Finite Group Databasebeta

← #13161135

Subgroups — #13161135

Order 1728.

920 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 920 conjugacy classes of subgroups — too many to draw. Only its 144 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#11723987C211yesmaximal, characteristic
2#11723503C211yesmaximal, characteristic
3C3×#63662C211yesmaximal, characteristic, complemented
4C32×#835C211yesmaximal, characteristic, complemented
5#11722057C211yesmaximal, characteristic, complemented
6#11722055C211yesmaximal, characteristic, complemented
7#11722024C211yesmaximal, characteristic
8#10592510—31nomaximal
9#10591555—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
12C2×#6289—21no—
13#88859—21no—
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15#88857—21no—
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20#88806—21no—
21#64155—31no—
22#64155—34no—
23#64155—64no—
24C4⋊C4×C3⋊D3—31no—
25#63900—34no—
26C32×#103—31no—
27#63662D311yescharacteristic, complemented
28#63647—31no—
29#63645—31no—
30#63634—31no—
31C3×#835—34no—
32C3×#835D314yescomplemented
33#63575—34no—
34#63575—34no—
35#63557—34no—
36C22×C3⋊(C3⋊D3)D411yescharacteristic
37C33×Q8—21no—
38C33×Q8—21no—
39C2×C33×C4—21no—
40C2×C33×C4C2311yescharacteristic
41C2×C3⋊(C3⋊Dic3)—21no—
42C2×C3⋊(C3⋊Dic3)D411yescharacteristic
43C4×C3⋊(C3⋊D3)—21no—
44C4×C3⋊(C3⋊D3)—21no—
45C2×C3×C3⋊Dic3—21no—
46C2×C32×Dic3—21no—
47#6289—42no—
48C3×C3⋊(C3⋊C8)—21no—
49C32×C3⋊C8—21no—
50#4811—94no—
51#4811—91no—
52C2×C32×Q8—24no—
53C2×C32×Q8C2×D314yes—
54C2×C32×Q8C2×D311yescharacteristic
55C2×C4×C3⋊D3—31no—
56C2×C4×C3⋊D3—34no—
57C2×C4×C3⋊D3—61no—
58C2×C4×C3⋊D3—64no—
59C2×#624—64no—
60C2×C32×C8—31no—
61C32×C4⋊C4—31no—
62#3752—61no—
63#3752—64no—
64#3752—68no—
65C4⋊(C3⋊Dic3)—31no—
66C4⋊(C3⋊Dic3)—34no—
67C4⋊(C3⋊Dic3)—64no—
68C4⋊(C3⋊Dic3)C2×D311yescharacteristic
69#3750—61no—
70#3750—61no—
71#3750—64no—
72#3750—68no—
73C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
74C3×C4⋊Dic3—34no—
75C3×C4⋊Dic3C2×D314yes—
76C2×C3×C3⋊C8—34no—
77C2×C3×C3⋊C8C2×D314yes—
78#3724—64no—
79C22×C33C2×D411yescharacteristic
80C2×C3⋊(C3⋊D3)—21no—
81C2×C3⋊(C3⋊D3)—21no—
82C2×C3⋊(C3⋊D3)Q1612yes—
83C33×C4—41no—
84C33×C4#4111yescharacteristic
85C33×C4#4111yescharacteristic
86C3⋊(C3⋊Dic3)—21no—
87C3⋊(C3⋊Dic3)—41no—
88C3×C3⋊Dic3—41no—
89C32×Dic3—41no—
90#946—94no—
91#946—91no—
92#946—184no—
93D3×C4⋊C4—94no—
94D3×C4⋊C4—91no—
95C3×#103—34no—
96C3×#103—91no—
97#835—34no—
98#835C3⋊D311yescharacteristic, complemented
99(C2×D3)⋊C8—94no—
100(C2×D3)⋊C8—91no—
101#818—94no—
102#818—91no—
103#807—94no—
104#807—91no—
105C22×C3⋊D3—31no—
106C22×C3⋊D3—34no—
107C22×C3⋊D3—64no—
108C32×Q8—21no—
109C32×Q8—21no—
110C32×Q8—24no—
111C32×Q8—24no—
112C32×Q8—44no—
113C32×Q8—44no—
114C2×C32×C4—28no—
115C2×C32×C4—24no—
116C2×C32×C4—21no—
117C2×C32×C4—24no—
118C2×C32×C4—61no—
119C2×C32×C4C22×D314yes—
120C2×C32×C4C22×D311yescharacteristic
121C2×C3⋊Dic3—21no—
122C2×C3⋊Dic3—31no—
123C2×C3⋊Dic3—34no—
124C2×C3⋊Dic3—64no—
125C2×C3⋊Dic3—68no—
126C2×C3⋊Dic3—64no—
127C2×C3⋊Dic3—61no—
128C4×C3⋊D3—61no—
129C4×C3⋊D3—61no—
130C4×C3⋊D3—64no—
131C4×C3⋊D3—64no—
132C4×C3⋊D3—124no—
133C4×C3⋊D3—124no—
134C4×C3⋊D3—122no—
135C2×C3×Dic3—24no—
136C2×C3×Dic3—64no—
137#624—128no—
138C32×C8—61no—
139C3⋊(C3⋊C8)—21no—
140C3×C3⋊C8—24no—
141C3×C3⋊C8—64no—
142#484—271noSylow
143C2×C33#12311yescharacteristic
144C2×C33C2×Q1611yescharacteristic
145C2×C33AGL(1,ℤ/8)11yescharacteristic
146C3⋊(C3⋊D3)—22no—
147C2×C3×Q8—24no—
148C2×C3×Q8D3214yes—
149C2×C3×Q8C2×C3⋊D311yescharacteristic
150C2×C4×D3—94no—
151C2×C4×D3—91no—
152C2×C4×D3—184no—
153C2×C4×D3—181no—
154C2×C4×D3—184no—
155C2×C3×C8—34no—
156C2×C3×C8—91no—
157C3×C4⋊C4—34no—
158C3×C4⋊C4—91no—
159(C2×D3)⋊C4—184no—
160(C2×D3)⋊C4—181no—
161(C2×D3)⋊C4—188no—
162C4⋊Dic3—34no—
163C4⋊Dic3—94no—
164C4⋊Dic3—91no—
165C4⋊Dic3—184no—
166C4⋊Dic3C2×C3⋊D311yescharacteristic
167Dic3⋊C4—181no—
168Dic3⋊C4—184no—
169Dic3⋊C4—184no—
170Dic3⋊C4—181no—
171Dic3⋊C4—188no—
172C2×C3⋊C8—34no—
173C2×C3⋊C8C2×C3⋊D311yescharacteristic
174C22×C32—24no—
175C22×C32D3×D414yes—
176C22×C32D3×D411yescharacteristic
177C2×C3⋊D3—32no—
178C2×C3⋊D3—38no—
179C2×C3⋊D3—68no—
180C2×C3⋊D3—61no—
181C2×C3⋊D3—61no—
182C2×C3⋊D3—64no—
183C2×C3⋊D3—64no—
184C2×C3⋊D3—124no—
185C2×C3⋊D3—124no—
186C32×C4—24no—
187C32×C4—24no—
188C32×C4—48no—
189C32×C4—44no—
190C32×C4—41no—
191C32×C4—121no—
192C32×C4#23514yes—
193C32×C4#23511yescharacteristic
194C32×C4#23714yes—
195C32×C4#23711yescharacteristic
196C3⋊Dic3—41no—
197C3⋊Dic3—61no—
198C3⋊Dic3—64no—
199C3⋊Dic3—124no—
200C3⋊Dic3—128no—
201C3⋊Dic3—124no—
202C3⋊Dic3—121no—
203C3×Dic3—44no—
204C3×Dic3—124no—
205C22⋊Q8—271no—
206C2×C4⋊C4—271no—
207#107—271no—
208#107—271no—
209#103—31no—
210#103—91no—
211C22⋊C8—271no—
212C33#48411yescharacteristic, complemented, Sylow
213C22×D3—94no—
214C22×D3—91no—
215C22×D3—184no—
216C3×Q8—24no—
217C3×Q8—24no—
218C3×Q8—21no—
219C3×Q8—21no—
220C3×Q8—44no—
221C3×Q8—44no—
222C2×C3×C4—28no—
223C2×C3×C4—24no—
224C2×C3×C4—21no—
225C2×C3×C4—24no—
226C2×C3×C4—64no—
227C2×C3×C4—181no—
228C2×C3×C4C2×D3214yes—
229C2×C3×C4C22×C3⋊D311yescharacteristic
230C2×Dic3—21no—
231C2×Dic3—64no—
232C2×Dic3—94no—
233C2×Dic3—91no—
234C2×Dic3—184no—
235C2×Dic3—188no—
236C2×Dic3—184no—
237C2×Dic3—181no—
238C4×D3—184no—
239C4×D3—184no—
240C4×D3—181no—
241C4×D3—181no—
242C4×D3—364no—
243C4×D3—364no—
244C4×D3—362no—
245C4×D3—368no—
246C3×C8—64no—
247C3×C8—181no—
248C3⋊C8—21no—
249C3⋊C8—64no—
250C2×C32—24no—
251C2×C32—24no—
252C2×C32—24no—
253C2×C32#88314yes—
254C2×C32#88311yescharacteristic
255C2×C32#91414yes—
256C2×C32#91411yescharacteristic
257C2×C32D3×Q1614yes—
258C2×C32D3×Q1611yescharacteristic
259C3⋊D3—62no—
260C3⋊D3—68no—
261C3⋊D3—128no—
262C2×Q8D3×C3⋊D311yescharacteristic
263C22×C4—271no—
264C22×C4—541no—
265C2×C8—31no—
266C2×C8—91no—
267C4⋊C4—31no—
268C4⋊C4—91no—
269C4⋊C4—271no—
270C4⋊C4—541no—
271C4⋊C4—541no—
272C22⋊C4—541no—
273C22×C3—24no—
274C22×C3#381114yes—
275C22×C3D4×C3⋊D311yescharacteristic
276C2×D3—92no—
277C2×D3—98no—
278C2×D3—188no—
279C2×D3—181no—
280C2×D3—181no—
281C2×D3—184no—
282C2×D3—184no—
283C2×D3—364no—
284C2×D3—364no—
285C3×C4—24no—
286C3×C4—24no—
287C3×C4—48no—
288C3×C4—41no—
289C3×C4—44no—
290C3×C4—124no—
291C3×C4—361no—
292C3×C4#380414yes—
293C3×C4#380414yes—
294C3×C4#382811yescharacteristic
295C3×C4#383011yescharacteristic
296Dic3—41no—
297Dic3—124no—
298Dic3—181no—
299Dic3—184no—
300Dic3—364no—
301Dic3—368no—
302Dic3—361no—
303Dic3—364no—
304C32—24no—
305C32#481114yescomplemented
306C32#481111yescharacteristic, complemented
307C23—271no—
308Q8—21no—
309Q8—21no—
310C2×C4—21no—
311C2×C4—61no—
312C2×C4—181no—
313C2×C4—271no—
314C2×C4—541no—
315C2×C4—541no—
316C2×C4—541no—
317C2×C4—1082no—
318C2×C4C2×D3×C3⋊D311yescharacteristic
319C8—61no—
320C8—181no—
321C2×C3—24no—
322C2×C3—24no—
323C2×C3—24no—
324C2×C3#6394114yes—
325C2×C3#6394414yes—
326C2×C3#6397514yes—
327C2×C3#6409211yescharacteristic
328C2×C3#6412311yescharacteristic
329C2×C3Q16×C3⋊D311yescharacteristic
330D3—188no—
331D3—182no—
332D3—368no—
333C22—272no—
334C22—541no—
335C22—541no—
336C22#8903911yescharacteristic
337C4—41no—
338C4—121no—
339C4—361no—
340C4—541no—
341C4—1081no—
342C4#8902811yescharacteristic
343C4#8903111yescharacteristic
344C3—24no—
345C3#1059155514yescomplemented
346C3#1059251011yescharacteristic, complemented
347C2—542no—
348C2#1172357211yescharacteristic
349C2#1172357711yescharacteristic
350C2#1172362811yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC22×C3⋊(C3⋊D3)D4
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#484
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
3. C3×#63662C2semidirect
4. C32×#835C2semidirect
5. #11722057C2semidirect
6. #11722055C2semidirect
27. #63662D3semidirect
32. C3×#835D3semidirect
98. #835C3⋊D3semidirect
212. C33#484semidirect
305. C32#4811semidirect
306. C32#4811semidirect
345. C3#10591555semidirect
346. C3#10592510semidirect

Maximal subgroups

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

subgroupindex
#117239872
#117235032
C3×#636622
C32×#8352
#117220572
#117220552
#117220242
#105925103
#105915553