Finite Group Databasebeta

← #13159938

Subgroups — #13159938

Order 1728.

980 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 980 conjugacy classes of subgroups — too many to draw. Only its 148 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×C3×#3743C211yesmaximal, characteristic, complemented
2#11723526C211yesmaximal, characteristic, complemented
3#11723522C211yesmaximal, characteristic, complemented
4C2×#88790C211yesmaximal, characteristic, complemented
5#11722191C211yesmaximal, characteristic, complemented
6C32×#821C211yesmaximal, characteristic, complemented
7#11722054C211yesmaximal, characteristic, complemented
8#10592481—31nomaximal
9#10590827—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×#6292—21no—
14C2×#6290—21no—
15C2×C33×C8C2211yescharacteristic
16C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C3×#3743—22no—
19C32×C4⋊Dic3C2211yescharacteristic
20#88804—21no—
21(C2×D3)⋊(C3⋊Dic3)—21no—
22#88790—22no—
23#64147—31no—
24C2×#3753—31no—
25C2×#3743D311yescharacteristic, complemented
26#64104—31no—
27C2×C3×ΓL(1,𝔽25)—34no—
28#63909—34no—
29#63908—34no—
30C2×#3714—34no—
31C32×#102—31no—
32#63648—31no—
33#63648—34no—
34#63648—64no—
35#63635—31no—
36C3×#821D314yescomplemented
37#63573—34no—
38C22×C3⋊(C3⋊D3)—21no—
39C22×C3×C3⋊D3D411yescharacteristic
40C22×C32×D3—21no—
41C2×C33×C4C2311yescharacteristic
42C2×C3⋊(C3⋊Dic3)—21no—
43C4⋊(C3⋊(C3⋊D3))—21no—
44C4⋊(C3⋊(C3⋊D3))—21no—
45C2×C3×C3⋊Dic3D411yescharacteristic
46C3×C4×C3⋊D3—21no—
47C3×C4×C3⋊D3—21no—
48C2×C32×Dic3—21no—
49C32×D12—21no—
50C32×D12—21no—
51#6292—42no—
52#6290—42no—
53C33×C8—21no—
54C3×C3⋊(C3⋊C8)D412yescomplemented
55#5135—91no—
56#4782—94no—
57C2×C32×D4—31no—
58C2×#638—61no—
59C2×C4⋊(C3⋊D3)—31no—
60C2×C4⋊(C3⋊D3)—34no—
61C2×C4⋊(C3⋊D3)—64no—
62C2×C4⋊(C3⋊D3)—61no—
63C2×C4×C3⋊D3C2×D311yescharacteristic
64C2×C3×D12C2×D314yes—
65C2×C3×C4×D3—34no—
66C2×#626—64no—
67C2×#625—64no—
68C2×C32×C8—24no—
69C2×C32×C8C2×D314yes—
70C2×C32×C8C2×D311yescharacteristic
71C32×C4⋊C4—31no—
72#3757—61no—
73#3753—62no—
74#3752—61no—
75C4⋊(C3⋊Dic3)—31no—
76C4⋊(C3⋊Dic3)—34no—
77C4⋊(C3⋊Dic3)—64no—
78C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
79#3743—22no—
80C3×C4⋊Dic3C2×D314yes—
81C2×C3×C3⋊C8—34no—
82C3×ΓL(1,𝔽25)—68no—
83#3722—64no—
84(C2×D3)⋊Dic3—64no—
85#3714—68no—
86C22×C33C2×D411yescharacteristic
87C2×C3⋊(C3⋊D3)—41no—
88C2×C3⋊(C3⋊D3)—41no—
89C2×C3⋊(C3⋊D3)—41no—
90C2×C3×C3⋊D3—21no—
91C2×C3×C3⋊D3—21no—
92C2×C3×C3⋊D3—21no—
93C2×C32×D3—41no—
94C2×C32×D3—41no—
95C2×C32×D3—41no—
96C33×C4C2×D411yescharacteristic
97C33×C4#4111yescharacteristic
98C3⋊(C3⋊Dic3)—41no—
99C3×C3⋊Dic3—21no—
100C32×Dic3—41no—
101C2×C3×M16—91no—
102#938—94no—
103C2×#213—94no—
104#930—91no—
105#930—91no—
106C2×D24—91no—
107C2×ΓL(1,𝔽25)—34no—
108#895—94no—
109C3×#102—34no—
110#821—94no—
111#821—91no—
112#821—184no—
113#821C3⋊D311yescharacteristic, complemented
114#817—91no—
115#808—94no—
116C23×C32—61no—
117C22×C3⋊D3—61no—
118C22×C3⋊D3—64no—
119C22×C3⋊D3—124no—
120C22×C3⋊D3D1211yescharacteristic
121C22×C3×D3—24no—
122C22×C3×D3—34no—
123C32×D4—61no—
124C32×D4—61no—
125C2×C32×C4—24no—
126C2×C32×C4—61no—
127C2×C32×C4C22×D314yes—
128C2×C32×C4C22×D311yescharacteristic
129#638—122no—
130C2×C3⋊Dic3—61no—
131C2×C3⋊Dic3—64no—
132C2×C3⋊Dic3—124no—
133C2×C3⋊Dic3D1211yescharacteristic
134C4⋊(C3⋊D3)—68no—
135C4⋊(C3⋊D3)—64no—
136C4⋊(C3⋊D3)—61no—
137C4⋊(C3⋊D3)—61no—
138C4⋊(C3⋊D3)—64no—
139C4⋊(C3⋊D3)—124no—
140C4⋊(C3⋊D3)—122no—
141C4×C3⋊D3—21no—
142C4×C3⋊D3—21no—
143C2×C3×Dic3—24no—
144C2×C3×Dic3—34no—
145C3×D12—24no—
146C3×D12—24no—
147C3×C4×D3—64no—
148C3×C4×D3—64no—
149#626—128no—
150#625—128no—
151C32×C8—21no—
152C32×C8—24no—
153C32×C8—44no—
154C3⋊(C3⋊C8)#6712yescomplemented
155C3×C3⋊C8—38no—
156#469—271noSylow
157C2×C33#12111yescharacteristic
158C2×C33AGL(1,ℤ/8)11yescharacteristic
159C2×C33AGL(1,ℤ/8)11yescharacteristic
160C3⋊(C3⋊D3)—81no—
161C3×C3⋊D3—41no—
162C32×D3—81no—
163C2×C3×D4—34no—
164C22×C3×C4—91no—
165C2×#67—181no—
166C2×#67—181no—
167C2×#67—184no—
168C2×D12—94no—
169C2×D12—91no—
170C2×D12—184no—
171C2×D12—184no—
172C2×D12C2×C3⋊D311yescharacteristic
173C2×C4×D3—34no—
174C3×M16—182no—
175C2×C3×C8—24no—
176C2×C3×C8—91no—
177C2×C3×C8D3214yes—
178C2×C3×C8C2×C3⋊D311yescharacteristic
179C3×C4⋊C4—34no—
180#217—184no—
181#213—188no—
182(C2×D3)⋊C4—181no—
183(C2×D3)⋊C4—184no—
184(C2×D3)⋊C4—181no—
185C4⋊Dic3—94no—
186C4⋊Dic3—91no—
187C4⋊Dic3—184no—
188C4⋊Dic3C2×C3⋊D311yescharacteristic
189C2×C3⋊C8—34no—
190D24—182no—
191ΓL(1,𝔽25)—68no—
192C22×C32—24no—
193C22×C32—121no—
194C22×C32—121no—
195C22×C32—121no—
196C22×C32C2×D1211yescharacteristic
197C22×C32D3×D414yes—
198C2×C3⋊D3—21no—
199C2×C3⋊D3—21no—
200C2×C3⋊D3—21no—
201C2×C3⋊D3—128no—
202C2×C3⋊D3—124no—
203C2×C3⋊D3—121no—
204C2×C3⋊D3—121no—
205C2×C3⋊D3—121no—
206C2×C3⋊D3—124no—
207C2×C3⋊D3—124no—
208C2×C3⋊D3—244no—
209C2×C3⋊D3—244no—
210C2×C3×D3—44no—
211C2×C3×D3—44no—
212C2×C3×D3—44no—
213C2×C3×D3—64no—
214C2×C3×D3—64no—
215C2×C3×D3—64no—
216C32×C4—24no—
217C32×C4—24no—
218C32×C4—121no—
219C32×C4#23514yes—
220C32×C4#23511yescharacteristic
221C32×C4D3×D414yes—
222C32×C4C2×#6711yescharacteristic
223C3⋊Dic3—21no—
224C3⋊Dic3—121no—
225C3⋊Dic3—124no—
226C3⋊Dic3—244no—
227C3×Dic3—44no—
228C3×Dic3—64no—
229C2×D8—271no—
230C2×M16—91no—
231#121—271no—
232#121—271no—
233#106—271no—
234#102—31no—
235#102—271no—
236C33#46911yescharacteristic, complemented, Sylow
237C23×C3—64no—
238C23×C3—91no—
239C22×D3—21no—
240C22×D3—34no—
241C22×D3—184no—
242C22×D3—181no—
243C22×D3—364no—
244C3×D4—64no—
245C3×D4—64no—
246C2×C3×C4—24no—
247C2×C3×C4—64no—
248C2×C3×C4—91no—
249C2×C3×C4—181no—
250C2×C3×C4—181no—
251C2×C3×C4C2×D3214yes—
252C2×C3×C4C22×C3⋊D311yescharacteristic
253#67—362no—
254#67—362no—
255#67—368no—
256C2×Dic3—21no—
257C2×Dic3—34no—
258C2×Dic3—184no—
259C2×Dic3—181no—
260C2×Dic3—364no—
261D12—21no—
262D12—21no—
263D12—188no—
264D12—184no—
265D12—184no—
266D12—181no—
267D12—181no—
268D12—364no—
269D12—368no—
270C4×D3—64no—
271C4×D3—64no—
272C3×C8—24no—
273C3×C8—21no—
274C3×C8—44no—
275C3×C8—92no—
276C3⋊C8—38no—
277C2×C32—24no—
278C2×C32—24no—
279C2×C32—24no—
280C2×C32—241no—
281C2×C32#88414yes—
282C2×C32#90811yescharacteristic
283C2×C32#90811yescharacteristic
284C2×C32#91114yes—
285C2×C32#91414yes—
286C2×C32#93011yescharacteristic
287C3⋊D3—41no—
288C3⋊D3—248no—
289C3⋊D3—241no—
290C3⋊D3—244no—
291C3×D3—84no—
292C3×D3—124no—
293C2×D4—31no—
294C2×D4—271no—
295C2×D4—541no—
296C2×D4—541no—
297C22×C4—91no—
298D8—542no—
299M16—182no—
300C2×C8—91no—
301C2×C8D3×C3⋊D311yescharacteristic
302C4⋊C4—31no—
303C4⋊C4—271no—
304C22⋊C4—541no—
305C22⋊C4—541no—
306C22×C3—24no—
307C22×C3—124no—
308C22×C3—124no—
309C22×C3—124no—
310C22×C3—181no—
311C22×C3—181no—
312C22×C3—181no—
313C22×C3D3×D1214yes—
314C22×C3D4×C3⋊D311yescharacteristic
315C2×D3—41no—
316C2×D3—41no—
317C2×D3—41no—
318C2×D3—64no—
319C2×D3—64no—
320C2×D3—64no—
321C2×D3—368no—
322C2×D3—364no—
323C2×D3—364no—
324C2×D3—361no—
325C2×D3—361no—
326C2×D3—361no—
327C2×D3—364no—
328C2×D3—724no—
329C2×D3—724no—
330C3×C4—24no—
331C3×C4—24no—
332C3×C4—124no—
333C3×C4—181no—
334C3×C4#379814yes—
335C3×C4D3×#6714yes—
336C3×C4#382811yescharacteristic
337C3×C4D4×C3⋊D311yescharacteristic
338Dic3—41no—
339Dic3—64no—
340Dic3—364no—
341Dic3—361no—
342Dic3—724no—
343C32—24no—
344C32#478214yescomplemented
345C32#513511yescharacteristic, complemented
346C23—61no—
347C23—91no—
348C23—541no—
349D4—61no—
350D4—61no—
351D4—541no—
352D4—541no—
353D4—1082no—
354D4—1082no—
355C2×C4—61no—
356C2×C4—91no—
357C2×C4—181no—
358C2×C4—181no—
359C2×C4—541no—
360C2×C4C2×D3×C3⋊D311yescharacteristic
361C8—21no—
362C8—92no—
363C2×C3—24no—
364C2×C3—24no—
365C2×C3—24no—
366C2×C3—244no—
367C2×C3—361no—
368C2×C3#6379414yes—
369C2×C3#6379514yes—
370C2×C3#6390614yes—
371C2×C3#6409311yescharacteristic
372C2×C3#6412011yescharacteristic
373C2×C3#6412311yescharacteristic
374D3—81no—
375D3—124no—
376D3—728no—
377D3—724no—
378D3—721no—
379C22—121no—
380C22—121no—
381C22—121no—
382C22—181no—
383C22—181no—
384C22—181no—
385C22—1081no—
386C22—1081no—
387C22—1081no—
388C22C3⋊D3×D1211yescharacteristic
389C4—121no—
390C4—181no—
391C4—1081no—
392C4#8902011yescharacteristic
393C4C3⋊D3×#6711yescharacteristic
394C3—24no—
395C3#1059082714yescomplemented
396C3#1059248111yescharacteristic, complemented
397C2—241no—
398C2—361no—
399C2—2161no—
400C2#1172331911yescharacteristic
401C2#1172332011yescharacteristic
402C2#1172352011yescharacteristic
subgroupG/H
centerC22C3⋊D3×D12
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#469
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3743C2semidirect
2. #11723526C2semidirect
3. #11723522C2semidirect
4. C2×#88790C2semidirect
5. #11722191C2semidirect
6. C32×#821C2semidirect
7. #11722054C2semidirect
25. C2×#3743D3semidirect
36. C3×#821D3semidirect
54. C3×C3⋊(C3⋊C8)D4semidirect
113. #821C3⋊D3semidirect
154. C3⋊(C3⋊C8)#67semidirect
236. C33#469semidirect
344. C32#4782semidirect
345. C32#5135semidirect
395. C3#10590827semidirect
396. C3#10592481semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37432
#117235262
#117235222
C2×#887902
#117221912
C32×#8212
#117220542
#105924813
#105908273