3×C4—41no—
432C3×C4—62no—
433C3×C4—62no—
434C3×C4—92no—
435C3×C4—181no—
436C3×C4—181no—
437C3×C4—181no—
438C3×C4#369912yes—
439C3×C4#380412yes—
440Dic3—22no—
441Dic3—22no—
442Dic3—32no—
443Dic3—61no—
444Dic3—61no—
445Dic3—61no—
446Dic3—182no—
447Dic3—182no—
448Dic3—362no—
449C32#562411yescharacteristic
450C9—21no—
451C9#547911yescharacteristic, complemented
452C23—272no—
453C23C2×D3×D911yescharacteristic
454Q8—62no—
455D4—544no—
456D4—542no—
457C2×C4—22no—
458C2×C4—32no—
459C2×C4—32no—
460C2×C4—91no—
461C2×C4—91no—
462C2×C4—91no—
463C2×C4—91no—
464C2×C4—182no—
465C2×C4—182no—
466C2×C4—272no—
467C2×C4—544no—
468C2×C4C2×D3×D911yescharacteristic
469C2×C4C2×D3×D911yescharacteristic
470C2×C3—22no—
471C2×C3—22no—
472C2×C3—21no—
473C2×C3—21no—
474C2×C3—21no—
475C2×C3—42no—
476C2×C3C2×#369911yescharacteristic
477C2×C3C2×#370111yescharacteristic
478C2×C3#41×D911yescharacteristic
479C2×C3C2×#379911yescharacteristic
480C2×C3D3×#23511yescharacteristic
481C2×C3C2×#380411yescharacteristic
482D3—362no—
483D3—362no—
484D3—722no—
485C22—21no—
486C22—21no—
487C22—542no—
488C22—542no—
489C22—542no—
490C22#8863512yes—
491C22C22×D3×D911yescharacteristic
492C4—21no—
493C4—62no—
494C4—62no—
495C4—92no—
496C4—181no—
497C4—181no—
498C4—181no—
499C4—542no—
500C4#8866212yes—
501C3—21no—
502C3#1058972611yescharacteristic, complemented
503C3#1059495511yescharacteristic
504C2—22no—
505C2—1082no—
506C2C2×#8863511yescharacteristic
507C2D9×#23511yescharacteristic
508C2C2×#8866211yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
prime radical#11721819C2
hypercenterC22×C4D3×D9
hypocenterC3×C9#529
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C9×#924C2semidirect
3. C2×C3×C4×Dic9C2semidirect
4. #11721255C2semidirect
5. #11721241C2semidirect
6. #11720992C2semidirect
7. #11720953C2semidirect
8. #11720935C2semidirect
9. #11720934C2semidirect
10. Dic6×Dic9C2semidirect
11. #11720899C2semidirect
53. C2×C4×Dic9D3semidirect
156. #924D9semidirect
324. C3×C9#529semidirect
451. C9#5479semidirect
502. C3#10589726semidirect

Maximal subgroups

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

subgroupindex
#117218192
C9×#9242
C2×C3×C4×Dic92
#117212552
#117212412
#117209922
#117209532
#117209352
#117209342
Dic6×Dic92
#117208992
#105949553
#105897263