Finite Group Databasebeta

← #13140904

Subgroups — #13140904

Order 1728.

378 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 69 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#64156⋊C3C211yesmaximal, characteristic
2#11721093C211yesmaximal, characteristic, complemented
3#11720583C211yesmaximal, characteristic, complemented
4#63662⋊C3C211yesmaximal, characteristic, complemented
5#11720318C211yesmaximal, characteristic
6#11720310C211yesmaximal, characteristic, complemented
7#11720308C211yesmaximal, characteristic
8#10591614—31nomaximal
9#10591539—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
12C4⋊#1084C2211yescharacteristic
13C2×#6180C2211yescharacteristic
14#3752⋊C3—21no—
15#3752⋊C3—21no—
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#88449—21no—
20#88448—21no—
21#64156D311yescharacteristic
22C3×#947—31no—
23#63872—31no—
24#63869—31no—
25#63662D311yescharacteristic, complemented
26C3×#835—31no—
27C3×#835—31no—
28C3×#835—31no—
29C3×#835—61no—
30#63564—31no—
31#63564—31no—
32#63563—31no—
33#63562—31no—
34#63553—31no—
35Dic3×C3⋊C8—31no—
36C22×(C3⋊D3)⋊C3—21no—
37Q8×H(3,𝔽3)—21no—
38Q8×H(3,𝔽3)—21no—
39C2×C4×H(3,𝔽3)—21no—
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×#1084—21no—
42C2×(C3⋊Dic3)⋊C3—21no—
43C2×(C3⋊Dic3)⋊C3D411yescharacteristic
44C2×(C3⋊Dic3)⋊C3D411yescharacteristic
45C4⋊((C3⋊D3)⋊C3)—21no—
46C4⋊((C3⋊D3)⋊C3)—21no—
47#6180—21no—
48(C3⋊(C3⋊C8))⋊C3—21no—
49#5056—91no—
50#4800—91no—
51#4800—91no—
52C2×C32×Q8—21no—
53C2×C32×Q8C2×D311yescharacteristic
54C2×C32×Q8C2×D311yescharacteristic
55C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
56C2×C3×D12—31no—
57#3752—21no—
58#3752—21no—
59C4⋊(C3⋊Dic3)C2×D311yescharacteristic
60C4×C3⋊Dic3C2×D311yescharacteristic
61C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
62C3×(C2×D3)⋊C4—61no—
63C3×(C2×D3)⋊C4—61no—
64C3×C4⋊Dic3—31no—
65C3×C4⋊Dic3—31no—
66C3×C4⋊Dic3—31no—
67C3×C4⋊Dic3—61no—
68C3×C4×Dic3—31no—
69C2×C3×C3⋊C8—31no—
70C2×C3×C3⋊C8—31no—
71C2×C3×C3⋊C8—31no—
72C2×C3×C3⋊C8—61no—
73#3724—61no—
74Dic3⋊Dic3—61no—
75Dic3⋊Dic3—61no—
76Dic3⋊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)—41no—
82C4×H(3,𝔽3)#4111yescharacteristic
83C4×H(3,𝔽3)#4111yescharacteristic
84#1084—41no—
85(C3⋊Dic3)⋊C3—21no—
86(C3⋊Dic3)⋊C3—21no—
87(C3⋊Dic3)⋊C3—41no—
88C3×#124—91no—
89#947—31no—
90#947—91no—
91#890—91no—
92#890—91no—
93#870—91no—
94C3×#103—91no—
95C3×#103—91no—
96#835—31no—
97#835—31no—
98#835—31no—
99#835—61no—
100#835—91no—
101#832—91no—
102#832—91no—
103#821—91no—
104#821—91no—
105C8×Dic3—91no—
106C8×Dic3—91no—
107#809—91no—
108#809—91no—
109C4×C3⋊C8—91no—
110C22×C3⋊D3—21no—
111C22×C3×D3—61no—
112C32×Q8—22no—
113C32×Q8—21no—
114C32×Q8—21no—
115C32×Q8—21no—
116C32×Q8—21no—
117C32×Q8—41no—
118C2×C32×C4—21no—
119C2×C32×C4—21no—
120C2×C32×C4—21no—
121C2×C32×C4—41no—
122C2×C32×C4C22×D311yescharacteristic
123C2×C32×C4C22×D311yescharacteristic
124C2×C3⋊Dic3—21no—
125C2×C3⋊Dic3#6711yescharacteristic
126C2×C3⋊Dic3#6711yescharacteristic
127C4⋊(C3⋊D3)—21no—
128C4⋊(C3⋊D3)—21no—
129C2×C3×Dic3—31no—
130C2×C3×Dic3—31no—
131C2×C3×Dic3—61no—
132C2×C3×Dic3—61no—
133C2×C3×Dic3—61no—
134C2×C3×Dic3—121no—
135C3×D12—61no—
136C3×D12—61no—
137C3⋊(C3⋊C8)—21no—
138C3×C3⋊C8—61no—
139C3×C3⋊C8—61no—
140C3×C3⋊C8—61no—
141C3×C3⋊C8—121no—
142#488—271noSylow
143C2×H(3,𝔽3)#12411yescharacteristic
144C2×H(3,𝔽3)#13511yescharacteristic
145C2×H(3,𝔽3)#13511yescharacteristic
146(C3⋊D3)⋊C3—81no—
147C2×C3×Q8—31no—
148C2×C3×Q8—31no—
149C2×C3×Q8—61no—
150C2×C3×Q8D3211yescharacteristic
151C2×C3×D4—91no—
152C2×D12—31no—
153C2×D12—91no—
154C2×C3×C8—91no—
155C2×C3×C8—91no—
156C3×C4⋊C4—91no—
157C3×C4⋊C4—91no—
158C3×C22⋊C4—181no—
159C3×C22⋊C4—181no—
160C3×C42—91no—
161(C2×D3)⋊C4—61no—
162(C2×D3)⋊C4—61no—
163(C2×D3)⋊C4—181no—
164(C2×D3)⋊C4—181no—
165C4⋊Dic3—31no—
166C4⋊Dic3—31no—
167C4⋊Dic3—31no—
168C4⋊Dic3—61no—
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—91no—
178C2×C3⋊C8—31no—
179C2×C3⋊C8—31no—
180C2×C3⋊C8—31no—
181C2×C3⋊C8—61no—
182C2×C3⋊C8—91no—
183C22×C32—21no—
184C22×C32D3×D411yescharacteristic
185C22×C32C2×#6711yescharacteristic
186C2×C3⋊D3—41no—
187C2×C3⋊D3—41no—
188C2×C3⋊D3—41no—
189C2×C3×D3—121no—
190C2×C3×D3—121no—
191C2×C3×D3—121no—
192C32×C4—21no—
193C32×C4—21no—
194C32×C4—42no—
195C32×C4—41no—
196C32×C4—41no—
197C32×C4#23511yescharacteristic
198C32×C4#23711yescharacteristic
199C32×C4#23711yescharacteristic
200C32×C4#23711yescharacteristic
201C3⋊Dic3—21no—
202C3⋊Dic3—21no—
203C3⋊Dic3—41no—
204C3×Dic3—61no—
205C3×Dic3—61no—
206C3×Dic3—121no—
207C3×Dic3—121no—
208C3×Dic3—121no—
209C3×Dic3—241no—
210#125—271no—
211#124—91no—
212#103—91no—
213#103—91no—
214#102—271no—
215#102—271no—
216C4×C8—271no—
217H(3,𝔽3)#48811yescharacteristic, complemented, Sylow
218C23×C3—181no—
219C22×D3—61no—
220C22×D3—181no—
221C3×Q8—21no—
222C3×Q8—21no—
223C3×Q8—61no—
224C3×Q8—61no—
225C3×Q8—61no—
226C3×Q8—61no—
227C3×Q8—62no—
228C3×Q8—121no—
229C3×D4—181no—
230C3×D4—181no—
231C2×C3×C4—21no—
232C2×C3×C4—31no—
233C2×C3×C4—31no—
234C2×C3×C4—61no—
235C2×C3×C4—61no—
236C2×C3×C4—61no—
237C2×C3×C4—91no—
238C2×C3×C4—91no—
239C2×C3×C4—121no—
240C2×C3×C4—181no—
241C2×C3×C4—181no—
242C2×C3×C4C2×D3211yescharacteristic
243C2×Dic3—31no—
244C2×Dic3—31no—
245C2×Dic3—61no—
246C2×Dic3—61no—
247C2×Dic3—61no—
248C2×Dic3—91no—
249C2×Dic3—91no—
250C2×Dic3—121no—
251C2×Dic3—181no—
252D12—61no—
253D12—61no—
254D12—181no—
255D12—181no—
256C3×C8—181no—
257C3×C8—181no—
258C3⋊C8—61no—
259C3⋊C8—61no—
260C3⋊C8—61no—
261C3⋊C8—121no—
262C3⋊C8—181no—
263C2×C32—21no—
264C2×C32—21no—
265C2×C32—21no—
266C2×C32#88511yescharacteristic
267C2×C32#91611yescharacteristic
268C2×C32#91911yescharacteristic
269C2×C32#93611yescharacteristic
270C2×C32#95011yescharacteristic
271C2×C32#95011yescharacteristic
272C3⋊D3—81no—
273C3×D3—241no—
274C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
275C2×D4—91no—
276C2×C8—91no—
277C2×C8—91no—
278C4⋊C4—91no—
279C4⋊C4—91no—
280C4⋊C4—541no—
281C4⋊C4—541no—
282C22⋊C4—181no—
283C22⋊C4—181no—
284C42—91no—
285C22×C3—31no—
286C22×C3—31no—
287C22×C3—61no—
288C22×C3—361no—
289C22×C3—361no—
290C22×C3—361no—
291C22×C3D3×#6711yescharacteristic
292C2×D3—121no—
293C2×D3—121no—
294C2×D3—121no—
295C2×D3—361no—
296C2×D3—361no—
297C2×D3—361no—
298C3×C4—31no—
299C3×C4—31no—
300C3×C4—31no—
301C3×C4—31no—
302C3×C4—41no—
303C3×C4—61no—
304C3×C4—61no—
305C3×C4—121no—
306C3×C4—121no—
307C3×C4—122no—
308C3×C4—181no—
309C3×C4—181no—
310C3×C4—361no—
311C3×C4—361no—
312C3×C4#380411yescharacteristic
313C3×C4#380511yescharacteristic
314Dic3—61no—
315Dic3—61no—
316Dic3—121no—
317Dic3—121no—
318Dic3—121no—
319Dic3—181no—
320Dic3—181no—
321Dic3—241no—
322Dic3—361no—
323C32—21no—
324C32#480011yescharacteristic, complemented
325C32#505611yescharacteristic, complemented
326C23—181no—
327Q8—21no—
328Q8—21no—
329D4—181no—
330D4—181no—
331C2×C4—21no—
332C2×C4—91no—
333C2×C4—91no—
334C2×C4—181no—
335C2×C4—181no—
336C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
337C8—181no—
338C8—181no—
339C2×C3—31no—
340C2×C3—31no—
341C2×C3—31no—
342C2×C3—31no—
343C2×C3—31no—
344C2×C3—31no—
345C2×C3—61no—
346C2×C3—61no—
347C2×C3—61no—
348C2×C3—721no—
349C2×C3#6394711yescharacteristic
350C2×C3#6394911yescharacteristic
351C2×C3#6395911yescharacteristic
352D3—241no—
353D3—721no—
354C22—361no—
355C22—361no—
356C22—361no—
357C22#8867611yescharacteristic
358C4—41no—
359C4—181no—
360C4—181no—
361C4—361no—
362C4—361no—
363C4#8867111yescharacteristic
364C4#8867211yescharacteristic
365C3—31no—
366C3—31no—
367C3—61no—
368C3#1059161411yescharacteristic
369C2—721no—
370C2#1172123511yescharacteristic
371C2#1172123911yescharacteristic
372C2#1172130411yescharacteristic
subgroupG/H
centerC22#88676
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×Q8×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C22
hypercenterC2×Q8((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×Q8

Complements

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

NH ≅ G/Ntype
2. #11721093C2semidirect
3. #11720583C2semidirect
4. #63662⋊C3C2semidirect
6. #11720310C2semidirect
25. #63662D3semidirect
217. H(3,𝔽3)#488semidirect
324. C32#4800semidirect
325. C32#5056semidirect

Maximal subgroups

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

subgroupindex
#64156⋊C32
#117210932
#117205832
#63662⋊C32
#117203182
#117203102
#117203082
#105916143
#105915393