Finite Group Databasebeta

← #13139885

Subgroups — #13139885

Order 1728.

558 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 77 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11722535C211yesmaximal, characteristic
2#11721818C211yesmaximal, characteristic, complemented
3C9×#929C211yesmaximal, characteristic, complemented
4C3×#63491C211yesmaximal, characteristic, complemented
5#11720332C211yesmaximal, characteristic, complemented
6#11720330C211yesmaximal, characteristic, complemented
7#11720329C211yesmaximal, characteristic, complemented
8#10591398—31nomaximal
9#10588733—31nomaximal
10C22×C3×C4×C9C2211yescharacteristic, complemented
11C2×#6233C2211yescharacteristic
12C2×C9×#67C2211yescharacteristic
13C2×C3×#611C2211yescharacteristic
14#88668—21no—
15D9×#67—21no—
16D3×#611—21no—
17#88663—21no—
18#88662—21no—
19#88658—21no—
20#88551C2211yescharacteristic, complemented
21#88536—21no—
22#88534—21no—
23C9×#217C2211yescharacteristic, complemented
24C3×#3676C2211yescharacteristic, complemented
25C9×(C2×D3)⋊C4—21no—
26C3×(C2×D9)⋊C4—21no—
27C9×Dic3⋊C4—21no—
28C3×Dic9⋊C4—21no—
29#64330—31no—
30#64138—31no—
31C3×#929—31no—
32C3×#929D311yescharacteristic
33#63710—31no—
34#63581—31no—
35#63580—31no—
36#63580—31no—
37C9×#123—31no—
38#63491—31no—
39#63491—61no—
40#63491D311yescharacteristic, complemented
41#63393—31no—
42#63365—31no—
43#63365—31no—
44C23×C3×C9C2311yescharacteristic
45C22×C3⋊D9D411yescharacteristic
46C22×D3×C9D411yescharacteristic
47C22×C3×D9D411yescharacteristic
48C2×D3×D9—21no—
49C2×C3×C4×C9—21no—
50C2×C3×C4×C9—21no—
51C2×C3×C4×C9—21no—
52#6233—21no—
53#6233—21no—
54C2×C3⋊Dic9—21no—
55C2×C3⋊Dic9D411yescharacteristic
56C9×#67—21no—
57C9×#67—21no—
58C3×#611—21no—
59C3×#611—21no—
60C2×C9×Dic3—21no—
61C2×C9×Dic3D411yescharacteristic
62C2×C3×Dic9—21no—
63C2×C3×Dic9D411yescharacteristic
64#6195—21no—
65#6194—21no—
66D3×Dic9—21no—
67#6192—21no—
68(C3⋊D9)⋊C4—21no—
69Dic3×D9—21no—
70#6189—21no—
71#5053—91no—
72#5053—91no—
73C22×C32×C4C2×D311yescharacteristic
74C2×#638—31no—
75C2×C3×#67—31no—
76C2×C3×#67C2×D311yescharacteristic
77#3811—61no—
78D3×#67—61no—
79D3×#67—61no—
80#3805—61no—
81#3804—61no—
82#3804—61no—
83#3757—31no—
84#3752—61no—
85#3750—61no—
86C3×#217—31no—
87C3×#217C2×D311yescharacteristic
88C3×(C2×D3)⋊C4—21no—
89C3×(C2×D3)⋊C4—61no—
90C3×Dic3⋊C4—21no—
91C3×Dic3⋊C4—61no—
92C2×D4×C9—31no—
93C22×C4×C9—21no—
94C22×C4×C9C2×D311yescharacteristic, complemented
95C2×#611—31no—
96C2×#611—61no—
97C2×#611—61no—
98C2×#611C2×D311yescharacteristic
99#3699—61no—
100#3699—61no—
101D4×D9—61no—
102D4×D9—61no—
103#3697—61no—
104C9×C4⋊C4—61no—
105C9×C22⋊C4—31no—
106C9×C22⋊C4—61no—
107#3676—31no—
108#3676—61no—
109#3676C2×D311yescharacteristic, complemented
110(C2×D9)⋊C4—21no—
111(C2×D9)⋊C4—61no—
112(C2×D9)⋊C4—62no—
113Dic9⋊C4—21no—
114Dic9⋊C4—61no—
115Dic9⋊C4—62no—
116C22×C3×C9—41no—
117C22×C3×C9C2×D411yescharacteristic
118C22×C3×C9C2×D411yescharacteristic
119C22×C3×C9C2×D411yescharacteristic
120C2×C3⋊D9—21no—
121C2×C3⋊D9—41no—
122C2×D3×C9—21no—
123C2×D3×C9—41no—
124C2×C3×D9—21no—
125C2×C3×D9—41no—
126D3×D9—81no—
127C3×C4×C9—41no—
128C3⋊Dic9—21no—
129C3⋊Dic9—41no—
130C9×Dic3—21no—
131C9×Dic3—41no—
132C3×Dic9—21no—
133C3×Dic9—41no—
134#1004—91no—
135#1004—91no—
136C3×#123—31no—
137C3×#123—91no—
138#929—31no—
139#929—91no—
140#929—91no—
141#929—181no—
142#929D911yescharacteristic, complemented
143#834—91no—
144#834—91no—
145#806—91no—
146#806—91no—
147#806—91no—
148#806—91no—
149C23×C32C22×D311yescharacteristic
150C22×C3⋊D3—31no—
151C22×C3×D3—31no—
152C22×C3×D3#6711yescharacteristic
153C2×D32—61no—
154C2×C32×C4—21no—
155C2×C32×C4—21no—
156C2×C32×C4—21no—
157#638—61no—
158#638—61no—
159C2×C3⋊Dic3—31no—
160C2×C3⋊Dic3—61no—
161C3×#67—21no—
162C3×#67—21no—
163C3×#67—61no—
164C3×#67—61no—
165C2×C3×Dic3—21no—
166C2×C3×Dic3—31no—
167C2×C3×Dic3—61no—
168C2×C3×Dic3#6711yescharacteristic
169#627—61no—
170#626—61no—
171#626—61no—
172#625—61no—
173#624—61no—
174D3×Dic3—61no—
175D3×Dic3—61no—
176C23×C9—21no—
177C23×C9—31no—
178C23×C9C22×D311yescharacteristic
179C22×D9—31no—
180C22×D9—61no—
181C22×D9—61no—
182C22×D9#6711yescharacteristic
183D4×C9—61no—
184D4×C9—61no—
185C2×C4×C9—22no—
186C2×C4×C9—21no—
187C2×C4×C9—21no—
188C2×C4×C9—21no—
189C2×C4×C9—31no—
190C2×C4×C9—41no—
191C2×C4×C9—41no—
192C2×C4×C9—61no—
193#611—21no—
194#611—21no—
195#611—61no—
196#611—61no—
197#611—61no—
198#611—61no—
199#611—121no—
200#611—121no—
201C2×Dic9—21no—
202C2×Dic9—31no—
203C2×Dic9—61no—
204C2×Dic9—61no—
205C2×Dic9—62no—
206C2×Dic9—61no—
207C2×Dic9#6711yescharacteristic
208D36—61no—
209C4×D9—61no—
210C4×D9—61no—
211Dic18—61no—
212#458—271noSylow
213C2×C3×C9—21no—
214C2×C3×C9—21no—
215C2×C3×C9—21no—
216C2×C3×C9#12011yescharacteristic
217C3⋊D9—41no—
218D3×C9—41no—
219C3×D9—41no—
220C2×C3×D4—31no—
221C2×C3×D4—91no—
222C22×C3×C4—21no—
223C22×C3×C4C2×D911yescharacteristic, complemented
224C22×C3×C4D3211yescharacteristic
225C2×#67—31no—
226C2×#67—91no—
227C2×#67—91no—
228C2×#67—181no—
229C2×#67—181no—
230C2×#67—181no—
231C2×#67C2×D911yescharacteristic
232#237—181no—
233#237—181no—
234#237—181no—
235#237—181no—
236D3×D4—181no—
237D3×D4—181no—
238D3×D4—181no—
239D3×D4—181no—
240#235—181no—
241#235—181no—
242C3×C4⋊C4—61no—
243C3×C4⋊C4—181no—
244C3×C22⋊C4—31no—
245C3×C22⋊C4—61no—
246C3×C22⋊C4—91no—
247C3×C22⋊C4—181no—
248#217—31no—
249#217—91no—
250#217—91no—
251#217—181no—
252#217C2×D911yescharacteristic, complemented
253(C2×D3)⋊C4—21no—
254(C2×D3)⋊C4—61no—
255(C2×D3)⋊C4—181no—
256(C2×D3)⋊C4—181no—
257(C2×D3)⋊C4—182no—
258Dic3⋊C4—21no—
259Dic3⋊C4—61no—
260Dic3⋊C4—181no—
261Dic3⋊C4—181no—
262Dic3⋊C4—182no—
263C22×C32—41no—
264C22×C32D3×D411yescharacteristic
265C22×C32D3×D411yescharacteristic
266C22×C32C2×#6711yescharacteristic
267C2×C3⋊D3—61no—
268C2×C3⋊D3—121no—
269C2×C3×D3—21no—
270C2×C3×D3—41no—
271C2×C3×D3—61no—
272C2×C3×D3—121no—
273D32—241no—
274C32×C4—41no—
275C3⋊Dic3—61no—
276C3⋊Dic3—121no—
277C3×Dic3—21no—
278C3×Dic3—41no—
279C3×Dic3—61no—
280C3×Dic3—121no—
281C22×C9—21no—
282C22×C9—21no—
283C22×C9—21no—
284C22×C9—41no—
285C22×C9—42no—
286C22×C9—61no—
287C22×C9—121no—
288C22×C9D3×D411yescharacteristic
289C22×C9D3×D411yescharacteristic
290C22×C9C2×#6711yescharacteristic
291C2×D9—21no—
292C2×D9—41no—
293C2×D9—61no—
294C2×D9—121no—
295C2×D9—121no—
296C2×D9—122no—
297C2×D9—241no—
298C4×C9—42no—
299C4×C9—41no—
300C4×C9—61no—
301C4×C9—121no—
302Dic9—21no—
303Dic9—41no—
304Dic9—61no—
305Dic9—121no—
306Dic9—122no—
307Dic9—121no—
308H(5,𝔽2)—271no—
309#123—31no—
310#123—91no—
311#123—271no—
312#99—271no—
313#99—271no—
314#99—271no—
315C3×C9#45811yescharacteristic, complemented, Sylow
316C23×C3—21no—
317C23×C3—31no—
318C23×C3—91no—
319C23×C3C22×D911yescharacteristic
320C23×C3C2×D3211yescharacteristic
321C22×D3—31no—
322C22×D3—91no—
323C22×D3—91no—
324C22×D3—181no—
325C22×D3—181no—
326C22×D3—181no—
327C22×D3#61111yescharacteristic
328C3×D4—61no—
329C3×D4—61no—
330C3×D4—181no—
331C3×D4—181no—
332C2×C3×C4—22no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—31no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—61no—
343C2×C3×C4—91no—
344C2×C3×C4—181no—
345#67—21no—
346#67—21no—
347#67—61no—
348#67—61no—
349#67—181no—
350#67—181no—
351#67—181no—
352#67—181no—
353#67—181no—
354#67—181no—
355#67—181no—
356#67—181no—
357#67—361no—
358#67—361no—
359C2×Dic3—21no—
360C2×Dic3—31no—
361C2×Dic3—61no—
362C2×Dic3—91no—
363C2×Dic3—91no—
364C2×Dic3—181no—
365C2×Dic3—181no—
366C2×Dic3—181no—
367C2×Dic3—182no—
368C2×Dic3—181no—
369C2×Dic3—181no—
370C2×Dic3#61111yescharacteristic
371D12—181no—
372D12—181no—
373C4×D3—181no—
374C4×D3—181no—
375C4×D3—181no—
376C4×D3—181no—
377Dic6—181no—
378Dic6—181no—
379C2×C32—21no—
380C2×C32—21no—
381C2×C32—21no—
382C2×C32#93711yescharacteristic
383C3⋊D3—121no—
384C3×D3—41no—
385C3×D3—121no—
386C2×C9—22no—
387C2×C9—21no—
388C2×C9—21no—
389C2×C9—21no—
390C2×C9—21no—
391C2×C9—41no—
392C2×C9—41no—
393C2×C9—121no—
394C2×C9#93711yescharacteristic
395D9—41no—
396D9—121no—
397D9—241no—
398#41—541no—
399#41—541no—
400#41—541no—
401C2×D4—31no—
402C2×D4—91no—
403C2×D4—271no—
404C2×D4—541no—
405C2×D4—541no—
406C2×D4—541no—
407C22×C4D3×D911yescharacteristic, complemented
408C4⋊C4—61no—
409C4⋊C4—181no—
410C4⋊C4—541no—
411C22⋊C4—31no—
412C22⋊C4—61no—
413C22⋊C4—91no—
414C22⋊C4—181no—
415C22⋊C4—271no—
416C22⋊C4—541no—
417C22×C3—21no—
418C22×C3—21no—
419C22×C3—21no—
420C22×C3—41no—
421C22×C3—41no—
422C22×C3—42no—
423C22×C3—61no—
424C22×C3—121no—
425C22×C3—181no—
426C22×C3—361no—
427C22×C3D4×D911yescharacteristic
428C22×C3D4×D911yescharacteristic
429C22×C3C2×#61111yescharacteristic
430C22×C3D3×#6711yescharacteristic
431C22×C3D3×#6711yescharacteristic
432C22×C3#381111yescharacteristic
433C2×D3—21no—
434C2×D3—41no—
435C2×D3—61no—
436C2×D3—121no—
437C2×D3—181no—
438C2×D3—181no—
439C2×D3—361no—
440C2×D3—361no—
441C2×D3—361no—
442C2×D3—362no—
443C2×D3—721no—
444C2×D3—721no—
445C3×C4—42no—
446C3×C4—41no—
447C3×C4—41no—
448C3×C4—61no—
449C3×C4—121no—
450C3×C4—181no—
451C3×C4—361no—
452Dic3—21no—
453Dic3—41no—
454Dic3—61no—
455Dic3—121no—
456Dic3—181no—
457Dic3—181no—
458Dic3—361no—
459Dic3—361no—
460Dic3—362no—
461Dic3—361no—
462C32#505311yescharacteristic
463C9—21no—
464C9#505311yescharacteristic, complemented
465C23—31no—
466C23—91no—
467C23—271no—
468C23—541no—
469C23C2×D3×D911yescharacteristic
470Q8—541no—
471D4—61no—
472D4—61no—
473D4—181no—
474D4—181no—
475D4—541no—
476D4—541no—
477D4—541no—
478D4—541no—
479D4—541no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—31no—
484C2×C4—61no—
485C2×C4—91no—
486C2×C4—181no—
487C2×C4—271no—
488C2×C4—541no—
489C2×C4—541no—
490C2×C4—541no—
491C2×C4—541no—
492C2×C3—22no—
493C2×C3—21no—
494C2×C3—21no—
495C2×C3—21no—
496C2×C3—21no—
497C2×C3—21no—
498C2×C3—21no—
499C2×C3—21no—
500C2×C3—41no—
501C2×C3—41no—
502C2×C3—121no—
503C2×C3—361no—
504C2×C3#6349911yescharacteristic
505C2×C3#6398311yescharacteristic
506D3—41no—
507D3—121no—
508D3—361no—
509D3—361no—
510D3—721no—
511C22—41no—
512C22—61no—
513C22—121no—
514C22—181no—
515C22—361no—
516C22—541no—
517C22—1081no—
518C22—2161no—
519C22D3×#61111yescharacteristic
520C22D9×#6711yescharacteristic
521C22#8866811yescharacteristic
522C4—41no—
523C4—61no—
524C4—121no—
525C4—181no—
526C4—361no—
527C4—541no—
528C4—1081no—
529C3—21no—
530C3#1058873311yescharacteristic, complemented
531C3#1059139811yescharacteristic
532C2—21no—
533C2—21no—
534C2—21no—
535C2—121no—
536C2—361no—
537C2—1081no—
538C2#1172129211yescharacteristic
subgroupG/H
centerC2#11721292
commutator, cosocleC23×C3×C9C23
FrattiniC23×C3C2×D32
FittingC22×C3×C4×C9C22
socleC2×C32#937
prime radical#11722535C2
hypercenterC22×C4D3×D9
hypocenterC3×C9#458
LCSG ▷ C23×C3×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. #11721818C2semidirect
3. C9×#929C2semidirect
4. C3×#63491C2semidirect
5. #11720332C2semidirect
6. #11720330C2semidirect
7. #11720329C2semidirect
10. C22×C3×C4×C9C22semidirect
20. #88551C22semidirect
23. C9×#217C22semidirect
24. C3×#3676C22semidirect
40. #63491D3semidirect
94. C22×C4×C9C2×D3semidirect
109. #3676C2×D3semidirect
142. #929D9semidirect
223. C22×C3×C4C2×D9semidirect
252. #217C2×D9semidirect
315. C3×C9#458semidirect
407. C22×C4D3×D9semidirect
464. C9#5053semidirect
530. C3#10588733semidirect

Maximal subgroups

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

subgroupindex
#117225352
#117218182
C9×#9292
C3×#634912
#117203322
#117203302
#117203292
#105913983
#105887333