Finite Group Databasebeta

← #13159940

Subgroups — #13159940

Order 1728.

992 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 992 conjugacy classes of subgroups — too many to draw. Only its 156 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×C32×D24C211yesmaximal, characteristic, complemented
2#11723529C211yesmaximal, characteristic, complemented
3#11723499C211yesmaximal, characteristic, complemented
4C2×#88789C211yesmaximal, characteristic, complemented
5#11722184C211yesmaximal, characteristic, complemented
6C3×#63648C211yesmaximal, characteristic, complemented
7#11722031C211yesmaximal, characteristic, complemented
8#10592849—31nomaximal
9#10590749—34nomaximal
10C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
11C2×C32×D12C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×#6290—21no—
14C2×#6290—21no—
15C2×D3×C3⋊Dic3—21no—
16C2×C33×C8C2211yescharacteristic
17C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
18C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
19C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
20C32×D24—22no—
21(C2×D3)⋊(C3⋊Dic3)—21no—
22#88789—22no—
23C2×C32×D8—31no—
24#64149—31no—
25D4×C3⋊Dic3—31no—
26C2×#3753—31no—
27C2×C3×D24D314yescomplemented
28#63914—34no—
29#63898—34no—
30C2×#3713—34no—
31#63659—31no—
32#63648D311yescharacteristic, complemented
33#63641—31no—
34#63641—34no—
35#63641—64no—
36C3×#821—34no—
37#63562—34no—
38C22×C3×C3⋊D3—21no—
39C22×C32×D3—21no—
40C22×C32×D3—21no—
41C2×C33×C4C2311yescharacteristic
42C2×C3⋊(C3⋊Dic3)D411yescharacteristic
43C2×C3⋊(C3⋊Dic3)D411yescharacteristic
44C2×C3×C3⋊Dic3—21no—
45C3×C4⋊(C3⋊D3)—21no—
46C3×C4⋊(C3⋊D3)—21no—
47C32×D12—21no—
48C32×D12—21no—
49C32×D12—21no—
50C32×D12D412yescomplemented
51#6290—42no—
52#6290—42no—
53D3×C3⋊Dic3—42no—
54C33×C8—21no—
55C3⋊(C3⋊(C3⋊C8))—21no—
56#5150—94no—
57#4786—91no—
58C2×C32×D4—31no—
59C2×C32×D4—31no—
60C2×#638—61no—
61C2×#638—61no—
62C22×C3⋊Dic3—61no—
63C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
64C2×C3×D12—34no—
65C2×C3×D12C2×D314yes—
66C2×C3×D12C2×D314yes—
67C2×#625—64no—
68C2×#625—64no—
69C2×D3×Dic3—64no—
70C32×D8—62no—
71C2×C32×C8—24no—
72C2×C32×C8C2×D314yes—
73C2×C32×C8C2×D311yescharacteristic
74#3757—61no—
75#3753—62no—
76C4⋊(C3⋊Dic3)C2×D311yescharacteristic
77C4×C3⋊Dic3—31no—
78C4×C3⋊Dic3—34no—
79C4×C3⋊Dic3—64no—
80C2×C3⋊(C3⋊C8)—31no—
81C2×C3⋊(C3⋊C8)—34no—
82C2×C3⋊(C3⋊C8)—64no—
83C3×C4⋊Dic3—34no—
84C3×D24—28no—
85(C2×D3)⋊Dic3—64no—
86#3713—68no—
87C22×C33C2×D411yescharacteristic
88C2×C3×C3⋊D3—41no—
89C2×C3×C3⋊D3—41no—
90C2×C3×C3⋊D3—41no—
91C2×C32×D3—22no—
92C2×C32×D3—41no—
93C2×C32×D3—41no—
94C2×C32×D3—41no—
95C2×C32×D3—41no—
96C2×C32×D3—41no—
97C33×C4C2×D411yescharacteristic
98C33×C4#4111yescharacteristic
99C3⋊(C3⋊Dic3)—21no—
100C3⋊(C3⋊Dic3)D812yescomplemented
101C3×C3⋊Dic3—41no—
102C2×C3×D8—34no—
103#940—94no—
104#940—91no—
105D4×Dic3—94no—
106C2×#213—94no—
107C2×#213—91no—
108C2×D24C3⋊D311yescharacteristic, complemented
109#893—91no—
110C3×#102—91no—
111#832—94no—
112#821—34no—
113Dic3⋊C8—94no—
114Dic3⋊C8—91no—
115Dic3⋊C8—184no—
116#809—91no—
117C23×C32—61no—
118C23×C32—61no—
119C22×C3⋊D3—21no—
120C22×C3×D3—24no—
121C22×C3×D3—24no—
122C22×C3×D3—64no—
123C32×D4—32no—
124C32×D4—61no—
125C32×D4—61no—
126C32×D4—61no—
127C2×C32×C4—24no—
128C2×C32×C4C22×D314yes—
129C2×C32×C4C22×D311yescharacteristic
130#638—122no—
131#638—122no—
132C2×C3⋊Dic3—21no—
133C2×C3⋊Dic3—31no—
134C2×C3⋊Dic3—31no—
135C2×C3⋊Dic3—34no—
136C2×C3⋊Dic3—34no—
137C2×C3⋊Dic3—64no—
138C2×C3⋊Dic3—64no—
139C2×C3⋊Dic3—122no—
140C4⋊(C3⋊D3)—21no—
141C4⋊(C3⋊D3)—21no—
142C2×C3×Dic3—64no—
143C3×D12—24no—
144C3×D12—24no—
145C3×D12—24no—
146C3×D12—64no—
147C3×D12—64no—
148C3×D12#6718yescomplemented
149#625—128no—
150#625—128no—
151D3×Dic3—128no—
152C32×C8—21no—
153C32×C8—24no—
154C32×C8—44no—
155C3⋊(C3⋊C8)—61no—
156C3⋊(C3⋊C8)—64no—
157C3⋊(C3⋊C8)—124no—
158#459—271noSylow
159C2×C33#12111yescharacteristic
160C2×C33C2×D811yescharacteristic
161C2×C33AGL(1,ℤ/8)11yescharacteristic
162C3×C3⋊D3—81no—
163C32×D3—42no—
164C32×D3—81no—
165C2×C3×D4—34no—
166C2×C3×D4—34no—
167C2×C3×D4—91no—
168C2×#67—181no—
169C2×#67—181no—
170C2×#67—184no—
171C2×#67—184no—
172C22×Dic3—184no—
173C2×D12—34no—
174C2×D12C2×C3⋊D311yescharacteristic
175C2×D12C2×C3⋊D311yescharacteristic
176C2×C4×D3—181no—
177C3×D8—68no—
178C2×C3×C8—24no—
179C2×C3×C8D3214yes—
180C2×C3×C8C2×C3⋊D311yescharacteristic
181C3×C4⋊C4—91no—
182#217—184no—
183#213—188no—
184#213—182no—
185(C2×D3)⋊C4—181no—
186C4⋊Dic3—34no—
187C4×Dic3—94no—
188C4×Dic3—91no—
189C4×Dic3—184no—
190C2×C3⋊C8—94no—
191C2×C3⋊C8—91no—
192C2×C3⋊C8—184no—
193D24—22no—
194C22×C32—24no—
195C22×C32—62no—
196C22×C32—121no—
197C22×C32—121no—
198C22×C32—121no—
199C22×C32—121no—
200C22×C32—121no—
201C22×C32D3×D414yes—
202C22×C32D3×D411yescharacteristic
203C2×C3⋊D3—41no—
204C2×C3⋊D3—41no—
205C2×C3⋊D3—41no—
206C2×C3×D3—28no—
207C2×C3×D3—44no—
208C2×C3×D3—44no—
209C2×C3×D3—44no—
210C2×C3×D3—44no—
211C2×C3×D3—44no—
212C2×C3×D3—124no—
213C2×C3×D3—124no—
214C2×C3×D3—124no—
215C32×C4—24no—
216C32×C4—24no—
217C32×C4#23511yescharacteristic
218C32×C4D3×D411yescharacteristic
219C32×C4#23714yes—
220C32×C4C2×#6714yes—
221C3⋊Dic3—32no—
222C3⋊Dic3—38no—
223C3⋊Dic3—41no—
224C3⋊Dic3—68no—
225C3⋊Dic3—61no—
226C3⋊Dic3—64no—
227C3⋊Dic3—124no—
228C3×Dic3—124no—
229C2×D8—31no—
230C2×D8—271no—
231#127—271no—
232C4×D4—271no—
233C4⋊C8—271no—
234#102—91no—
235#102—271no—
236C33#45911yescharacteristic, complemented, Sylow
237C23×C3—64no—
238C23×C3—64no—
239C23×C3—181no—
240C22×D3—21no—
241C22×D3—21no—
242C22×D3—64no—
243C3×D4—38no—
244C3×D4—64no—
245C3×D4—64no—
246C3×D4—64no—
247C3×D4—181no—
248C3×D4—181no—
249C2×C3×C4—24no—
250C2×C3×C4—181no—
251C2×C3×C4C2×D3214yes—
252C2×C3×C4C22×C3⋊D311yescharacteristic
253#67—362no—
254#67—368no—
255#67—362no—
256#67—368no—
257C2×Dic3—64no—
258C2×Dic3—94no—
259C2×Dic3—91no—
260C2×Dic3—94no—
261C2×Dic3—91no—
262C2×Dic3—184no—
263C2×Dic3—184no—
264C2×Dic3—368no—
265D12—21no—
266D12—21no—
267D12—21no—
268D12—64no—
269D12—64no—
270D12#63812yescomplemented
271C4×D3—362no—
272C3×C8—24no—
273C3×C8—21no—
274C3×C8—44no—
275C3⋊C8—184no—
276C3⋊C8—181no—
277C3⋊C8—364no—
278C2×C32—24no—
279C2×C32—24no—
280C2×C32—24no—
281C2×C32—122no—
282C2×C32—241no—
283C2×C32#88411yescharacteristic
284C2×C32D3×D814yes—
285C2×C32D3×D811yescharacteristic
286C2×C32#91114yes—
287C2×C32#91411yescharacteristic
288C2×C32#93914yes—
289C3⋊D3—81no—
290C3×D3—48no—
291C3×D3—84no—
292C3×D3—244no—
293C2×D4—31no—
294C2×D4—31no—
295C2×D4—91no—
296C2×D4—541no—
297C2×D4—541no—
298C22×C4—541no—
299D8—62no—
300D8—542no—
301C2×C8—271no—
302C2×C8D3×C3⋊D311yescharacteristic
303C4⋊C4—91no—
304C22⋊C4—541no—
305C42—271no—
306C22×C3—24no—
307C22×C3—68no—
308C22×C3—124no—
309C22×C3—124no—
310C22×C3—124no—
311C22×C3—124no—
312C22×C3—124no—
313C22×C3—361no—
314C22×C3—361no—
315C22×C3—361no—
316C22×C3#380214yes—
317C22×C3D4×C3⋊D311yescharacteristic
318C2×D3—22no—
319C2×D3—41no—
320C2×D3—41no—
321C2×D3—41no—
322C2×D3—41no—
323C2×D3—41no—
324C2×D3—124no—
325C2×D3—124no—
326C2×D3—124no—
327C3×C4—24no—
328C3×C4—24no—
329C3×C4—361no—
330C3×C4#379514yes—
331C3×C4D3×#6714yes—
332C3×C4#383011yescharacteristic
333C3×C4C2×#63811yescharacteristic
334Dic3—98no—
335Dic3—92no—
336Dic3—124no—
337Dic3—188no—
338Dic3—184no—
339Dic3—181no—
340Dic3—364no—
341C32—24no—
342C32#478611yescharacteristic, complemented
343C32#515014yescomplemented
344C23—61no—
345C23—61no—
346C23—181no—
347D4—32no—
348D4—61no—
349D4—61no—
350D4—61no—
351D4—181no—
352D4—181no—
353D4—1082no—
354D4—1082no—
355C2×C4—181no—
356C2×C4—271no—
357C2×C4—271no—
358C2×C4—1082no—
359C2×C4C2×D3×C3⋊D311yescharacteristic
360C8—21no—
361C8—541no—
362C2×C3—24no—
363C2×C3—24no—
364C2×C3—24no—
365C2×C3—128no—
366C2×C3—244no—
367C2×C3—721no—
368C2×C3#6379714yes—
369C2×C3#6379814yes—
370C2×C3#6388514yes—
371C2×C3D8×C3⋊D311yescharacteristic
372C2×C3#6412011yescharacteristic
373C2×C3#6414811yescharacteristic
374D3—42no—
375D3—81no—
376D3—244no—
377C22—62no—
378C22—121no—
379C22—121no—
380C22—121no—
381C22—121no—
382C22—121no—
383C22—361no—
384C22—361no—
385C22—361no—
386C22#8902611yescharacteristic
387C4—272no—
388C4—361no—
389C4—541no—
390C4#8901411yescharacteristic
391C4D3×#63811yescharacteristic
392C3—24no—
393C3#1059074914yescomplemented
394C3#1059284911yescharacteristic, complemented
395C2—122no—
396C2—241no—
397C2—721no—
398C2#1172332211yescharacteristic
399C2#1172332411yescharacteristic
400C2#1172347411yescharacteristic
subgroupG/H
centerC22#89026
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#459
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×C32×D24C2semidirect
2. #11723529C2semidirect
3. #11723499C2semidirect
4. C2×#88789C2semidirect
5. #11722184C2semidirect
6. C3×#63648C2semidirect
7. #11722031C2semidirect
27. C2×C3×D24D3semidirect
32. #63648D3semidirect
50. C32×D12D4semidirect
100. C3⋊(C3⋊Dic3)D8semidirect
108. C2×D24C3⋊D3semidirect
148. C3×D12#67semidirect
236. C33#459semidirect
270. D12#638semidirect
342. C32#4786semidirect
343. C32#5150semidirect
393. C3#10590749semidirect
394. C3#10592849semidirect

Maximal subgroups

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

subgroupindex
C2×C32×D242
#117235292
#117234992
C2×#887892
#117221842
C3×#636482
#117220312
#105928493
#105907493