Finite Group Databasebeta

← #13161378

Subgroups — #13161378

Order 1728.

920 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 920 conjugacy classes of subgroups — too many to draw. Only its 152 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#11723990C211yesmaximal, characteristic, complemented
2C3×#64158C211yesmaximal, characteristic, complemented
3C32×#948C211yesmaximal, characteristic, complemented
4C2×#88791C211yesmaximal, characteristic, complemented
5#11722058C211yesmaximal, characteristic, complemented
6#11722027C211yesmaximal, characteristic, complemented
7#11722026C211yesmaximal, characteristic, complemented
8#10592841—31nomaximal
9#10591677—34nomaximal
10C33×#41C2211yescharacteristic
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13#88862—21no—
14#88860—21no—
15#88855C2211yescharacteristic
16C3×#3755—21no—
17C3×#3753—21no—
18C3×#3748C2211yescharacteristic
19C32×#208—21no—
20C32×#208C2211yescharacteristic
21C2×C32×C3⋊C8—21no—
22C2×C32×C3⋊C8C2211yescharacteristic
23#88791—21no—
24#88791—21no—
25C32×#131—31no—
26#64158—31no—
27#64158—34no—
28#64158—64no—
29#64158D311yescharacteristic, complemented
30C2×#3745—31no—
31C3×#949—34no—
32C3×#948D314yescomplemented
33C2×#3714—34no—
34#63650—31no—
35#63650—31no—
36#63646—31no—
37#63577—34no—
38#63559—34no—
39#63558—34no—
40C22×C3⋊(C3⋊D3)—21no—
41C22×C3×C3⋊D3—21no—
42C33×Q8D411yescharacteristic
43C33×D4—21no—
44C33×D4D411yescharacteristic
45C2×C33×C4—21no—
46C2×C33×C4C2311yescharacteristic
47C4⋊(C3⋊(C3⋊D3))—21no—
48C4⋊(C3⋊(C3⋊D3))—21no—
49C3×C4⋊(C3⋊D3)—21no—
50C3×C4⋊(C3⋊D3)—21no—
51C3⋊(C3⋊(C3⋊C8))—21no—
52C3×C3⋊(C3⋊C8)—21no—
53C32×C3⋊C8—21no—
54C32×C3⋊C8D411yescharacteristic
55C32×C3⋊C8D411yescharacteristic
56#5199—91no—
57#5142—94no—
58C32×#41—24no—
59C32×#41C2×D314yes—
60C32×#41C2×D311yescharacteristic
61C2×C4⋊(C3⋊D3)—31no—
62C2×C4⋊(C3⋊D3)—34no—
63C2×C4⋊(C3⋊D3)—64no—
64C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
65C2×C3×D12—34no—
66C32×M16—31no—
67C32×M16—61no—
68C2×C32×C8—31no—
69C2×C32×C8—61no—
70#3755—21no—
71#3755—68no—
72#3755—64no—
73#3755—61no—
74#3753—21no—
75#3753—68no—
76#3753—64no—
77#3753—61no—
78#3748—31no—
79#3748—34no—
80#3748—64no—
81#3748C2×D311yescharacteristic
82#3745—61no—
83#3745—61no—
84C3×#215—64no—
85C3×#213—64no—
86C3×#208—24no—
87C3×#208—34no—
88C3×#208C2×D314yes—
89C2×C3×C3⋊C8—24no—
90C2×C3×C3⋊C8C2×D314yes—
91#3714—64no—
92#3714—64no—
93C22×C33—21no—
94C22×C33#4111yescharacteristic
95C2×C3⋊(C3⋊D3)—41no—
96C2×C3⋊(C3⋊D3)—81no—
97C2×C3×C3⋊D3—41no—
98C2×C3×C3⋊D3—81no—
99C33×C4—21no—
100C33×C4C2×D411yescharacteristic
101C33×C4C2×D411yescharacteristic
102C3×AGL(1,ℤ/8)—91no—
103C3×#131—34no—
104#949—34no—
105#949—94no—
106#949—91no—
107#949—184no—
108#948C3⋊D311yescharacteristic, complemented
109C2×#213—91no—
110C2×D24—94no—
111#833—91no—
112#823—94no—
113#823—94no—
114#823—91no—
115#822—91no—
116#819—94no—
117C22×C3⋊D3—21no—
118C22×C3⋊D3—61no—
119C22×C3⋊D3—64no—
120C22×C3⋊D3—124no—
121C22×C3×D3—64no—
122C32×Q8—24no—
123C32×Q8D1214yes—
124C32×Q8#6711yescharacteristic
125C32×D4—21no—
126C32×D4—24no—
127C32×D4—24no—
128C32×D4—44no—
129C32×D4D1214yes—
130C32×D4#6711yescharacteristic
131C2×C32×C4—21no—
132C2×C32×C4—24no—
133C2×C32×C4—24no—
134C2×C32×C4—44no—
135C2×C32×C4C22×D314yes—
136C2×C32×C4C22×D311yescharacteristic
137C4⋊(C3⋊D3)—21no—
138C4⋊(C3⋊D3)—21no—
139C4⋊(C3⋊D3)—68no—
140C4⋊(C3⋊D3)—64no—
141C4⋊(C3⋊D3)—61no—
142C4⋊(C3⋊D3)—61no—
143C4⋊(C3⋊D3)—64no—
144C4⋊(C3⋊D3)—124no—
145C3×D12—64no—
146C3×D12—64no—
147C32×C8—31no—
148C32×C8—31no—
149C32×C8—61no—
150C3⋊(C3⋊C8)—21no—
151C3⋊(C3⋊C8)—68no—
152C3⋊(C3⋊C8)—64no—
153C3⋊(C3⋊C8)—61no—
154C3×C3⋊C8—24no—
155C3×C3⋊C8—64no—
156C3×C3⋊C8#6714yes—
157C3×C3⋊C8#6714yes—
158#472—271noSylow
159C2×C33—21no—
160C2×C33—41no—
161C2×C33#12111yescharacteristic
162C3⋊(C3⋊D3)—81no—
163C3×C3⋊D3—81no—
164C3×#41—24no—
165C3×#41D3214yes—
166C3×#41C2×C3⋊D311yescharacteristic
167C2×C3×D4—91no—
168C2×D12—34no—
169C2×D12—94no—
170C2×D12—91no—
171C2×D12—184no—
172C3×SD16—181no—
173C3×D8—181no—
174C3×M16—34no—
175C3×M16—64no—
176C3×M16—91no—
177C2×C3×C8—34no—
178C2×C3×C8—64no—
179#215—64no—
180#215—188no—
181#215—184no—
182#215—181no—
183#213—64no—
184#213—188no—
185#213—184no—
186#213—181no—
187#213—181no—
188#213—181no—
189#208—21no—
190#208—34no—
191#208—94no—
192#208—91no—
193#208—184no—
194#208C2×C3⋊D311yescharacteristic
195C2×C3⋊C8—21no—
196C2×C3⋊C8C2×C3⋊D311yescharacteristic
197D24—184no—
198D24—184no—
199C22×C32—21no—
200C22×C32—24no—
201C22×C32—24no—
202C22×C32—44no—
203C22×C32#23514yes—
204C22×C32#23711yescharacteristic
205C2×C3⋊D3—41no—
206C2×C3⋊D3—81no—
207C2×C3⋊D3—128no—
208C2×C3⋊D3—124no—
209C2×C3⋊D3—121no—
210C2×C3⋊D3—248no—
211C2×C3⋊D3—244no—
212C2×C3⋊D3—241no—
213C2×C3×D3—124no—
214C2×C3×D3—244no—
215C32×C4—21no—
216C32×C4—24no—
217C32×C4—24no—
218C32×C4—24no—
219C32×C4—44no—
220C32×C4C2×D1214yes—
221C32×C4D3×D411yescharacteristic
222C32×C4C2×#6714yes—
223C32×C4C2×#6711yescharacteristic
224AGL(1,ℤ/8)—91no—
225AGL(1,ℤ/8)—271no—
226C2×D8—271no—
227#131—31no—
228#108—271no—
229#100—271no—
230#100—271no—
231C33#47211yescharacteristic, complemented, Sylow
232C23×C3—181no—
233C22×D3—64no—
234C22×D3—184no—
235C22×D3—181no—
236C22×D3—364no—
237C3×Q8—24no—
238C3×Q8#62614yes—
239C3×Q8C4⋊(C3⋊D3)11yescharacteristic
240C3×D4—24no—
241C3×D4—21no—
242C3×D4—24no—
243C3×D4—44no—
244C3×D4—181no—
245C3×D4—181no—
246C3×D4#62614yes—
247C3×D4C4⋊(C3⋊D3)11yescharacteristic
248C2×C3×C4—24no—
249C2×C3×C4—21no—
250C2×C3×C4—24no—
251C2×C3×C4—44no—
252C2×C3×C4C2×D3214yes—
253C2×C3×C4C22×C3⋊D311yescharacteristic
254D12—64no—
255D12—64no—
256D12—188no—
257D12—184no—
258D12—184no—
259D12—181no—
260D12—181no—
261D12—364no—
262C3×C8—34no—
263C3×C8—34no—
264C3×C8—64no—
265C3×C8—181no—
266C3⋊C8—21no—
267C3⋊C8—64no—
268C3⋊C8—188no—
269C3⋊C8—184no—
270C3⋊C8—181no—
271C3⋊C8#63811yescharacteristic
272C3⋊C8#63811yescharacteristic
273C2×C32—21no—
274C2×C32—24no—
275C2×C32—24no—
276C2×C32—44no—
277C2×C32—41no—
278C2×C32—44no—
279C2×C32—84no—
280C2×C32#93014yes—
281C2×C32#93911yescharacteristic
282C3⋊D3—81no—
283C3⋊D3—248no—
284C3⋊D3—244no—
285C3⋊D3—241no—
286C3×D3—244no—
287#41D3×C3⋊D311yescharacteristic
288C2×D4—91no—
289C2×D4—271no—
290SD16—181no—
291SD16—541no—
292D8—181no—
293D8—541no—
294D8—541no—
295D8—541no—
296M16—31no—
297M16—61no—
298M16—91no—
299M16—271no—
300C2×C8—31no—
301C2×C8—61no—
302C22×C3—21no—
303C22×C3—24no—
304C22×C3—24no—
305C22×C3—44no—
306C22×C3—361no—
307C22×C3—721no—
308C22×C3#380414yes—
309C22×C3#382811yescharacteristic
310C2×D3—124no—
311C2×D3—244no—
312C2×D3—368no—
313C2×D3—361no—
314C2×D3—364no—
315C2×D3—728no—
316C2×D3—721no—
317C2×D3—724no—
318C3×C4—21no—
319C3×C4—24no—
320C3×C4—24no—
321C3×C4—24no—
322C3×C4—44no—
323C3×C4C2×#62614yes—
324C3×C4D3×#6714yes—
325C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
326C3×C4C2×#63811yescharacteristic
327C32—24no—
328C32#514214yescomplemented
329C32#519911yescharacteristic, complemented
330C23—181no—
331C23—541no—
332Q8#629211yescharacteristic
333D4—21no—
334D4—181no—
335D4—181no—
336D4—541no—
337D4—541no—
338D4#629211yescharacteristic
339C2×C4—21no—
340C2×C4C2×D3×C3⋊D311yescharacteristic
341C8—31no—
342C8—31no—
343C8—61no—
344C8—181no—
345C8—541no—
346C2×C3—21no—
347C2×C3—24no—
348C2×C3—24no—
349C2×C3—44no—
350C2×C3—41no—
351C2×C3—44no—
352C2×C3—84no—
353C2×C3—721no—
354C2×C3#6397814yes—
355C2×C3#6413911yescharacteristic
356D3—244no—
357D3—728no—
358D3—721no—
359D3—724no—
360C22—21no—
361C22—361no—
362C22—721no—
363C22—1081no—
364C22—2161no—
365C22#8903111yescharacteristic
366C4—21no—
367C4C2×#629211yescharacteristic
368C4D3×#63811yescharacteristic
369C3—24no—
370C3#1059167714yescomplemented
371C3#1059284111yescharacteristic, complemented
372C2—21no—
373C2—41no—
374C2—721no—
375C2—2161no—
376C2#1172363211yescharacteristic
subgroupG/H
centerC2#11723632
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×#41C22
socleC2×C33#121
hypercenter#41D3×C3⋊D3
hypocenterC33#472
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
1. #11723990C2semidirect
2. C3×#64158C2semidirect
3. C32×#948C2semidirect
4. C2×#88791C2semidirect
5. #11722058C2semidirect
6. #11722027C2semidirect
7. #11722026C2semidirect
29. #64158D3semidirect
32. C3×#948D3semidirect
108. #948C3⋊D3semidirect
231. C33#472semidirect
328. C32#5142semidirect
329. C32#5199semidirect
370. C3#10591677semidirect
371. C3#10592841semidirect

Maximal subgroups

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

subgroupindex
#117239902
C3×#641582
C32×#9482
C2×#887912
#117220582
#117220272
#117220262
#105928413
#105916773