Finite Group Databasebeta

← #72245981

Subgroups — #72245981

Order 3456.

1259 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 1259 conjugacy classes of subgroups — too many to draw. Only its 153 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#13164864C211yesmaximal, characteristic
2#10589525⋊C3C213yesmaximal, complemented
3C2×#63422⋊C3C213yesmaximal, complemented
4#11920440C311yesmaximal, characteristic, complemented
5C3×#84120—31nomaximal
6#11722579C413yescomplemented
7#11722576C411yescharacteristic, complemented
8C2×#88691C2213yes—
9#11721693—41nomaximal
10#63422⋊C3C2211yescharacteristic, complemented
11#63422⋊C3C2213yescomplemented
12C32×#585D311yescharacteristic
13C3×Dic3×#41—31no—
14C3×#5819—31no—
15#10593120—21no—
16#10593120C2×C311yescharacteristic
17C3×#5236—33no—
18C3×#5236—33no—
19C2×C3×#837—33no—
20C2×C3×#837—33no—
21#10589525C2×C313yescomplemented
22C2×#63422C2×C313yescomplemented
23#41×3−1+2—41no—
24#88691C2×C413yescomplemented
25#88691C2311yescharacteristic
26C2×#6204C2×C413yescomplemented
27#3667⋊C3—43no—
28C2×3−1+2⋊C8—43no—
29C3×#2901—91no—
30#84120—31no—
31C32×2−1+4—31no—
32C32×2−1+4Dic313yes—
33C32×H(5,𝔽2)—33no—
34C32×H(5,𝔽2)Dic311yescharacteristic
35C2×C32×#41—31no—
36C2×C32×#41—33no—
37C2×C32×#41C2×D313yes—
38C3×#948—63no—
39C3×#948—61no—
40C3×Q8×Dic3—61no—
41C3×D4×Dic3—63no—
42C3×#926—33no—
43C2×C3×C4×Dic3—33no—
44C2×C3×#208—33no—
45C2×C3×#208—33no—
462−1+4⋊C9—23no—
472−1+4⋊C9C3×C413yescomplemented
48#63700—21no—
49#63700C3×C411yescharacteristic, complemented
50C2×#41⋊C9—23no—
51C2×#41⋊C9C22×C313yes—
52C3×#837—33no—
53C3×#837—36no—
54C3×#837—33no—
55C3×#837—31no—
56C3×#837—33no—
57#63510—41no—
58#63422C22×C311yescharacteristic, complemented
59#63422C22×C313yescomplemented
60Q8×3−1+2—41no—
61D4×3−1+2—43no—
62C2×C4×3−1+2—43no—
63#6204C22×C411yescharacteristic
643−1+2⋊C8—41no—
653−1+2⋊C8—43no—
66C3×#585—31no—
67C3×#585C3×D311yescharacteristic
68C3×#568—91no—
69C3×C4×#41—91no—
70Dic3×#41—31no—
71#5819—31no—
72C3×#421—93no—
73C3×#421—93no—
74C2×C3×#104—93no—
75C2×C3×#104—93no—
76#5236—33no—
77#5236—33no—
78C2×#837—33no—
79C2×#837—33no—
80C32×#41—31no—
81C32×#41—33no—
82C32×#41—33no—
83C32×#41—33no—
84C32×#41—33no—
85C32×#41—36no—
86C32×#41—63no—
87C32×#41C2×Dic313yes—
88C32×#41S411yescharacteristic
89C32×#41C22×D311yescharacteristic
90C2×C32×Q8—31no—
91C2×C32×Q8—33no—
92C2×C32×Q8C2×Dic313yes—
93C2×C32×D4—33no—
94C2×C32×D4—33no—
95C2×C32×D4—33no—
96C2×C32×D4—63no—
97C22×C32×C4—33no—
98C22×C32×C4—61no—
99C22×C3×Dic3—63no—
100C3×#217—63no—
101C3×C4⋊Dic3—63no—
102C3×C4×Dic3—31no—
103C3×C4×Dic3—33no—
104C3×C4×Dic3—63no—
105C3×#208—33no—
106C3×#208—31no—
107C3×#208—63no—
108C3×#208—63no—
109C2×C3×C3⋊C8—63no—
110C2×C3×C3⋊C8—63no—
111C9×#41—41no—
112C9×#41—81no—
113#41⋊C9—23no—
114#41⋊C9—21no—
115#41⋊C9C2×C3×C413yescomplemented
116#41⋊C9C23×C311yescharacteristic
117C2×Q8⋊C9—23no—
118C2×Q8⋊C9C2×C3×C413yescomplemented
119#3667—43no—
120C2×C9⋊C8—43no—
121#2901—91noSylow
122C22×3−1+2—43no—
123C4×3−1+2—43no—
124C4×3−1+2—41no—
125C3×2−1+4—31no—
126C3×2−1+4—31no—
127C3×2−1+4—33no—
128C3×2−1+4—61no—
129C3×2−1+4C3×Dic313yes—
130C3×H(5,𝔽2)—33no—
131C3×H(5,𝔽2)—33no—
132C3×H(5,𝔽2)—31no—
133C3×H(5,𝔽2)—63no—
134C3×H(5,𝔽2)C3×Dic311yescharacteristic
135C2×C3×#41—31no—
136C2×C3×#41—31no—
137C2×C3×#41—33no—
138C2×C3×#41—33no—
139C2×C3×#41—33no—
140C2×C3×#41—61no—
141C2×C3×#41—63no—
142C2×C3×#41C2×C3×D313yes—
143C3×#131—183no—
144C3×#131—181no—
145C2×C3×M16—93no—
146C2×C3×M16—93no—
147C3×C4×Q8—181no—
148C3×C4×D4—183no—
149C3×#117—93no—
150C2×C3×C42—93no—
151#948—63no—
152#948—61no—
153Q8×Dic3—61no—
154D4×Dic3—63no—
155#926—33no—
156C2×C4×Dic3—33no—
157C2×#208—33no—
158C2×#208—33no—
159C3×#104—93no—
160C3×#104—91no—
161C3×#104—96no—
162C3×#104—93no—
163C3×#104—93no—
164#837—33no—
165#837—36no—
166#837—33no—
167#837—31no—
168#837—33no—
169C23×C32—33no—
170C23×C32—61no—
171C32×Q8—33no—
172C32×Q8—33no—
173C32×Q8SL(2,ℤ/4)11yescharacteristic
174C32×Q8C22×Dic311yescharacteristic
175C32×D4—33no—
176C32×D4—31no—
177C32×D4—36no—
178C32×D4—33no—
179C32×D4—63no—
180C32×D4SL(2,ℤ/4)13yes—
181C2×C32×C4—33no—
182C2×C32×C4—33no—
183C2×C32×C4—33no—
184C2×C32×C4—33no—
185C2×C32×C4—31no—
186C2×C32×C4—63no—
187C2×C32×C4C2×S413yes—
188C2×C3×Dic3—61no—
189C2×C3×Dic3—63no—
190C2×C3×Dic3—126no—
191C3×C3⋊C8—61no—
192C3×C3⋊C8—63no—
193Q8×C9—41no—
194Q8×C9—81no—
195D4×C9—43no—
196D4×C9—83no—
197C2×C4×C9—43no—
198C2×C4×C9—83no—
199Q8⋊C9—21no—
200Q8⋊C9C22×C3×C411yescharacteristic
201C9⋊C8—41no—
202C9⋊C8—43no—
203#585AGL(1,ℤ/9)11yescharacteristic
204#568—91no—
205C4×#41—91no—
206#421—93no—
207#421—93no—
208C2×#104—93no—
209C2×#104—93no—
210C2×3−1+2—41no—
211C2×3−1+2—83no—
212C3×#41—33no—
213C3×#41—31no—
214C3×#41—33no—
215C3×#41—31no—
216C3×#41—33no—
217C3×#41—33no—
218C3×#41—33no—
219C3×#41—33no—
220C3×#41—36no—
221C3×#41—33no—
222C3×#41—33no—
223C3×#41—36no—
224C3×#41—33no—
225C3×#41—31no—
226C3×#41—31no—
227C3×#41—63no—
228C3×#41—63no—
229C3×#41—63no—
230C3×#41—63no—
231C3×#41—66no—
232C3×#41—61no—
233C3×#41—63no—
234C3×#41—63no—
235C3×#41—63no—
236C3×#41—63no—
237C3×#41C2×C3×Dic313yes—
238C3×#41C3×S411yescharacteristic
239C3×#41C22×C3×D311yescharacteristic
240C2×C3×Q8—31no—
241C2×C3×Q8—31no—
242C2×C3×Q8—33no—
243C2×C3×Q8—33no—
244C2×C3×Q8—33no—
245C2×C3×Q8—61no—
246C2×C3×Q8—63no—
247C2×C3×Q8C2×C3×Dic313yes—
248C2×C3×D4—33no—
249C2×C3×D4—33no—
250C2×C3×D4—33no—
251C2×C3×D4—33no—
252C2×C3×D4—33no—
253C2×C3×D4—33no—
254C2×C3×D4—63no—
255C2×C3×D4—63no—
256C2×C3×D4—63no—
257C2×C3×D4—63no—
258C2×C3×D4—66no—
259C2×C3×D4—63no—
260C22×C3×C4—33no—
261C22×C3×C4—33no—
262C22×C3×C4—63no—
263C22×C3×C4—61no—
264C22×C3×C4—63no—
265C22×C3×C4—183no—
266C22×Dic3—63no—
267C3×M16—93no—
268C3×M16—91no—
269C3×M16—183no—
270C3×M16—183no—
271C2×C3×C8—183no—
272C2×C3×C8—183no—
273C3×C4⋊C4—183no—
274C3×C22⋊C4—183no—
275C3×C42—91no—
276C3×C42—93no—
277C3×C42—183no—
278#217—63no—
279C4⋊Dic3—63no—
280C4×Dic3—31no—
281C4×Dic3—33no—
282C4×Dic3—63no—
283#208—33no—
284#208—31no—
285#208—63no—
286#208—63no—
287C2×C3⋊C8—63no—
288C2×C3⋊C8—63no—
289C22×C32—31no—
290C22×C32—33no—
291C22×C32—123no—
292C22×C32—241no—
293C22×C32C2×SL(2,ℤ/4)13yes—
294C32×C4—31no—
295C32×C4—33no—
296C32×C4C2×SL(2,ℤ/4)13yes—
297C32×C4C22×S411yescharacteristic
298C3×Dic3—62no—
299C3×Dic3—123no—
300C22×C9—43no—
301C22×C9—83no—
302C4×C9—43no—
303C4×C9—41no—
304C4×C9—83no—
305C4×C9—81no—
3062−1+4—31no—
3072−1+43−1+2⋊C413yes—
308H(5,𝔽2)—33no—
309H(5,𝔽2)3−1+2⋊C411yescharacteristic
310C2×#41—31no—
311C2×#41—33no—
312C2×#41C2×AGL(1,ℤ/9)13yes—
313#131—183no—
314#131—181no—
315C2×M16—93no—
316C2×M16—93no—
317C4×Q8—181no—
318C4×D4—183no—
319#117—93no—
320C2×C42—93no—
321#104—93no—
322#104—96no—
323#104—93no—
324#104—91no—
325#104—93no—
3263−1+2—41noSylow
327C23×C3—33no—
328C23×C3—33no—
329C23×C3—63no—
330C23×C3—61no—
331C23×C3—63no—
332C3×Q8—33no—
333C3×Q8—33no—
334C3×Q8—33no—
335C3×Q8—33no—
336C3×Q8—31no—
337C3×Q8—31no—
338C3×Q8—63no—
339C3×Q8—63no—
340C3×Q8C3×SL(2,ℤ/4)11yescharacteristic
341C3×Q8C22×C3×Dic311yescharacteristic
342C3×D4—33no—
343C3×D4—31no—
344C3×D4—36no—
345C3×D4—33no—
346C3×D4—36no—
347C3×D4—33no—
348C3×D4—31no—
349C3×D4—33no—
350C3×D4—33no—
351C3×D4—63no—
352C3×D4—63no—
353C3×D4—61no—
354C3×D4—66no—
355C3×D4—63no—
356C3×D4—63no—
357C3×D4—63no—
358C3×D4—63no—
359C3×D4C3×SL(2,ℤ/4)13yes—
360C2×C3×C4—31no—
361C2×C3×C4—33no—
362C2×C3×C4—33no—
363C2×C3×C4—33no—
364C2×C3×C4—33no—
365C2×C3×C4—33no—
366C2×C3×C4—33no—
367C2×C3×C4—33no—
368C2×C3×C4—33no—
369C2×C3×C4—31no—
370C2×C3×C4—33no—
371C2×C3×C4—61no—
372C2×C3×C4—63no—
373C2×C3×C4—63no—
374C2×C3×C4—63no—
375C2×C3×C4—63no—
376C2×C3×C4—63no—
377C2×C3×C4—66no—
378C2×C3×C4—63no—
379C2×C3×C4—181no—
380C2×C3×C4—183no—
381C2×C3×C4—366no—
382C2×C3×C4C2×C3×S413yes—
383C2×Dic3—61no—
384C2×Dic3—63no—
385C2×Dic3—126no—
386C3×C8—183no—
387C3×C8—181no—
388C3⋊C8—61no—
389C3⋊C8—63no—
390C2×C32—23no—
391C2×C32—61no—
392C2×C32—63no—
393C2×C32C22×SL(2,ℤ/4)11yescharacteristic
394C2×C9—41no—
395C2×C9—83no—
396C2×C9—81no—
397C2×C9—163no—
398#41—31no—
399#41—33no—
400#41—33no—
401#41—33no—
402#41—33no—
403#41—36no—
404#41—63no—
405#41C2×3−1+2⋊C413yes—
406#41((C22⋊C9)⋊C2)⋊C311yescharacteristic
407#41C22×AGL(1,ℤ/9)11yescharacteristic
408C2×Q8—31no—
409C2×Q8—33no—
410C2×Q8C2×3−1+2⋊C413yes—
411C2×D4—33no—
412C2×D4—33no—
413C2×D4—33no—
414C2×D4—63no—
415C22×C4—33no—
416C22×C4—61no—
417C22×C4—183no—
418M16—93no—
419M16—91no—
420M16—183no—
421M16—183no—
422C2×C8—183no—
423C2×C8—183no—
424C4⋊C4—183no—
425C22⋊C4—183no—
426C42—91no—
427C42—93no—
428C42—183no—
429C22×C3—31no—
430C22×C3—33no—
431C22×C3—31no—
432C22×C3—33no—
433C22×C3—33no—
434C22×C3—61no—
435C22×C3—63no—
436C22×C3—123no—
437C22×C3—123no—
438C22×C3—243no—
439C22×C3—241no—
440C22×C3—243no—
441C22×C3C2×C3×SL(2,ℤ/4)13yes—
442C3×C4—31no—
443C3×C4—33no—
444C3×C4—31no—
445C3×C4—33no—
446C3×C4—31no—
447C3×C4—33no—
448C3×C4—61no—
449C3×C4—63no—
450C3×C4—182no—
451C3×C4—363no—
452C3×C4C2×C3×SL(2,ℤ/4)13yes—
453C3×C4C22×C3×S411yescharacteristic
454Dic3—62no—
455Dic3—123no—
456C32#8554111yescharacteristic
457C9—41no—
458C9—81no—
459C23—33no—
460C23—61no—
461Q8—33no—
462Q8—33no—
463Q8#1094⋊C411yescharacteristic
464Q8C22×3−1+2⋊C411yescharacteristic
465D4—33no—
466D4—31no—
467D4—36no—
468D4—33no—
469D4—63no—
470D4#1094⋊C413yes—
471C2×C4—33no—
472C2×C4—31no—
473C2×C4—33no—
474C2×C4—33no—
475C2×C4—33no—
476C2×C4—63no—
477C2×C4—181no—
478C2×C4—183no—
479C2×C4—366no—
480C2×C4C2×((C22⋊C9)⋊C2)⋊C313yes—
481C8—183no—
482C8—181no—
483C2×C3—23no—
484C2×C3—31no—
485C2×C3—63no—
486C2×C3—61no—
487C2×C3—63no—
488C2×C3—61no—
489C2×C3—63no—
490C2×C3—121no—
491C2×C3—123no—
492C2×C3C22×C3×SL(2,ℤ/4)11yescharacteristic
493C22—31no—
494C22—33no—
495C22—123no—
496C22—241no—
497C22C2×#1094⋊C413yes—
498C4—31no—
499C4—33no—
500C4—182no—
501C4—363no—
502C4C2×#1094⋊C413yes—
503C4C22×((C22⋊C9)⋊C2)⋊C311yescharacteristic
504C3—31no—
505C3C3×#8554111yescharacteristic
506C2—23no—
507C2—61no—
508C2—63no—
509C2C22×#1094⋊C411yescharacteristic
subgroupG/H
centerC4C22×((C22⋊C9)⋊C2)⋊C3
commutator, hypocenterQ8⋊C9C22×C3×C4
FrattiniC3×C4C22×C3×S4
Fitting, prime radicalC32×#585D3
socleC2×C3C22×C3×SL(2,ℤ/4)
cosocle#41⋊C9C23×C3
hypercenter#41((C22⋊C9)⋊C2)⋊C3
LCSG ▷ Q8⋊C9
UCSC1 ◁ C4 ◁ #41

Complements

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

NH ≅ G/Ntype
2. #10589525⋊C3C2semidirect
3. C2×#63422⋊C3C2semidirect
4. #11920440C3semidirect
6. #11722579C4semidirect
7. #11722576C4semidirect
10. #63422⋊C3C22semidirect
11. #63422⋊C3C22semidirect
21. #10589525C2×C3semidirect
22. C2×#63422C2×C3semidirect
24. #88691C2×C4semidirect
26. C2×#6204C2×C4semidirect
47. 2−1+4⋊C9C3×C4semidirect
49. #63700C3×C4semidirect
58. #63422C22×C3semidirect
59. #63422C22×C3semidirect
115. #41⋊C9C2×C3×C4semidirect
118. C2×Q8⋊C9C2×C3×C4semidirect

Maximal subgroups

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

subgroupindex
#131648642
#10589525⋊C32
C2×#63422⋊C32
#119204403
C3×#841203
#117216934