Finite Group Databasebeta

← #13159276

Subgroups — #13159276

Order 1728.

980 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 980 conjugacy classes of subgroups — too many to draw. Only its 146 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#11723929C211yesmaximal, characteristic, complemented
2#11723529C211yesmaximal, characteristic
3C3×#63636C211yesmaximal, characteristic, complemented
4C32×#809C211yesmaximal, characteristic, complemented
5#11722034C211yesmaximal, characteristic, complemented
6#11722032C211yesmaximal, characteristic, complemented
7#11722018C211yesmaximal, characteristic, complemented
8#10592459—31nomaximal
9#10590552—34nomaximal
10C2×#6308—21no—
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×#6290—21no—
14C2×#6290—21no—
15C33×C4⋊C4C2211yescharacteristic
16C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
17#88857—21no—
18C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
19C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21#64149—31no—
22#64097—31no—
23#64097—34no—
24#64097—64no—
25#63914—34no—
26C32×#102—31no—
27#63659—31no—
28#63648—31no—
29C8×C3⋊Dic3—31no—
30#63636D311yescharacteristic, complemented
31C3×#809—34no—
32C3×#809D314yescomplemented
33#63563—34no—
34#63563—34no—
35#63553—34no—
36C22×C3×C3⋊D3—21no—
37C22×C32×D3—21no—
38C2×C33×C4—21no—
39C2×C33×C4C2311yescharacteristic
40C2×C3⋊(C3⋊Dic3)—21no—
41C2×C3⋊(C3⋊Dic3)D411yescharacteristic
42C2×C3⋊(C3⋊Dic3)D411yescharacteristic
43#6308—42no—
44C3×C4⋊(C3⋊D3)—21no—
45C3×C4⋊(C3⋊D3)—21no—
46C32×D12—21no—
47C32×D12—21no—
48#6290—42no—
49#6290—42no—
50C3×C3⋊(C3⋊C8)—21no—
51C32×C3⋊C8—21no—
52#4760—94no—
53#4760—91no—
54C2×C32×D4—31no—
55C2×#638—61no—
56C2×#638—61no—
57C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
58C2×#634—61no—
59C2×#634—64no—
60C2×#634—68no—
61C2×C3×D12—34no—
62C2×C3×D12C2×D314yes—
63C2×#625—64no—
64C2×#625—64no—
65C2×C32×C8—31no—
66C32×C4⋊C4—24no—
67C32×C4⋊C4C2×D314yes—
68C32×C4⋊C4C2×D311yescharacteristic
69C4⋊(C3⋊Dic3)—31no—
70C4⋊(C3⋊Dic3)—34no—
71C4⋊(C3⋊Dic3)—64no—
72#3750—61no—
73#3750—64no—
74#3750—68no—
75C4×C3⋊Dic3—31no—
76C4×C3⋊Dic3—34no—
77C4×C3⋊Dic3—64no—
78C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
79C2×C3×C3⋊C8—34no—
80C2×C3×C3⋊C8C2×D314yes—
81C22×C33C2×D411yescharacteristic
82C2×C3×C3⋊D3—41no—
83C2×C3×C3⋊D3—41no—
84C2×C3×C3⋊D3—41no—
85C2×C32×D3—41no—
86C2×C32×D3—41no—
87C2×C32×D3—41no—
88C33×C4—41no—
89C33×C4#4111yescharacteristic
90C33×C4#4111yescharacteristic
91C3⋊(C3⋊Dic3)—41no—
92C3⋊(C3⋊Dic3)D812yes—
93C3⋊(C3⋊Dic3)SD1612yes—
94#940—94no—
95#940—91no—
96#888—94no—
97#888—91no—
98#888—184no—
99C3×#102—34no—
100C3×#102—91no—
101#832—94no—
102#832—91no—
103#821—94no—
104#821—91no—
105C8×Dic3—94no—
106C8×Dic3—91no—
107#809—34no—
108#809C3⋊D311yescharacteristic, complemented
109C23×C32—61no—
110C22×C3⋊D3—21no—
111C22×C3×D3—24no—
112C22×C3×D3—64no—
113C32×D4—61no—
114C32×D4—61no—
115C2×C32×C4—28no—
116C2×C32×C4—24no—
117C2×C32×C4—21no—
118C2×C32×C4—24no—
119C2×C32×C4C22×D314yes—
120C2×C32×C4C22×D311yescharacteristic
121#638—122no—
122#638—122no—
123C2×C3⋊Dic3—31no—
124C2×C3⋊Dic3—31no—
125C2×C3⋊Dic3—34no—
126C2×C3⋊Dic3—34no—
127C2×C3⋊Dic3—64no—
128C2×C3⋊Dic3—64no—
129C2×C3⋊Dic3—68no—
130C2×C3⋊Dic3—64no—
131C2×C3⋊Dic3—61no—
132C4⋊(C3⋊D3)—21no—
133C4⋊(C3⋊D3)—21no—
134#634—122no—
135#634—128no—
136#634—1216no—
137C3×D12—24no—
138C3×D12—24no—
139C3×D12—64no—
140C3×D12—64no—
141#625—128no—
142#625—128no—
143C32×C8—61no—
144C3⋊(C3⋊C8)—21no—
145C3×C3⋊C8—24no—
146C3×C3⋊C8—64no—
147#486—271noSylow
148C2×C33#12411yescharacteristic
149C2×C33C2×D811yescharacteristic
150C2×C33C2×SD1611yescharacteristic
151C3×C3⋊D3—81no—
152C32×D3—81no—
153C2×C3×D4—34no—
154C2×C3×D4—91no—
155C2×#67—181no—
156C2×#67—181no—
157C2×#67—184no—
158C2×#67—184no—
159C2×D12—34no—
160C2×D12C2×C3⋊D311yescharacteristic
161C2×Dic6—184no—
162C2×Dic6—181no—
163C2×Dic6—188no—
164C2×C3×C8—34no—
165C2×C3×C8—91no—
166C3×C4⋊C4—24no—
167C3×C4⋊C4D3214yes—
168C3×C4⋊C4C2×C3⋊D311yescharacteristic
169C4⋊Dic3—94no—
170C4⋊Dic3—91no—
171C4⋊Dic3—184no—
172Dic3⋊C4—184no—
173Dic3⋊C4—181no—
174Dic3⋊C4—188no—
175C4×Dic3—94no—
176C4×Dic3—91no—
177C4×Dic3—184no—
178C2×C3⋊C8—34no—
179C2×C3⋊C8C2×C3⋊D311yescharacteristic
180C22×C32—24no—
181C22×C32—121no—
182C22×C32—121no—
183C22×C32—121no—
184C22×C32D3×D414yes—
185C22×C32D3×D411yescharacteristic
186C2×C3⋊D3—41no—
187C2×C3⋊D3—41no—
188C2×C3⋊D3—41no—
189C2×C3×D3—44no—
190C2×C3×D3—44no—
191C2×C3×D3—44no—
192C2×C3×D3—124no—
193C2×C3×D3—124no—
194C2×C3×D3—124no—
195C32×C4—24no—
196C32×C4—24no—
197C32×C4—48no—
198C32×C4—44no—
199C32×C4—41no—
200C32×C4#23514yes—
201C32×C4#23511yescharacteristic
202C32×C4#23714yes—
203C32×C4#23711yescharacteristic
204C3⋊Dic3—32no—
205C3⋊Dic3—32no—
206C3⋊Dic3—38no—
207C3⋊Dic3—38no—
208C3⋊Dic3—68no—
209C3⋊Dic3—68no—
210C3⋊Dic3—128no—
211C3⋊Dic3—124no—
212C3⋊Dic3—121no—
213C4⋊Q8—271no—
214#127—271no—
215#102—31no—
216#102—91no—
217#102—271no—
218#102—271no—
219C4×C8—271no—
220C33#48611yescharacteristic, complemented, Sylow
221C23×C3—64no—
222C23×C3—181no—
223C22×D3—21no—
224C22×D3—64no—
225C3×D4—64no—
226C3×D4—64no—
227C3×D4—181no—
228C3×D4—181no—
229C2×C3×C4—28no—
230C2×C3×C4—24no—
231C2×C3×C4—21no—
232C2×C3×C4—24no—
233C2×C3×C4C2×D3214yes—
234C2×C3×C4C22×C3⋊D311yescharacteristic
235#67—362no—
236#67—362no—
237#67—368no—
238#67—368no—
239C2×Dic3—94no—
240C2×Dic3—94no—
241C2×Dic3—91no—
242C2×Dic3—91no—
243C2×Dic3—184no—
244C2×Dic3—184no—
245C2×Dic3—188no—
246C2×Dic3—184no—
247C2×Dic3—181no—
248D12—21no—
249D12—21no—
250D12—64no—
251D12—64no—
252Dic6—362no—
253Dic6—368no—
254Dic6—3616no—
255C3×C8—64no—
256C3×C8—181no—
257C3⋊C8—21no—
258C3⋊C8—64no—
259C2×C32—24no—
260C2×C32—24no—
261C2×C32—24no—
262C2×C32—241no—
263C2×C32#88514yes—
264C2×C32#88511yescharacteristic
265C2×C32D3×D814yes—
266C2×C32D3×D811yescharacteristic
267C2×C32D3×SD1614yes—
268C2×C32D3×SD1611yescharacteristic
269C3⋊D3—81no—
270C3×D3—84no—
271C3×D3—244no—
272C2×Q8—541no—
273C2×D4—31no—
274C2×D4—91no—
275C2×D4—541no—
276C2×D4—541no—
277C2×C8—31no—
278C2×C8—91no—
279C4⋊C4—271no—
280C4⋊C4—541no—
281C4⋊C4D3×C3⋊D311yescharacteristic
282C42—271no—
283C22×C3—24no—
284C22×C3—124no—
285C22×C3—124no—
286C22×C3—124no—
287C22×C3—361no—
288C22×C3—361no—
289C22×C3—361no—
290C22×C3#381114yes—
291C22×C3D4×C3⋊D311yescharacteristic
292C2×D3—41no—
293C2×D3—41no—
294C2×D3—41no—
295C2×D3—124no—
296C2×D3—124no—
297C2×D3—124no—
298C3×C4—24no—
299C3×C4—24no—
300C3×C4—48no—
301C3×C4—41no—
302C3×C4—44no—
303C3×C4#379514yes—
304C3×C4#379514yes—
305C3×C4#382811yescharacteristic
306C3×C4#383011yescharacteristic
307Dic3—92no—
308Dic3—92no—
309Dic3—98no—
310Dic3—98no—
311Dic3—188no—
312Dic3—188no—
313Dic3—368no—
314Dic3—361no—
315Dic3—364no—
316C32—24no—
317C32#476014yescomplemented
318C32#476011yescharacteristic, complemented
319C23—61no—
320C23—181no—
321Q8—1082no—
322D4—61no—
323D4—61no—
324D4—181no—
325D4—181no—
326D4—1082no—
327D4—1082no—
328C2×C4—21no—
329C2×C4—271no—
330C2×C4—271no—
331C2×C4—541no—
332C2×C4C2×D3×C3⋊D311yescharacteristic
333C8—61no—
334C8—181no—
335C2×C3—24no—
336C2×C3—24no—
337C2×C3—24no—
338C2×C3—244no—
339C2×C3—721no—
340C2×C3#6391314yes—
341C2×C3#6392614yes—
342C2×C3#6394014yes—
343C2×C3#6409411yescharacteristic
344C2×C3D8×C3⋊D311yescharacteristic
345C2×C3SD16×C3⋊D311yescharacteristic
346D3—81no—
347D3—244no—
348C22—121no—
349C22—121no—
350C22—121no—
351C22—361no—
352C22—361no—
353C22—361no—
354C22#8903911yescharacteristic
355C4—41no—
356C4—272no—
357C4—272no—
358C4—1081no—
359C4#8901411yescharacteristic
360C4#8902111yescharacteristic
361C3—24no—
362C3#1059055214yescomplemented
363C3#1059245911yescharacteristic, complemented
364C2—241no—
365C2—721no—
366C2#1172352811yescharacteristic
367C2#1172354711yescharacteristic
368C2#1172357111yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radical#11723529C2
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#486
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #11723929C2semidirect
3. C3×#63636C2semidirect
4. C32×#809C2semidirect
5. #11722034C2semidirect
6. #11722032C2semidirect
7. #11722018C2semidirect
30. #63636D3semidirect
32. C3×#809D3semidirect
108. #809C3⋊D3semidirect
220. C33#486semidirect
317. C32#4760semidirect
318. C32#4760semidirect
362. C3#10590552semidirect
363. C3#10592459semidirect

Maximal subgroups

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

subgroupindex
#117239292
#117235292
C3×#636362
C32×#8092
#117220342
#117220322
#117220182
#105924593
#105905523