Finite Group Databasebeta

← #13161258

Subgroups — #13161258

Order 1728.

904 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 904 conjugacy classes of subgroups — too many to draw. Only its 142 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#11723988C211yesmaximal, characteristic
2#11723536C211yesmaximal, characteristic, complemented
3C3×#63662C211yesmaximal, characteristic, complemented
4C32×#835C211yesmaximal, characteristic, complemented
5#11722036C211yesmaximal, characteristic
6#11722035C211yesmaximal, characteristic
7#11722021C211yesmaximal, characteristic, complemented
8#10592493—31nomaximal
9#10591632—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×#6293—21no—
13#88859—21no—
14#88859—21no—
15C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20#88806—21no—
21#64156—31no—
22#64156—34no—
23#64156—64no—
24#64097—31no—
25#63919—34no—
26C32×#103—31no—
27#63662D311yescharacteristic, complemented
28#63648—31no—
29#63642—31no—
30#63636—31no—
31C3×#835—34no—
32C3×#835D314yescomplemented
33#63564—34no—
34#63564—34no—
35#63555—34no—
36C22×C3⋊(C3⋊D3)—21no—
37C33×Q8—21no—
38C33×Q8—21no—
39C2×C33×C4—21no—
40C2×C33×C4C2311yescharacteristic
41C2×C3⋊(C3⋊Dic3)D411yescharacteristic
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#6293—42no—
48C3×C3⋊(C3⋊C8)—21no—
49C32×C3⋊C8—21no—
50#4794—94no—
51#4794—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)—64no—
58C2×#634—61no—
59C2×#627—64no—
60C2×C32×C8—31no—
61C32×C4⋊C4—31no—
62#3752—61no—
63#3752—61no—
64#3752—64no—
65#3752—64no—
66#3752—68no—
67#3752—68no—
68C4⋊(C3⋊Dic3)C2×D311yescharacteristic
69#3750—61no—
70C4×C3⋊Dic3—31no—
71C4×C3⋊Dic3—34no—
72C4×C3⋊Dic3—64no—
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)—41no—
81C2×C3⋊(C3⋊D3)—41no—
82C2×C3⋊(C3⋊D3)—41no—
83C33×C4—41no—
84C33×C4#4111yescharacteristic
85C33×C4#4111yescharacteristic
86C3⋊(C3⋊Dic3)—21no—
87C3⋊(C3⋊Dic3)—21no—
88C3×C3⋊Dic3—41no—
89C32×Dic3—41no—
90#947—94no—
91#947—91no—
92#947—184no—
93#888—94no—
94#888—91no—
95C3×#103—34no—
96C3×#103—91no—
97#835—34no—
98#835C3⋊D311yescharacteristic, complemented
99#821—94no—
100#821—91no—
101#815—94no—
102#815—91no—
103#809—94no—
104#809—91no—
105C22×C3⋊D3—68no—
106C22×C3⋊D3—64no—
107C22×C3⋊D3—61no—
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—31no—
124C2×C3⋊Dic3—34no—
125C2×C3⋊Dic3—34no—
126C2×C3⋊Dic3—64no—
127C2×C3⋊Dic3—64no—
128C4⋊(C3⋊D3)—61no—
129C4⋊(C3⋊D3)—61no—
130C4⋊(C3⋊D3)—64no—
131C4⋊(C3⋊D3)—64no—
132C4⋊(C3⋊D3)—124no—
133C4⋊(C3⋊D3)—124no—
134#634—122no—
135C2×C3×Dic3—24no—
136C2×C3×Dic3—64no—
137#627—128no—
138C32×C8—61no—
139C3⋊(C3⋊C8)—21no—
140C3×C3⋊C8—24no—
141C3×C3⋊C8—64no—
142#489—271noSylow
143C2×C33#12411yescharacteristic
144C2×C33AGL(1,ℤ/8)11yescharacteristic
145C2×C33#13711yescharacteristic
146C3⋊(C3⋊D3)—81no—
147C2×C3×Q8—24no—
148C2×C3×Q8D3214yes—
149C2×C3×Q8C2×C3⋊D311yescharacteristic
150C2×D12—94no—
151C2×D12—91no—
152C2×D12—184no—
153C2×Dic6—181no—
154C2×Dic6—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—184no—
161(C2×D3)⋊C4—181no—
162(C2×D3)⋊C4—181no—
163(C2×D3)⋊C4—188no—
164(C2×D3)⋊C4—188no—
165C4⋊Dic3—34no—
166C4⋊Dic3C2×C3⋊D311yescharacteristic
167Dic3⋊C4—181no—
168Dic3⋊C4—184no—
169C4×Dic3—94no—
170C4×Dic3—91no—
171C4×Dic3—184no—
172C2×C3⋊C8—34no—
173C2×C3⋊C8C2×C3⋊D311yescharacteristic
174C22×C32—24no—
175C22×C32D3×D414yes—
176C22×C32D3×D411yescharacteristic
177C2×C3⋊D3—128no—
178C2×C3⋊D3—128no—
179C2×C3⋊D3—128no—
180C2×C3⋊D3—124no—
181C2×C3⋊D3—121no—
182C2×C3⋊D3—124no—
183C2×C3⋊D3—121no—
184C2×C3⋊D3—124no—
185C2×C3⋊D3—121no—
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—61no—
199C3⋊Dic3—64no—
200C3⋊Dic3—64no—
201C3⋊Dic3—124no—
202C3⋊Dic3—124no—
203C3×Dic3—44no—
204C3×Dic3—124no—
205C4⋊Q8—271no—
206#124—271no—
207#103—31no—
208#103—91no—
209#102—271no—
210#102—271no—
211#97—271no—
212C33#48911yescharacteristic, complemented, Sylow
213C22×D3—188no—
214C22×D3—184no—
215C22×D3—181no—
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—94no—
234C2×Dic3—91no—
235C2×Dic3—91no—
236C2×Dic3—184no—
237C2×Dic3—184no—
238D12—184no—
239D12—184no—
240D12—181no—
241D12—181no—
242D12—364no—
243D12—364no—
244Dic6—362no—
245Dic6—368no—
246C3×C8—64no—
247C3×C8—181no—
248C3⋊C8—21no—
249C3⋊C8—64no—
250C2×C32—24no—
251C2×C32—24no—
252C2×C32—24no—
253C2×C32#88514yes—
254C2×C32#88511yescharacteristic
255C2×C32#91414yes—
256C2×C32#91411yescharacteristic
257C2×C32#91814yes—
258C2×C32#91811yescharacteristic
259C3⋊D3—248no—
260C3⋊D3—244no—
261C3⋊D3—241no—
262C2×Q8—541no—
263C2×Q8D3×C3⋊D311yescharacteristic
264C2×D4—271no—
265C2×C8—31no—
266C2×C8—91no—
267C4⋊C4—31no—
268C4⋊C4—91no—
269C4⋊C4—541no—
270C22⋊C4—541no—
271C22⋊C4—541no—
272C42—271no—
273C22×C3—24no—
274C22×C3#381114yes—
275C22×C3D4×C3⋊D311yescharacteristic
276C2×D3—368no—
277C2×D3—368no—
278C2×D3—368no—
279C2×D3—361no—
280C2×D3—361no—
281C2×D3—361no—
282C2×D3—364no—
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—181no—
300Dic3—184no—
301Dic3—184no—
302Dic3—364no—
303Dic3—364no—
304C32—24no—
305C32#479414yescomplemented
306C32#479411yescharacteristic, complemented
307C23—541no—
308Q8—21no—
309Q8—21no—
310Q8—1082no—
311D4—541no—
312D4—541no—
313C2×C4—21no—
314C2×C4—61no—
315C2×C4—181no—
316C2×C4—271no—
317C2×C4—271no—
318C2×C4C2×D3×C3⋊D311yescharacteristic
319C8—61no—
320C8—181no—
321C2×C3—24no—
322C2×C3—24no—
323C2×C3—24no—
324C2×C3#6394114yes—
325C2×C3#6394514yes—
326C2×C3#6395314yes—
327C2×C3#6409411yescharacteristic
328C2×C3#6412311yescharacteristic
329C2×C3#6412711yescharacteristic
330D3—728no—
331D3—724no—
332D3—721no—
333C22—1081no—
334C22—1081no—
335C22—1081no—
336C22#8903911yescharacteristic
337C4—41no—
338C4—121no—
339C4—361no—
340C4—541no—
341C4—541no—
342C4#8902811yescharacteristic
343C4#8903111yescharacteristic
344C3—24no—
345C3#1059163214yescomplemented
346C3#1059249311yescharacteristic, complemented
347C2—2161no—
348C2#1172357211yescharacteristic
349C2#1172357811yescharacteristic
350C2#1172359211yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×C4⋊(C3⋊(C3⋊D3))C22
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#489
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
2. #11723536C2semidirect
3. C3×#63662C2semidirect
4. C32×#835C2semidirect
7. #11722021C2semidirect
27. #63662D3semidirect
32. C3×#835D3semidirect
98. #835C3⋊D3semidirect
212. C33#489semidirect
305. C32#4794semidirect
306. C32#4794semidirect
345. C3#10591632semidirect
346. C3#10592493semidirect

Maximal subgroups

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

subgroupindex
#117239882
#117235362
C3×#636622
C32×#8352
#117220362
#117220352
#117220212
#105924933
#105916323