Finite Group Databasebeta

← #13138806

Subgroups — #13138806

Order 1728.

380 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 67 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#11721113C211yesmaximal, characteristic
2#11721095C211yesmaximal, characteristic, complemented
3C8×#1084C211yesmaximal, characteristic, complemented
4#63648⋊C3C211yesmaximal, characteristic
5#63643⋊C3C211yesmaximal, characteristic, complemented
6#11720318C211yesmaximal, characteristic
7#11720313C211yesmaximal, characteristic, complemented
8#10590872—31nomaximal
9#10590863—31nomaximal
10C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
11C2×#634⋊C3C2211yescharacteristic
12C2×C8×H(3,𝔽3)C2211yescharacteristic
13C4×#1084C2211yescharacteristic
14C2×#6180C2211yescharacteristic
15C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17#88450—21no—
18#88450—21no—
19#88448—21no—
20#88448—21no—
21#63869—31no—
22#63869—31no—
23#63864—31no—
24#63860—31no—
25#63648D311yescharacteristic
26#63643D311yescharacteristic, complemented
27C3×#821—31no—
28C3×#816—31no—
29C3×C8×Dic3—31no—
30C3×C8×Dic3—31no—
31C3×C8×Dic3—61no—
32#63567—31no—
33#63566—31no—
34#63564—31no—
35#63563—31no—
36C22×(C3⋊D3)⋊C3—21no—
37C2×C4×H(3,𝔽3)C2311yescharacteristic
38C2×#1084D411yescharacteristic
39C2×#1084D411yescharacteristic
40C2×(C3⋊Dic3)⋊C3—21no—
41C2×(C3⋊Dic3)⋊C3—21no—
42C2×(C3⋊Dic3)⋊C3—21no—
43C4⋊((C3⋊D3)⋊C3)—21no—
44C4⋊((C3⋊D3)⋊C3)—21no—
45#634⋊C3—21no—
46#634⋊C3—21no—
47C8×H(3,𝔽3)—21no—
48#6180—21no—
49#4800—91no—
50#4768—91no—
51#4697—91no—
52C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
53C2×#634C2×D311yescharacteristic
54C2×C3×D12—31no—
55C2×C3×Dic6—31no—
56C2×C32×C8—21no—
57C2×C32×C8C2×D311yescharacteristic
58C2×C32×C8C2×D311yescharacteristic
59C4⋊(C3⋊Dic3)C2×D311yescharacteristic
60C4⋊(C3⋊Dic3)C2×D311yescharacteristic
61C3×C4⋊Dic3—31no—
62C3×C4⋊Dic3—31no—
63C3×C4×Dic3—31no—
64C3×C4×Dic3—31no—
65C3×C4×Dic3—61no—
66C2×C3×C3⋊C8—31no—
67C2×C3×C3⋊C8—31no—
68C2×C3×C3⋊C8—61no—
69#3724—61no—
70#3724—61no—
71Dic3⋊Dic3—61no—
72Dic3⋊Dic3—61no—
73#3722—61no—
74#3722—61no—
75(C2×D3)⋊Dic3—61no—
76(C2×D3)⋊Dic3—61no—
77C22×H(3,𝔽3)C2×D411yescharacteristic
78C2×(C3⋊D3)⋊C3—41no—
79C2×(C3⋊D3)⋊C3—41no—
80C2×(C3⋊D3)⋊C3—41no—
81C4×H(3,𝔽3)#4111yescharacteristic
82C4×H(3,𝔽3)#4111yescharacteristic
83#1084—21no—
84#1084—21no—
85(C3⋊Dic3)⋊C3—41no—
86(C3⋊Dic3)⋊C3—41no—
87(C3⋊Dic3)⋊C3—41no—
88#947—91no—
89#936—91no—
90#890—91no—
91#890—91no—
92#875—91no—
93#870—91no—
94C3×#103—91no—
95C3×#102—91no—
96C3×C4×C8—91no—
97#835—91no—
98#832—91no—
99#821—31no—
100#821—91no—
101#821—91no—
102#816—31no—
103#816—91no—
104#816—91no—
105C8×Dic3—31no—
106C8×Dic3—31no—
107C8×Dic3—61no—
108#810—91no—
109#809—91no—
110C22×C3⋊D3—21no—
111C22×C3×D3—61no—
112C2×C32×C4—21no—
113C2×C32×C4C22×D311yescharacteristic
114C2×C32×C4C22×D311yescharacteristic
115C2×C3⋊Dic3—21no—
116C2×C3⋊Dic3—21no—
117C2×C3⋊Dic3—21no—
118C4⋊(C3⋊D3)—21no—
119C4⋊(C3⋊D3)—21no—
120#634—21no—
121#634—21no—
122C2×C3×Dic3—31no—
123C2×C3×Dic3—31no—
124C2×C3×Dic3—31no—
125C2×C3×Dic3—31no—
126C2×C3×Dic3—61no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—61no—
130C2×C3×Dic3—61no—
131C3×D12—61no—
132C3×D12—61no—
133C3×Dic6—61no—
134C3×Dic6—61no—
135C32×C8—22no—
136C32×C8—21no—
137C32×C8—21no—
138C3×C3⋊C8—62no—
139C3×C3⋊C8—61no—
140C3×C3⋊C8—61no—
141#488—271noSylow
142C2×H(3,𝔽3)#12411yescharacteristic
143C2×H(3,𝔽3)#13511yescharacteristic
144C2×H(3,𝔽3)#13511yescharacteristic
145(C3⋊D3)⋊C3—81no—
146C2×C3×Q8—91no—
147C2×C3×D4—91no—
148C2×D12—31no—
149C2×D12—91no—
150C2×Dic6—31no—
151C2×Dic6—91no—
152C2×C3×C8—31no—
153C2×C3×C8—31no—
154C2×C3×C8—61no—
155C2×C3×C8—91no—
156C2×C3×C8D3211yescharacteristic
157C3×C4⋊C4—91no—
158C3×C4⋊C4—91no—
159C3×C42—91no—
160#217—181no—
161#217—181no—
162(C2×D3)⋊C4—181no—
163(C2×D3)⋊C4—181no—
164(C2×D3)⋊C4—181no—
165(C2×D3)⋊C4—181no—
166C4⋊Dic3—31no—
167C4⋊Dic3—31no—
168C4⋊Dic3—91no—
169C4⋊Dic3—91no—
170C4⋊Dic3—181no—
171C4⋊Dic3—181no—
172Dic3⋊C4—181no—
173Dic3⋊C4—181no—
174Dic3⋊C4—181no—
175Dic3⋊C4—181no—
176C4×Dic3—31no—
177C4×Dic3—31no—
178C4×Dic3—61no—
179C2×C3⋊C8—31no—
180C2×C3⋊C8—31no—
181C2×C3⋊C8—61no—
182C22×C32—21no—
183C22×C32D3×D411yescharacteristic
184C22×C32D3×D411yescharacteristic
185C2×C3⋊D3—41no—
186C2×C3⋊D3—41no—
187C2×C3⋊D3—41no—
188C2×C3×D3—121no—
189C2×C3×D3—121no—
190C2×C3×D3—121no—
191C32×C4—21no—
192C32×C4—21no—
193C32×C4#23511yescharacteristic
194C32×C4#23511yescharacteristic
195C32×C4#23711yescharacteristic
196C32×C4#23711yescharacteristic
197C3⋊Dic3—41no—
198C3⋊Dic3—41no—
199C3⋊Dic3—41no—
200C3×Dic3—62no—
201C3×Dic3—62no—
202C3×Dic3—61no—
203C3×Dic3—61no—
204C3×Dic3—61no—
205C3×Dic3—61no—
206C3×Dic3—121no—
207C3×Dic3—121no—
208C3×Dic3—121no—
209#125—271no—
210#124—271no—
211#103—91no—
212#103—271no—
213#102—91no—
214#102—271no—
215C4×C8—91no—
216H(3,𝔽3)#48811yescharacteristic, complemented, Sylow
217C23×C3—181no—
218C22×D3—61no—
219C22×D3—181no—
220C3×Q8—181no—
221C3×Q8—181no—
222C3×D4—181no—
223C3×D4—181no—
224C2×C3×C4—31no—
225C2×C3×C4—31no—
226C2×C3×C4—61no—
227C2×C3×C4—91no—
228C2×C3×C4—91no—
229C2×C3×C4—181no—
230C2×C3×C4—181no—
231C2×C3×C4—181no—
232C2×C3×C4C2×D3211yescharacteristic
233C2×Dic3—31no—
234C2×Dic3—31no—
235C2×Dic3—31no—
236C2×Dic3—31no—
237C2×Dic3—61no—
238C2×Dic3—61no—
239C2×Dic3—61no—
240C2×Dic3—61no—
241C2×Dic3—61no—
242C2×Dic3—181no—
243C2×Dic3—181no—
244C2×Dic3—181no—
245D12—61no—
246D12—61no—
247D12—181no—
248D12—181no—
249Dic6—61no—
250Dic6—61no—
251Dic6—181no—
252Dic6—181no—
253C3×C8—21no—
254C3×C8—61no—
255C3×C8—61no—
256C3×C8—62no—
257C3×C8—181no—
258C3⋊C8—61no—
259C3⋊C8—61no—
260C3⋊C8—62no—
261C2×C32—21no—
262C2×C32—21no—
263C2×C32—21no—
264C2×C32#88511yescharacteristic
265C2×C32#88511yescharacteristic
266C2×C32#91211yescharacteristic
267C2×C32#91611yescharacteristic
268C2×C32#91611yescharacteristic
269C2×C32#91911yescharacteristic
270C3⋊D3—81no—
271C3×D3—241no—
272C2×Q8—91no—
273C2×D4—91no—
274C2×C8—91no—
275C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
276C4⋊C4—91no—
277C4⋊C4—91no—
278C4⋊C4—541no—
279C4⋊C4—541no—
280C22⋊C4—541no—
281C22⋊C4—541no—
282C42—91no—
283C22×C3—31no—
284C22×C3—31no—
285C22×C3—61no—
286C22×C3—361no—
287C22×C3—361no—
288C22×C3—361no—
289C22×C3#380211yescharacteristic
290C2×D3—121no—
291C2×D3—121no—
292C2×D3—121no—
293C2×D3—361no—
294C2×D3—361no—
295C2×D3—361no—
296C3×C4—31no—
297C3×C4—31no—
298C3×C4—31no—
299C3×C4—31no—
300C3×C4—61no—
301C3×C4—61no—
302C3×C4—181no—
303C3×C4—181no—
304C3×C4—361no—
305C3×C4—361no—
306C3×C4—361no—
307C3×C4#379811yescharacteristic
308C3×C4#380511yescharacteristic
309Dic3—61no—
310Dic3—61no—
311Dic3—61no—
312Dic3—61no—
313Dic3—62no—
314Dic3—62no—
315Dic3—121no—
316Dic3—121no—
317Dic3—121no—
318Dic3—361no—
319Dic3—361no—
320Dic3—361no—
321C32—21no—
322C32#476811yescharacteristic, complemented
323C32#480011yescharacteristic, complemented
324C23—181no—
325Q8—181no—
326Q8—181no—
327D4—181no—
328D4—181no—
329C2×C4—91no—
330C2×C4—91no—
331C2×C4—181no—
332C2×C4—181no—
333C2×C4—181no—
334C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
335C8—21no—
336C8—181no—
337C2×C3—31no—
338C2×C3—31no—
339C2×C3—31no—
340C2×C3—31no—
341C2×C3—31no—
342C2×C3—31no—
343C2×C3—61no—
344C2×C3—61no—
345C2×C3—61no—
346C2×C3—721no—
347C2×C3#6380911yescharacteristic
348C2×C3#6381011yescharacteristic
349C2×C3#6391511yescharacteristic
350D3—241no—
351D3—721no—
352C22—361no—
353C22—361no—
354C22—361no—
355C22#8865511yescharacteristic
356C4—181no—
357C4—181no—
358C4—361no—
359C4—361no—
360C4—361no—
361C4#8865011yescharacteristic
362C4#8867211yescharacteristic
363C3—31no—
364C3—31no—
365C3—61no—
366C3#1059086811yescharacteristic
367C2—721no—
368C2#1172084311yescharacteristic
369C2#1172084411yescharacteristic
370C2#1172114511yescharacteristic
subgroupG/H
centerC22#88655
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3#3802
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C22
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#488
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. #11721095C2semidirect
3. C8×#1084C2semidirect
5. #63643⋊C3C2semidirect
7. #11720313C2semidirect
26. #63643D3semidirect
216. H(3,𝔽3)#488semidirect
322. C32#4768semidirect
323. C32#4800semidirect

Maximal subgroups

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

subgroupindex
#117211132
#117210952
C8×#10842
#63648⋊C32
#63643⋊C32
#117203182
#117203132
#105908723
#105908633