Finite Group Databasebeta

← #13138840

Subgroups — #13138840

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#11721107C211yesmaximal, characteristic
2#11721081C211yesmaximal, characteristic
3#11720549C211yesmaximal, characteristic, complemented
4#63647⋊C3C211yesmaximal, characteristic
5#63644⋊C3C211yesmaximal, characteristic, complemented
6#11720326C211yesmaximal, characteristic, complemented
7#11720306C211yesmaximal, characteristic, complemented
8#10590713—31nomaximal
9#10590711—31nomaximal
10C2×#6229C2211yescharacteristic
11C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
12C2×C8×H(3,𝔽3)C2211yescharacteristic
13C4⋊#1084C2211yescharacteristic
14C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
15C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
17#88450—21no—
18#88450—21no—
19#88448—21no—
20#88447—21no—
21#63871—31no—
22#63856—31no—
23#63849—31no—
24#63841—31no—
25#63647D311yescharacteristic
26#63644D311yescharacteristic, complemented
27C3×(C2×D3)⋊C8—31no—
28C3×#817—31no—
29C3×#816—31no—
30C3×#816—31no—
31C3×#816—61no—
32#63576—31no—
33#63575—31no—
34#63566—31no—
35#63565—31no—
36C22×(C3⋊D3)⋊C3D411yescharacteristic
37C2×C4×H(3,𝔽3)C2311yescharacteristic
38C2×#1084—21no—
39C2×#1084—21no—
40#6229—21no—
41#6229—21no—
42C2×(C3⋊Dic3)⋊C3—21no—
43C2×(C3⋊Dic3)⋊C3—21no—
44C2×(C3⋊Dic3)⋊C3D411yescharacteristic
45C4×(C3⋊D3)⋊C3—21no—
46C4×(C3⋊D3)⋊C3—21no—
47C8×H(3,𝔽3)—21no—
48(C3⋊(C3⋊C8))⋊C3—21no—
49#4868—91no—
50#4815—91no—
51#4723—91no—
52C2×C4×C3⋊D3C2×D311yescharacteristic
53C2×C3×C4×D3—31no—
54C2×C3×Dic6—31no—
55C2×C3×Dic6—31no—
56C2×C3×Dic6—61no—
57C2×C32×C8—21no—
58C2×C32×C8C2×D311yescharacteristic
59C2×C32×C8C2×D311yescharacteristic
60C4⋊(C3⋊Dic3)C2×D311yescharacteristic
61C4⋊(C3⋊Dic3)C2×D311yescharacteristic
62C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
63C3×C4⋊Dic3—31no—
64C3×C4⋊Dic3—31no—
65C3×C4⋊Dic3—31no—
66C3×C4⋊Dic3—31no—
67C3×C4⋊Dic3—61no—
68C2×C3×C3⋊C8—31no—
69#3724—61no—
70Dic3⋊Dic3—61no—
71#3722—61no—
72#3722—61no—
73(C2×D3)⋊Dic3—61no—
74(C2×D3)⋊Dic3—61no—
75Dic32—61no—
76Dic32—61no—
77C22×H(3,𝔽3)C2×D411yescharacteristic
78C2×(C3⋊D3)⋊C3—21no—
79C2×(C3⋊D3)⋊C3—21no—
80C2×(C3⋊D3)⋊C3—21no—
81C4×H(3,𝔽3)#4111yescharacteristic
82C4×H(3,𝔽3)#4111yescharacteristic
83#1084—41no—
84#1084—41no—
85(C3⋊Dic3)⋊C3—21no—
86(C3⋊Dic3)⋊C3—41no—
87(C3⋊Dic3)⋊C3—41no—
88#946—91no—
89#926—91no—
90#924—91no—
91#897—91no—
92#892—91no—
93#892—91no—
94C3×#106—91no—
95C3×#103—91no—
96C3×C22⋊C8—91no—
97#835—91no—
98(C2×D3)⋊C8—31no—
99(C2×D3)⋊C8—91no—
100#818—91no—
101#817—31no—
102#817—91no—
103#816—31no—
104#816—31no—
105#816—61no—
106#816—91no—
107#810—91no—
108#807—91no—
109#807—91no—
110C22×C3⋊D3D1211yescharacteristic
111C22×C3×D3—31no—
112C2×C32×C4—21no—
113C2×C32×C4C22×D311yescharacteristic
114C2×C32×C4C22×D311yescharacteristic
115C2×C3⋊Dic3—21no—
116C2×C3⋊Dic3—21no—
117C2×C3⋊Dic3D1211yescharacteristic
118C4×C3⋊D3—21no—
119C4×C3⋊D3—21no—
120C2×C3×Dic3—31no—
121C2×C3×Dic3—61no—
122C2×C3×Dic3—61no—
123C2×C3×Dic3—61no—
124C2×C3×Dic3—61no—
125C2×C3×Dic3—61no—
126C2×C3×Dic3—61no—
127C2×C3×Dic3—121no—
128C2×C3×Dic3—121no—
129C3×C4×D3—61no—
130C3×C4×D3—61no—
131C3×Dic6—62no—
132C3×Dic6—61no—
133C3×Dic6—61no—
134C3×Dic6—61no—
135C3×Dic6—61no—
136C3×Dic6—121no—
137C32×C8—21no—
138C32×C8—21no—
139C32×C8—41no—
140C3⋊(C3⋊C8)—21no—
141C3×C3⋊C8—61no—
142#485—271noSylow
143C2×H(3,𝔽3)#12311yescharacteristic
144C2×H(3,𝔽3)#13511yescharacteristic
145C2×H(3,𝔽3)#13711yescharacteristic
146(C3⋊D3)⋊C3—41no—
147C2×C3×Q8—91no—
148C22×C3×C4—91no—
149C2×C4×D3—31no—
150C2×C4×D3—91no—
151C2×Dic6—31no—
152C2×Dic6—31no—
153C2×Dic6—61no—
154C2×C3×C8—31no—
155C2×C3×C8—31no—
156C2×C3×C8—61no—
157C2×C3×C8—91no—
158C2×C3×C8D3211yescharacteristic
159C3×C4⋊C4—91no—
160C3×C4⋊C4—91no—
161C3×C4⋊C4—91no—
162#217—181no—
163#217—181no—
164(C2×D3)⋊C4—181no—
165(C2×D3)⋊C4—181no—
166(C2×D3)⋊C4—181no—
167(C2×D3)⋊C4—181no—
168C4⋊Dic3—31no—
169C4⋊Dic3—31no—
170C4⋊Dic3—31no—
171C4⋊Dic3—31no—
172C4⋊Dic3—61no—
173C4⋊Dic3—91no—
174C4⋊Dic3—91no—
175C4⋊Dic3—181no—
176Dic3⋊C4—181no—
177Dic3⋊C4—181no—
178C4×Dic3—181no—
179C4×Dic3—181no—
180C4×Dic3—181no—
181C2×C3⋊C8—31no—
182C2×C3⋊C8—91no—
183C22×C32—21no—
184C22×C32C2×D1211yescharacteristic
185C22×C32D3×D411yescharacteristic
186C2×C3⋊D3—21no—
187C2×C3⋊D3—21no—
188C2×C3⋊D3—21no—
189C2×C3×D3—61no—
190C2×C3×D3—61no—
191C2×C3×D3—61no—
192C32×C4—21no—
193C32×C4—21no—
194C32×C4#23511yescharacteristic
195C32×C4#23711yescharacteristic
196C32×C4#23711yescharacteristic
197C32×C4#23911yescharacteristic
198C3⋊Dic3—21no—
199C3⋊Dic3—41no—
200C3⋊Dic3—41no—
201C3×Dic3—61no—
202C3×Dic3—122no—
203C3×Dic3—121no—
204C3×Dic3—121no—
205C3×Dic3—121no—
206C3×Dic3—121no—
207C3×Dic3—121no—
208C3×Dic3—121no—
209C3×Dic3—241no—
210C22⋊Q8—271no—
211#117—271no—
212#107—271no—
213#106—91no—
214#103—91no—
215#103—271no—
216C22⋊C8—91no—
217H(3,𝔽3)#48511yescharacteristic, complemented, Sylow
218C23×C3—91no—
219C22×D3—31no—
220C22×D3—91no—
221C3×Q8—181no—
222C3×Q8—181no—
223C2×C3×C4—31no—
224C2×C3×C4—31no—
225C2×C3×C4—61no—
226C2×C3×C4—91no—
227C2×C3×C4—181no—
228C2×C3×C4—181no—
229C2×C3×C4—181no—
230C2×C3×C4—181no—
231C2×C3×C4—181no—
232C2×C3×C4—181no—
233C2×C3×C4C2×D3211yescharacteristic
234C2×Dic3—31no—
235C2×Dic3—61no—
236C2×Dic3—61no—
237C2×Dic3—61no—
238C2×Dic3—61no—
239C2×Dic3—61no—
240C2×Dic3—61no—
241C2×Dic3—91no—
242C2×Dic3—121no—
243C2×Dic3—121no—
244C2×Dic3—181no—
245C2×Dic3—181no—
246C4×D3—61no—
247C4×D3—61no—
248C4×D3—181no—
249C4×D3—181no—
250Dic6—61no—
251Dic6—61no—
252Dic6—61no—
253Dic6—61no—
254Dic6—62no—
255Dic6—121no—
256C3×C8—21no—
257C3×C8—61no—
258C3×C8—61no—
259C3×C8—121no—
260C3×C8—181no—
261C3⋊C8—61no—
262C3⋊C8—181no—
263C2×C32—21no—
264C2×C32—21no—
265C2×C32—21no—
266C2×C32#88611yescharacteristic
267C2×C32#89411yescharacteristic
268C2×C32#90411yescharacteristic
269C2×C32#90911yescharacteristic
270C2×C32#91511yescharacteristic
271C2×C32#91611yescharacteristic
272C3⋊D3—41no—
273C3×D3—121no—
274C2×Q8—91no—
275C22×C4—91no—
276C2×C8—91no—
277C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
278C4⋊C4—91no—
279C4⋊C4—91no—
280C4⋊C4—91no—
281C4⋊C4—541no—
282C22⋊C4—541no—
283C22⋊C4—541no—
284C42—541no—
285C22×C3—31no—
286C22×C3—31no—
287C22×C3—61no—
288C22×C3—181no—
289C22×C3—181no—
290C22×C3—181no—
291C22×C3D3×D1211yescharacteristic
292C2×D3—61no—
293C2×D3—61no—
294C2×D3—61no—
295C2×D3—181no—
296C2×D3—181no—
297C2×D3—181no—
298C3×C4—31no—
299C3×C4—31no—
300C3×C4—31no—
301C3×C4—31no—
302C3×C4—61no—
303C3×C4—61no—
304C3×C4—181no—
305C3×C4—361no—
306C3×C4—361no—
307C3×C4—361no—
308C3×C4—361no—
309C3×C4#379611yescharacteristic
310C3×C4#380411yescharacteristic
311Dic3—61no—
312Dic3—121no—
313Dic3—121no—
314Dic3—121no—
315Dic3—121no—
316Dic3—122no—
317Dic3—121no—
318Dic3—121no—
319Dic3—181no—
320Dic3—241no—
321Dic3—361no—
322Dic3—361no—
323C32—21no—
324C32#472311yescharacteristic, complemented
325C32#486811yescharacteristic, complemented
326C23—91no—
327Q8—181no—
328Q8—181no—
329C2×C4—91no—
330C2×C4—181no—
331C2×C4—181no—
332C2×C4—181no—
333C2×C4—181no—
334C2×C4—181no—
335C2×C4—181no—
336C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
337C8—21no—
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—361no—
349C2×C3#6380011yescharacteristic
350C2×C3#6380611yescharacteristic
351C2×C3#6389011yescharacteristic
352D3—121no—
353D3—361no—
354C22—181no—
355C22—181no—
356C22—181no—
357C22#8865411yescharacteristic
358C4—181no—
359C4—361no—
360C4—361no—
361C4—361no—
362C4—361no—
363C4#8864911yescharacteristic
364C4#8867111yescharacteristic
365C3—31no—
366C3—31no—
367C3—61no—
368C3#1059070911yescharacteristic
369C2—361no—
370C2#1172085211yescharacteristic
371C2#1172085411yescharacteristic
372C2#1172113011yescharacteristic
subgroupG/H
centerC22#88654
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3D3×D12
prime radicalC22×(C3⋊D3)⋊C3D4
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#485
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
3. #11720549C2semidirect
5. #63644⋊C3C2semidirect
6. #11720326C2semidirect
7. #11720306C2semidirect
26. #63644D3semidirect
217. H(3,𝔽3)#485semidirect
324. C32#4723semidirect
325. C32#4868semidirect

Maximal subgroups

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

subgroupindex
#117211072
#117210812
#117205492
#63647⋊C32
#63644⋊C32
#117203262
#117203062
#105907133
#105907113