Finite Group Databasebeta

← #13160919

Subgroups — #13160919

Order 1728.

1048 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 1048 conjugacy classes of subgroups — too many to draw. Only its 154 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
1C2×#88860C211yesmaximal, characteristic
2C32×#936C211yesmaximal, characteristic, complemented
3#11723484C211yesmaximal, characteristic, complemented
4C2×#88795C211yesmaximal, characteristic, complemented
5C3×#63659C211yesmaximal, characteristic, complemented
6Dic3⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
7#11722037C211yesmaximal, characteristic, complemented
8#10592756—31nomaximal
9#10591432—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×#6289—21no—
14#88860—22no—
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×#217—21no—
19C32×#217—21no—
20C32×C4×Dic3C2211yescharacteristic
21#88804—21no—
22#88795—22no—
23C32×#124—31no—
24C2×#3755—31no—
25C2×#3753—31no—
26C2×#3753—34no—
27C2×#3753—64no—
28C4×C4⋊(C3⋊D3)—31no—
29C3×#936D314yescomplemented
30#63894—34no—
31C2×#3717—34no—
32#63662—31no—
33#63659D311yescharacteristic, complemented
34C4⋊(C3⋊(C3⋊C8))—31no—
35C3×#832—34no—
36Dic3⋊(C3⋊C8)—34no—
37#63565—34no—
38C23×C33—21no—
39C22×C3⋊(C3⋊D3)—21no—
40C33×D4—21no—
41C33×D4—21no—
42C2×C33×C4C2311yescharacteristic
43C4⋊(C3⋊(C3⋊D3))—21no—
44C4⋊(C3⋊(C3⋊D3))D412yes—
45C2×C3×C3⋊Dic3—21no—
46C2×C32×Dic3—21no—
47C2×C32×Dic3D411yescharacteristic
48C2×C32×Dic3D411yescharacteristic
49C32×Dic6—21no—
50C32×Dic6—21no—
51#6289—42no—
52C3⋊(C3⋊(C3⋊C8))—21no—
53C3×C3⋊(C3⋊C8)—21no—
54#5057—94no—
55#4787—91no—
56C2×C32×Q8—31no—
57C2×C32×D4—24no—
58C2×C32×D4C2×D314yes—
59C2×C32×D4C2×D311yescharacteristic
60C2×C4⋊(C3⋊D3)—31no—
61C2×C4⋊(C3⋊D3)—34no—
62C2×C4⋊(C3⋊D3)—64no—
63C2×C4×C3⋊D3—61no—
64C2×C3×Dic6C2×D314yes—
65C2×#624—64no—
66C32×C22⋊C4—61no—
67C32×C22⋊C4—61no—
68C32×C42—31no—
69#3755—62no—
70#3753—62no—
71#3753—68no—
72#3753—616no—
73#3752—61no—
74C4⋊(C3⋊Dic3)C2×D311yescharacteristic
75C2×C3⋊(C3⋊C8)—31no—
76C2×C3⋊(C3⋊C8)—34no—
77C2×C3⋊(C3⋊C8)—64no—
78C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
79C3×#217—24no—
80C3×#217—24no—
81C3×C4⋊Dic3—34no—
82C3×C4×Dic3C2×D314yes—
83C2×C3×C3⋊C8—34no—
84#3722—64no—
85#3717—68no—
86C22×C33—41no—
87C22×C33—41no—
88C22×C33—41no—
89C22×C33C2×D411yescharacteristic
90C2×C3⋊(C3⋊D3)—22no—
91C2×C3⋊(C3⋊D3)—41no—
92C2×C3⋊(C3⋊D3)—41no—
93C33×C4C2×D411yescharacteristic
94C33×C4#4111yescharacteristic
95C3×C3⋊Dic3—41no—
96C32×Dic3—21no—
97C32×Dic3—21no—
98C32×Dic3—41no—
99C3×#124—34no—
100C2×#215—94no—
101#936C3⋊D311yescharacteristic, complemented
102C2×#213—94no—
103C2×#213—91no—
104C2×#213—184no—
105C2×#204—91no—
106#893—91no—
107C4×D12—94no—
108C3×#102—91no—
109#835—94no—
110#832—34no—
111Dic3⋊C8—91no—
112#810—91no—
113C4⋊(C3⋊C8)—94no—
114C23×C32—21no—
115C23×C32—24no—
116C23×C32—44no—
117C22×C3⋊D3—61no—
118C22×C3⋊D3—64no—
119C22×C3⋊D3—124no—
120C32×Q8—61no—
121C32×Q8—61no—
122C32×D4—28no—
123C32×D4—21no—
124C32×D4—24no—
125C32×D4—21no—
126C32×D4—24no—
127C32×D4—44no—
128C2×C32×C4—24no—
129C2×C32×C4—31no—
130C2×C32×C4—31no—
131C2×C32×C4—61no—
132C2×C32×C4C22×D314yes—
133C2×C32×C4C22×D311yescharacteristic
134C2×C3⋊Dic3—21no—
135C4⋊(C3⋊D3)—32no—
136C4⋊(C3⋊D3)—38no—
137C4⋊(C3⋊D3)—68no—
138C4⋊(C3⋊D3)—61no—
139C4⋊(C3⋊D3)—64no—
140C4⋊(C3⋊D3)—124no—
141C4×C3⋊D3—122no—
142C2×C3×Dic3—24no—
143C2×C3×Dic3—64no—
144C2×C3×Dic3#6714yes—
145C2×C3×Dic3#6714yes—
146C3×Dic6—24no—
147C3×Dic6—24no—
148#624—128no—
149C3⋊(C3⋊C8)—21no—
150C3⋊(C3⋊C8)—68no—
151C3⋊(C3⋊C8)—64no—
152C3⋊(C3⋊C8)—61no—
153C3×C3⋊C8—64no—
154#463—271noSylow
155C2×C33—81no—
156C2×C33#12111yescharacteristic
157C2×C33#13511yescharacteristic
158C2×C33AGL(1,ℤ/8)11yescharacteristic
159C3⋊(C3⋊D3)—42no—
160C2×C3×Q8—34no—
161C2×C3×D4—24no—
162C2×C3×D4D3214yes—
163C2×C3×D4C2×C3⋊D311yescharacteristic
164C2×D12—94no—
165C2×D12—91no—
166C2×D12—184no—
167C2×C4×D3—184no—
168C2×C4×D3—181no—
169C2×Dic6C2×C3⋊D311yescharacteristic
170C2×C3×C8—91no—
171C3×C4⋊C4—91no—
172C3×C22⋊C4—64no—
173C3×C22⋊C4—64no—
174C3×C42—34no—
175#217—21no—
176#217—21no—
177#215—188no—
178#213—188no—
179#213—182no—
180#213—1816no—
181(C2×D3)⋊C4—184no—
182(C2×D3)⋊C4—181no—
183C4⋊Dic3—34no—
184C4×Dic3C2×C3⋊D311yescharacteristic
185C2×C3⋊C8—34no—
186C2×C3⋊C8—94no—
187C2×C3⋊C8—91no—
188C2×C3⋊C8—184no—
189#204—182no—
190C22×C32—24no—
191C22×C32—48no—
192C22×C32—41no—
193C22×C32—41no—
194C22×C32—41no—
195C22×C32—44no—
196C22×C32—44no—
197C22×C32—44no—
198C22×C32—84no—
199C22×C32—84no—
200C22×C32D3×D411yescharacteristic
201C22×C32C2×#6714yes—
202C2×C3⋊D3—62no—
203C2×C3⋊D3—68no—
204C2×C3⋊D3—128no—
205C2×C3⋊D3—121no—
206C2×C3⋊D3—121no—
207C2×C3⋊D3—124no—
208C2×C3⋊D3—124no—
209C2×C3⋊D3—244no—
210C2×C3⋊D3—244no—
211C32×C4—24no—
212C32×C4—24no—
213C32×C4—61no—
214C32×C4—61no—
215C32×C4—121no—
216C32×C4#23511yescharacteristic
217C32×C4D3×D414yes—
218C32×C4D3×D411yescharacteristic
219C32×C4#23714yes—
220C3⋊Dic3—41no—
221C3×Dic3—24no—
222C3×Dic3—24no—
223C3×Dic3—44no—
224C3×Dic3—124no—
225C2×SD16—271no—
226C2×D8—271no—
227#124—31no—
228C4×D4—271no—
229C4⋊C8—271no—
230#103—271no—
231#102—91no—
232C33#46311yescharacteristic, complemented, Sylow
233C23×C3—24no—
234C23×C3—21no—
235C23×C3—44no—
236C22×D3—184no—
237C22×D3—181no—
238C22×D3—364no—
239C3×Q8—64no—
240C3×Q8—64no—
241C3×D4—28no—
242C3×D4—24no—
243C3×D4—24no—
244C3×D4—21no—
245C3×D4—21no—
246C3×D4—44no—
247C2×C3×C4—24no—
248C2×C3×C4—34no—
249C2×C3×C4—34no—
250C2×C3×C4—64no—
251C2×C3×C4—181no—
252C2×C3×C4C2×D3214yes—
253C2×C3×C4C22×C3⋊D311yescharacteristic
254C2×Dic3—21no—
255C2×Dic3—64no—
256C2×Dic3#63811yescharacteristic
257C2×Dic3#63811yescharacteristic
258D12—98no—
259D12—92no—
260D12—188no—
261D12—184no—
262D12—181no—
263D12—364no—
264C4×D3—362no—
265C4×D3—368no—
266Dic6—21no—
267Dic6—21no—
268C3×C8—181no—
269C3⋊C8—64no—
270C3⋊C8—188no—
271C3⋊C8—184no—
272C3⋊C8—181no—
273C2×C32—24no—
274C2×C32—24no—
275C2×C32—24no—
276C2×C32—88no—
277C2×C32—81no—
278C2×C32—84no—
279C2×C32#88411yescharacteristic
280C2×C32#91111yescharacteristic
281C2×C32#91611yescharacteristic
282C2×C32#93814yes—
283C2×C32#94914yes—
284C2×C32#95014yes—
285C3⋊D3—122no—
286C3⋊D3—128no—
287C3⋊D3—248no—
288C2×Q8—31no—
289C2×D4—271no—
290C2×D4D3×C3⋊D311yescharacteristic
291C22×C4—541no—
292SD16—542no—
293D8—542no—
294C2×C8—91no—
295C2×C8—271no—
296C4⋊C4—91no—
297C22⋊C4—61no—
298C22⋊C4—61no—
299C22⋊C4—541no—
300C42—31no—
301C22×C3—24no—
302C22×C3—48no—
303C22×C3—41no—
304C22×C3—41no—
305C22×C3—41no—
306C22×C3—44no—
307C22×C3—44no—
308C22×C3—44no—
309C22×C3—84no—
310C22×C3—84no—
311C22×C3D3×#6714yes—
312C22×C3C2×#63811yescharacteristic
313C2×D3—182no—
314C2×D3—188no—
315C2×D3—368no—
316C2×D3—361no—
317C2×D3—361no—
318C2×D3—364no—
319C2×D3—364no—
320C2×D3—724no—
321C2×D3—724no—
322C3×C4—24no—
323C3×C4—24no—
324C3×C4—64no—
325C3×C4—64no—
326C3×C4—124no—
327C3×C4—361no—
328C3×C4#380414yes—
329C3×C4#381114yes—
330C3×C4D4×C3⋊D311yescharacteristic
331C3×C4#383011yescharacteristic
332Dic3—21no—
333Dic3—21no—
334Dic3—41no—
335Dic3—124no—
336C32—24no—
337C32#478711yescharacteristic, complemented
338C32#505714yescomplemented
339C23—21no—
340C23—541no—
341Q8—61no—
342Q8—61no—
343D4—21no—
344D4—21no—
345D4—272no—
346D4—541no—
347C2×C4—31no—
348C2×C4—31no—
349C2×C4—61no—
350C2×C4—181no—
351C2×C4—1082no—
352C2×C4C2×D3×C3⋊D311yescharacteristic
353C8—181no—
354C8—541no—
355C2×C3—24no—
356C2×C3—24no—
357C2×C3—24no—
358C2×C3—88no—
359C2×C3—84no—
360C2×C3—81no—
361C2×C3#6392514yes—
362C2×C3#6393514yes—
363C2×C3#6395814yes—
364C2×C3#6414711yescharacteristic
365C2×C3#6415811yescharacteristic
366C2×C3#6415911yescharacteristic
367D3—368no—
368D3—362no—
369D3—728no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—542no—
374C22—1081no—
375C22—1081no—
376C22D3×#63811yescharacteristic
377C4—61no—
378C4—61no—
379C4—121no—
380C4—361no—
381C4#8902811yescharacteristic
382C4#8903911yescharacteristic
383C3—24no—
384C3#1059143214yescomplemented
385C3#1059275611yescharacteristic, complemented
386C2—81no—
387C2—1082no—
388C2#1172354411yescharacteristic
389C2#1172355811yescharacteristic
390C2#1172359911yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radicalC2×#88860C2
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#463
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C32×#936C2semidirect
3. #11723484C2semidirect
4. C2×#88795C2semidirect
5. C3×#63659C2semidirect
6. Dic3⋊(C3⋊(C3⋊C8))C2semidirect
7. #11722037C2semidirect
29. C3×#936D3semidirect
33. #63659D3semidirect
101. #936C3⋊D3semidirect
232. C33#463semidirect
337. C32#4787semidirect
338. C32#5057semidirect
384. C3#10591432semidirect
385. C3#10592756semidirect

Maximal subgroups

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

subgroupindex
C2×#888602
C32×#9362
#117234842
C2×#887952
C3×#636592
Dic3⋊(C3⋊(C3⋊C8))2
#117220372
#105927563
#105914323