Finite Group Databasebeta

← #13141039

Subgroups — #13141039

Order 1728.

670 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 670 conjugacy classes of subgroups — too many to draw. Only its 81 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#11722672C211yesmaximal, characteristic, complemented
2C9×#949C211yesmaximal, characteristic, complemented
3#11721040C211yesmaximal, characteristic, complemented
4#11720890C211yesmaximal, characteristic, complemented
5C3×#63396C211yesmaximal, characteristic, complemented
6#11720292C211yesmaximal, characteristic, complemented
7#11720288C211yesmaximal, characteristic, complemented
8#10591667—31nomaximal
9#10588879—31nomaximal
10C3×C9×#41C2211yescharacteristic
11#88746—21no—
12D4×C3⋊D9—22no—
13D4×C3⋊D9—21no—
14#88736—21no—
15#88736C2211yescharacteristic
16C2×C4⋊(C3⋊D9)—21no—
17C2×C4⋊(C3⋊D9)C2211yescharacteristic
18C2×C9×D12C2211yescharacteristic, complemented
19C2×#6195—22no—
20C9×#215—21no—
21C9×#213—21no—
22C3×C4×Dic9C2211yescharacteristic, complemented
23C9×#208C2211yescharacteristic, complemented
24C3×#3667C2211yescharacteristic, complemented
25#88424—21no—
26#88422—21no—
27#64366—31no—
28C3×#949D311yescharacteristic
29#63911—31no—
30#63830—31no—
31#63715—31no—
32#63715—61no—
33C3×#837—31no—
34#63570—31no—
35#63559—31no—
36C9×AGL(1,ℤ/8)—31no—
37#63502—31no—
38#63495—31no—
39#63396D311yescharacteristic, complemented
40#63384—31no—
41#63383—31no—
42C22×C3⋊D9—22no—
43C22×C3⋊D9D412yes—
44C22×D3×C9—21no—
45C3×Q8×C9D411yescharacteristic, complemented
46C3×D4×C9—21no—
47C3×D4×C9D411yescharacteristic, complemented
48C2×C3×C4×C9—21no—
49C2×C3×C4×C9C2311yescharacteristic
50#6233—22no—
51#6233—21no—
52C4⋊(C3⋊D9)—22no—
53C4⋊(C3⋊D9)—21no—
54C4⋊(C3⋊D9)—21no—
55C4⋊(C3⋊D9)D411yescharacteristic, complemented
56C4×C3⋊D9—22no—
57C4×C3⋊D9—21no—
58#6226D411yescharacteristic, complemented
59C2×C3×Dic9—21no—
60C9×D12—21no—
61C9×D12—21no—
62#6195—44no—
63C9×C3⋊C8—21no—
64C3×C9⋊C8—21no—
65#5198—91no—
66#4822—91no—
67C32×#41C2×D311yescharacteristic
68#3832—61no—
69D4×C3⋊D3—62no—
70D4×C3⋊D3—61no—
71#3828—31no—
72#3828—61no—
73C2×C4⋊(C3⋊D3)—31no—
74C2×C4⋊(C3⋊D3)—61no—
75C2×C3×D12C2×D311yescharacteristic
76C2×#626—62no—
77C3×#215—21no—
78C3×#213—21no—
79C3×C4×Dic3—31no—
80C3×#208—31no—
81C3×#208C2×D311yescharacteristic
82#3716—61no—
83#3714—61no—
84C9×#41—21no—
85C9×#41C2×D311yescharacteristic
86C2×D4×C9—31no—
87C2×#611—62no—
88#3701—62no—
89#3701—61no—
90D4×D9—62no—
91D4×D9—64no—
92D4×D9—62no—
93D4×D9—61no—
94#3697—31no—
95#3697—61no—
96#3697—61no—
97#3697—62no—
98C2×D36—31no—
99C2×D36—61no—
100C2×D36—61no—
101C2×D36—62no—
102C9×SD16—61no—
103C9×D8—61no—
104C9×M16—31no—
105#3673—61no—
106#3672—61no—
107C4×Dic9C2×D311yescharacteristic, complemented
108#3667C2×D311yescharacteristic, complemented
109C22×C3×C9—21no—
110C22×C3×C9C2×D411yescharacteristic
111C2×C3⋊D9—22no—
112C2×C3⋊D9—21no—
113C2×C3⋊D9—42no—
114C2×C3⋊D9—82no—
115C2×D3×C9—42no—
116C2×D3×C9—41no—
117C3×C4×C9—21no—
118C3×C4×C9C2×D411yescharacteristic
119C3×C4×C9C2×D411yescharacteristic
120C3⋊Dic9—21no—
121C3×Dic9—22no—
122#1009—91no—
123#1009—91no—
124#1009—181no—
125C3×AGL(1,ℤ/8)—31no—
126#949D911yescharacteristic, complemented
127#940—91no—
128#932—91no—
129#908—91no—
130#874—91no—
131C3×#104—91no—
132#837—31no—
133#825—91no—
134#823—91no—
135#823—91no—
136#805—91no—
137C22×C3⋊D3—32no—
138C22×C3⋊D3—62no—
139C22×C3×D3—21no—
140C32×Q8#6711yescharacteristic
141C32×D4—21no—
142C32×D4#6711yescharacteristic
143C2×C32×C4—21no—
144C2×C32×C4C22×D311yescharacteristic
145#638—62no—
146#638—61no—
147C4⋊(C3⋊D3)—31no—
148C4⋊(C3⋊D3)—62no—
149C4⋊(C3⋊D3)—61no—
150C4⋊(C3⋊D3)—61no—
151C4×C3⋊D3—62no—
152C4×C3⋊D3—61no—
153#634—31no—
154C2×C3×Dic3—61no—
155C3×D12—21no—
156C3×D12—21no—
157#626—124no—
158C23×C9—61no—
159C22×D9—32no—
160C22×D9—62no—
161C22×D9—64no—
162C22×D9—62no—
163C3×C3⋊C8—21no—
164C3×C3⋊C8—61no—
165Q8×C9—21no—
166Q8×C9D1211yescharacteristic, complemented
167D4×C9—22no—
168D4×C9—21no—
169D4×C9—21no—
170D4×C9—61no—
171D4×C9—61no—
172D4×C9D1211yescharacteristic, complemented
173C2×C4×C9—22no—
174C2×C4×C9—21no—
175C2×C4×C9—21no—
176C2×C4×C9C22×D311yescharacteristic
177#611—64no—
178#611—62no—
179#611—62no—
180#611—61no—
181#611—124no—
182C2×Dic9—21no—
183D36—31no—
184D36—61no—
185D36—62no—
186D36—64no—
187D36—62no—
188D36—62no—
189D36—61no—
190D36—61no—
191C4×D9—64no—
192C4×D9—62no—
193C4×D9—62no—
194C4×D9—61no—
195Dic18—31no—
196Dic18—61no—
197C8×C9—61no—
198C9⋊C8—21no—
199#453—271noSylow
200C2×C3×C9—21no—
201C2×C3×C9—41no—
202C2×C3×C9#12011yescharacteristic
203C3⋊D9—42no—
204C3⋊D9—41no—
205D3×C9—81no—
206C3×#41—21no—
207C3×#41C2×D911yescharacteristic
208C3×#41D3211yescharacteristic
209C2×C3×D4—31no—
210C2×#67—182no—
211#239—182no—
212#239—181no—
213#239—181no—
214D3×D4—182no—
215D3×D4—182no—
216D3×D4—184no—
217D3×D4—182no—
218D3×D4—181no—
219D3×D4—181no—
220#235—91no—
221#235—91no—
222#235—181no—
223#235—181no—
224#235—181no—
225#235—182no—
226C2×D12—91no—
227C2×D12—91no—
228C2×D12—181no—
229C2×D12—181no—
230C2×D12—181no—
231C2×D12—182no—
232C2×D12—182no—
233C2×D12C2×D911yescharacteristic, complemented
234C3×SD16—61no—
235C3×D8—61no—
236C3×M16—31no—
237C3×M16—91no—
238C3×C42—91no—
239#215—21no—
240#214—181no—
241#213—21no—
242#213—181no—
243C4×Dic3—31no—
244#208—31no—
245#208C2×D911yescharacteristic, complemented
246D24—181no—
247#204—181no—
248C22×C32—21no—
249C22×C32D3×D411yescharacteristic
250C2×C3⋊D3—61no—
251C2×C3⋊D3—62no—
252C2×C3⋊D3—122no—
253C2×C3⋊D3—242no—
254C2×C3×D3—42no—
255C2×C3×D3—41no—
256C32×C4—21no—
257C32×C4D3×D411yescharacteristic
258C32×C4C2×#6711yescharacteristic
259C3⋊Dic3—61no—
260C3×Dic3—62no—
261C22×C9—22no—
262C22×C9—21no—
263C22×C9—21no—
264C22×C9—122no—
265C22×C9—121no—
266C22×C9D3×D411yescharacteristic
267C2×D9—62no—
268C2×D9—64no—
269C2×D9—61no—
270C2×D9—62no—
271C2×D9—122no—
272C2×D9—242no—
273C2×D9—244no—
274C2×D9—242no—
275C4×C9—22no—
276C4×C9—21no—
277C4×C9—21no—
278C4×C9—21no—
279C4×C9C2×D1211yescharacteristic
280C4×C9D3×D411yescharacteristic
281Dic9—22no—
282Dic9—62no—
283Dic9—61no—
284H(5,𝔽2)—271no—
285AGL(1,ℤ/8)—31no—
286AGL(1,ℤ/8)—271no—
287#127—271no—
288#104—91no—
289#104—271no—
290#100—271no—
291C3×C9#45311yescharacteristic, complemented, Sylow
292C23×C3—61no—
293C22×D3—21no—
294C22×D3—92no—
295C22×D3—92no—
296C22×D3—182no—
297C22×D3—182no—
298C22×D3—184no—
299C22×D3—182no—
300C3×Q8—21no—
301C3×Q8#61111yescharacteristic, complemented
302C3×Q8#62611yescharacteristic
303C3×D4—22no—
304C3×D4—21no—
305C3×D4—21no—
306C3×D4—21no—
307C3×D4—61no—
308C3×D4—61no—
309C3×D4#61111yescharacteristic, complemented
310C3×D4#62611yescharacteristic
311C2×C3×C4—22no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—21no—
315C2×C3×C4—181no—
316C2×C3×C4C22×D911yescharacteristic
317C2×C3×C4C2×D3211yescharacteristic
318#67—182no—
319#67—184no—
320#67—182no—
321#67—182no—
322#67—181no—
323#67—181no—
324#67—364no—
325C2×Dic3—61no—
326D12—21no—
327D12—21no—
328D12—91no—
329D12—91no—
330D12—181no—
331D12—182no—
332D12—182no—
333D12—184no—
334D12—182no—
335D12—182no—
336D12—181no—
337D12—181no—
338D12—181no—
339D12—181no—
340D12—364no—
341C4×D3—182no—
342C4×D3—184no—
343C4×D3—182no—
344C4×D3—182no—
345C4×D3—181no—
346C4×D3—181no—
347Dic6—91no—
348Dic6—91no—
349Dic6—181no—
350C3×C8—61no—
351C3×C8—181no—
352C3⋊C8—21no—
353C3⋊C8—61no—
354C2×C32—21no—
355C2×C32—41no—
356C2×C32#93711yescharacteristic
357C3⋊D3—122no—
358C3⋊D3—121no—
359C3×D3—81no—
360C2×C9—21no—
361C2×C9—21no—
362C2×C9—41no—
363C2×C9—42no—
364C2×C9—41no—
365C2×C9—241no—
366C2×C9#88211yescharacteristic
367D9—124no—
368D9—122no—
369D9—122no—
370D9—121no—
371#41—271no—
372#41—541no—
373#41—541no—
374#41D3×D911yescharacteristic
375C2×D4—31no—
376C2×D4—271no—
377C2×D4—541no—
378C2×D4—542no—
379C2×D4—542no—
380C2×D4—541no—
381SD16—61no—
382SD16—541no—
383D8—61no—
384D8—541no—
385M16—31no—
386M16—91no—
387C42—91no—
388C22×C3—22no—
389C22×C3—21no—
390C22×C3—21no—
391C22×C3—21no—
392C22×C3—122no—
393C22×C3—121no—
394C22×C3D4×D911yescharacteristic
395C22×C3#381111yescharacteristic
396C2×D3—42no—
397C2×D3—41no—
398C2×D3—182no—
399C2×D3—184no—
400C2×D3—181no—
401C2×D3—182no—
402C2×D3—181no—
403C2×D3—182no—
404C2×D3—362no—
405C2×D3—362no—
406C2×D3—722no—
407C2×D3—722no—
408C2×D3—724no—
409C2×D3—722no—
410C3×C4—22no—
411C3×C4—21no—
412C3×C4—21no—
413C3×C4—21no—
414C3×C4—21no—
415C3×C4—182no—
416C3×C4D4×D911yescharacteristic
417C3×C4C2×#61111yescharacteristic
418C3×C4C2×#62611yescharacteristic
419C3×C4#381111yescharacteristic
420Dic3—62no—
421Dic3—182no—
422Dic3—181no—
423Dic3—181no—
424C32#519811yescharacteristic
425C9—21no—
426C9#482211yescharacteristic, complemented
427C23—61no—
428C23—272no—
429C23—542no—
430Q8—271no—
431Q8#619511yescharacteristic, complemented
432D4—21no—
433D4—61no—
434D4—61no—
435D4—271no—
436D4—542no—
437D4—542no—
438D4—541no—
439D4—541no—
440D4—541no—
441D4—1084no—
442D4#619511yescharacteristic, complemented
443C2×C4—21no—
444C2×C4—181no—
445C2×C4—542no—
446C2×C4—541no—
447C2×C4C2×D3×D911yescharacteristic
448C8—61no—
449C8—181no—
450C2×C3—21no—
451C2×C3—21no—
452C2×C3—21no—
453C2×C3—41no—
454C2×C3—42no—
455C2×C3—41no—
456C2×C3—41no—
457C2×C3—241no—
458C2×C3#6349911yescharacteristic
459C2×C3#6398111yescharacteristic
460D3—81no—
461D3—364no—
462D3—362no—
463D3—362no—
464D3—361no—
465D3—362no—
466D3—361no—
467C22—21no—
468C22—122no—
469C22—121no—
470C22—541no—
471C22—542no—
472C22—1082no—
473C22—2162no—
474C22#8866811yescharacteristic
475C4—21no—
476C4—182no—
477C4—541no—
478C4C2×#619511yescharacteristic
479C4#8866811yescharacteristic
480C3—21no—
481C3#1058887911yescharacteristic, complemented
482C3#1059166711yescharacteristic
483C2—21no—
484C2—41no—
485C2—241no—
486C2—1082no—
487C2—1081no—
488C2#1172129111yescharacteristic
subgroupG/H
centerC2#11721291
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×#41C22
socleC2×C32#937
hypercenter#41D3×D9
hypocenterC3×C9#453
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
1. #11722672C2semidirect
2. C9×#949C2semidirect
3. #11721040C2semidirect
4. #11720890C2semidirect
5. C3×#63396C2semidirect
6. #11720292C2semidirect
7. #11720288C2semidirect
18. C2×C9×D12C22semidirect
22. C3×C4×Dic9C22semidirect
23. C9×#208C22semidirect
24. C3×#3667C22semidirect
39. #63396D3semidirect
45. C3×Q8×C9D4semidirect
47. C3×D4×C9D4semidirect
55. C4⋊(C3⋊D9)D4semidirect
58. #6226D4semidirect
107. C4×Dic9C2×D3semidirect
108. #3667C2×D3semidirect
126. #949D9semidirect
166. Q8×C9D12semidirect
172. D4×C9D12semidirect
233. C2×D12C2×D9semidirect
245. #208C2×D9semidirect
291. C3×C9#453semidirect
301. C3×Q8#611semidirect
309. C3×D4#611semidirect
426. C9#4822semidirect
431. Q8#6195semidirect
442. D4#6195semidirect
481. C3#10588879semidirect

Maximal subgroups

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

subgroupindex
#117226722
C9×#9492
#117210402
#117208902
C3×#633962
#117202922
#117202882
#105916673
#105888793