Finite Group Databasebeta

← #13137996

Subgroups — #13137996

Order 1728.

407 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 407 conjugacy classes of subgroups — too many to draw. Only its 71 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#64099⋊C3C211yesmaximal, characteristic, complemented
2#11721149C211yesmaximal, characteristic
3#11720531C211yesmaximal, characteristic
4#63635⋊C3C211yesmaximal, characteristic, complemented
5#11720324C211yesmaximal, characteristic
6#11720315C211yesmaximal, characteristic, complemented
7#11720311C211yesmaximal, characteristic, complemented
8#10590474—31nomaximal
9#10590469—31nomaximal
10C2×C4⋊#272C2211yescharacteristic
11C2×#6199—21no—
12C4⋊C4×H(3,𝔽3)C2211yescharacteristic
13C2×#6180C2211yescharacteristic
14C4⋊((C3⋊Dic3)⋊C3)—21no—
15C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17#3750⋊C3—21no—
18C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20#88451—21no—
21#64099D311yescharacteristic, complemented
22C3×#890—31no—
23#63898—31no—
24Dic3×D12—31no—
25#63635D311yescharacteristic, complemented
26C3×#809—31no—
27C3×#809—31no—
28C3×#809—61no—
29C3×#808—31no—
30#63575—31no—
31#63575—31no—
32#63572—31no—
33Dic3⋊(C3⋊C8)—31no—
34#63563—31no—
35#63562—31no—
36C22×#272—21no—
37C2×C4×H(3,𝔽3)—21no—
38C2×C4×H(3,𝔽3)C2311yescharacteristic
39C4⋊#272—21no—
40C4⋊#272Q812yes—
41C2×(C3⋊Dic3)⋊C3—21no—
42C2×(C3⋊Dic3)⋊C3—21no—
43C2×(C3⋊Dic3)⋊C3D411yescharacteristic
44C2×(C3⋊Dic3)⋊C3D411yescharacteristic
45#6199—42no—
46#6180—21no—
47(C3⋊(C3⋊C8))⋊C3—21no—
48#5066—91no—
49#4871—91no—
50#4766—91no—
51C2×C3×D12—31no—
52C2×C3×D12—31no—
53C2×C3×D12—61no—
54C2×D3×Dic3—61no—
55C2×D3×Dic3—61no—
56C32×C4⋊C4—21no—
57C32×C4⋊C4C2×D311yescharacteristic
58C32×C4⋊C4C2×D311yescharacteristic
59C4⋊(C3⋊Dic3)—21no—
60C4⋊(C3⋊Dic3)C2×D311yescharacteristic
61C4⋊(C3⋊Dic3)C2×D311yescharacteristic
62#3750—21no—
63C4×C3⋊Dic3C2×D311yescharacteristic
64C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
65C3×C4⋊Dic3—31no—
66C3×C4⋊Dic3—31no—
67C3×C4⋊Dic3—61no—
68C3×Dic3⋊C4—61no—
69C3×C4×Dic3—31no—
70C2×C3×C3⋊C8—31no—
71C2×C3×C3⋊C8—31no—
72C2×C3×C3⋊C8—31no—
73C2×C3×C3⋊C8—61no—
74(C2×D3)⋊Dic3—61no—
75(C2×D3)⋊Dic3—61no—
76C22×H(3,𝔽3)C2×D411yescharacteristic
77C2×#272—22no—
78C2×#272—41no—
79C2×#272—41no—
80C4×H(3,𝔽3)—41no—
81C4×H(3,𝔽3)C2×Q811yescharacteristic
82C4×H(3,𝔽3)#4111yescharacteristic
83(C3⋊Dic3)⋊C3—21no—
84(C3⋊Dic3)⋊C3—21no—
85(C3⋊Dic3)⋊C3—41no—
86(C3⋊Dic3)⋊C3—41no—
87C3×#125—91no—
88D4×Dic3—91no—
89#893—91no—
90#890—31no—
91#890—91no—
92C4×D12—91no—
93C3×#106—91no—
94C3×#102—91no—
95#832—91no—
96#821—91no—
97#818—91no—
98Dic3⋊C8—91no—
99Dic3⋊C8—91no—
100#809—31no—
101#809—31no—
102#809—61no—
103#809—91no—
104#808—31no—
105#808—91no—
106#807—91no—
107#807—91no—
108C4⋊(C3⋊C8)—91no—
109C22×C3×D3—61no—
110C22×C3×D3—61no—
111C22×C3×D3—121no—
112C2×C32×C4—21no—
113C2×C32×C4—21no—
114C2×C32×C4—21no—
115C2×C32×C4—41no—
116C2×C32×C4C22×D311yescharacteristic
117C2×C32×C4C22×D311yescharacteristic
118C2×C3⋊Dic3—21no—
119C2×C3⋊Dic3—21no—
120C2×C3⋊Dic3#6711yescharacteristic
121C2×C3⋊Dic3#6711yescharacteristic
122C2×C3×Dic3—31no—
123C2×C3×Dic3—31no—
124C2×C3×Dic3—61no—
125C2×C3×Dic3—61no—
126C3×D12—32no—
127C3×D12—32no—
128C3×D12—62no—
129C3×D12—61no—
130C3×D12—61no—
131C3×D12—121no—
132D3×Dic3—122no—
133D3×Dic3—122no—
134C3⋊(C3⋊C8)—21no—
135C3×C3⋊C8—61no—
136C3×C3⋊C8—61no—
137C3×C3⋊C8—61no—
138C3×C3⋊C8—121no—
139#478—271noSylow
140C2×H(3,𝔽3)C22⋊Q811yescharacteristic
141C2×H(3,𝔽3)#13511yescharacteristic
142C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
143#272—42no—
144C2×C3×D4—91no—
145C22×Dic3—181no—
146C2×D12—31no—
147C2×D12—31no—
148C2×D12—61no—
149C2×C4×D3—181no—
150C2×C4×D3—181no—
151C2×C3×C8—91no—
152C2×C3×C8—91no—
153C3×C4⋊C4—31no—
154C3×C4⋊C4—31no—
155C3×C4⋊C4—61no—
156C3×C4⋊C4—91no—
157C3×C4⋊C4—91no—
158C3×C4⋊C4—181no—
159C3×C4⋊C4—181no—
160C3×C4⋊C4D3211yescharacteristic
161C3×C42—91no—
162#217—181no—
163(C2×D3)⋊C4—181no—
164(C2×D3)⋊C4—181no—
165C4⋊Dic3—31no—
166C4⋊Dic3—31no—
167C4⋊Dic3—61no—
168C4⋊Dic3—91no—
169C4⋊Dic3—91no—
170C4⋊Dic3—181no—
171Dic3⋊C4—61no—
172Dic3⋊C4—181no—
173C4×Dic3—31no—
174C4×Dic3—91no—
175C2×C3⋊C8—31no—
176C2×C3⋊C8—31no—
177C2×C3⋊C8—31no—
178C2×C3⋊C8—61no—
179C2×C3⋊C8—91no—
180C22×C32—21no—
181C22×C32D3×D411yescharacteristic
182C22×C32C2×#6711yescharacteristic
183C2×C3×D3—62no—
184C2×C3×D3—62no—
185C2×C3×D3—122no—
186C2×C3×D3—121no—
187C2×C3×D3—121no—
188C2×C3×D3—121no—
189C2×C3×D3—121no—
190C2×C3×D3—241no—
191C2×C3×D3—241no—
192C32×C4—21no—
193C32×C4—21no—
194C32×C4—41no—
195C32×C4—41no—
196C32×C4—81no—
197C32×C4#23511yescharacteristic
198C32×C4D3×Q811yescharacteristic
199C32×C4D3×Q811yescharacteristic
200C32×C4#23911yescharacteristic
201C3⋊Dic3—21no—
202C3⋊Dic3—21no—
203C3⋊Dic3—41no—
204C3⋊Dic3—41no—
205C3×Dic3—61no—
206C3×Dic3—61no—
207C3×Dic3—121no—
208C3×Dic3—121no—
209#125—91no—
210C4×D4—271no—
211#107—271no—
212#106—91no—
213C4⋊C8—271no—
214#102—91no—
215#102—271no—
216H(3,𝔽3)#47811yescharacteristic, complemented, Sylow
217C23×C3—181no—
218C22×D3—61no—
219C22×D3—61no—
220C22×D3—121no—
221C3×D4—92no—
222C3×D4—181no—
223C2×C3×C4—21no—
224C2×C3×C4—31no—
225C2×C3×C4—31no—
226C2×C3×C4—61no—
227C2×C3×C4—61no—
228C2×C3×C4—61no—
229C2×C3×C4—91no—
230C2×C3×C4—91no—
231C2×C3×C4—121no—
232C2×C3×C4—181no—
233C2×C3×C4—181no—
234C2×C3×C4C2×D3211yescharacteristic
235C2×Dic3—31no—
236C2×Dic3—31no—
237C2×Dic3—61no—
238C2×Dic3—61no—
239C2×Dic3—91no—
240C2×Dic3—91no—
241C2×Dic3—181no—
242C2×Dic3—181no—
243C2×Dic3—362no—
244D12—32no—
245D12—32no—
246D12—62no—
247D12—61no—
248D12—61no—
249D12—121no—
250C4×D3—362no—
251C4×D3—362no—
252C3×C8—181no—
253C3×C8—181no—
254C3⋊C8—61no—
255C3⋊C8—61no—
256C3⋊C8—61no—
257C3⋊C8—121no—
258C3⋊C8—181no—
259C2×C32—21no—
260C2×C32—21no—
261C2×C32—21no—
262C2×C32#89611yescharacteristic
263C2×C32#91411yescharacteristic
264C2×C32#91611yescharacteristic
265C2×C32#94611yescharacteristic
266C2×C32#94911yescharacteristic
267C2×C32#95011yescharacteristic
268C3×D3—122no—
269C3×D3—122no—
270C3×D3—242no—
271C2×D4—91no—
272C22×C4—541no—
273C2×C8—91no—
274C2×C8—91no—
275C4⋊C4—91no—
276C4⋊C4—91no—
277C4⋊C4—181no—
278C4⋊C4—181no—
279C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
280C22⋊C4—541no—
281C42—91no—
282C22×C3—31no—
283C22×C3—31no—
284C22×C3—61no—
285C22×C3—182no—
286C22×C3—361no—
287C22×C3—361no—
288C22×C3D3×#6711yescharacteristic
289C2×D3—62no—
290C2×D3—62no—
291C2×D3—122no—
292C2×D3—121no—
293C2×D3—121no—
294C2×D3—121no—
295C2×D3—121no—
296C2×D3—241no—
297C2×D3—241no—
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—181no—
308C3×C4—181no—
309C3×C4—241no—
310C3×C4—361no—
311C3×C4—361no—
312C3×C4#379711yescharacteristic
313C3×C4#379911yescharacteristic
314Dic3—61no—
315Dic3—61no—
316Dic3—121no—
317Dic3—121no—
318Dic3—181no—
319Dic3—181no—
320Dic3—361no—
321Dic3—361no—
322C32—21no—
323C32#487111yescharacteristic, complemented
324C32#506611yescharacteristic, complemented
325C23—181no—
326D4—92no—
327D4—181no—
328C2×C4—21no—
329C2×C4—91no—
330C2×C4—91no—
331C2×C4—181no—
332C2×C4—181no—
333C2×C4—1082no—
334C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
335C8—181no—
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—362no—
347C2×C3#6388811yescharacteristic
348C2×C3#6393511yescharacteristic
349C2×C3#6393911yescharacteristic
350D3—122no—
351D3—122no—
352D3—242no—
353C22—182no—
354C22—361no—
355C22—361no—
356C22#8867611yescharacteristic
357C4—41no—
358C4—181no—
359C4—181no—
360C4—361no—
361C4—361no—
362C4#8865111yescharacteristic
363C4#8865211yescharacteristic
364C3—31no—
365C3—31no—
366C3—61no—
367C3#1059047011yescharacteristic
368C2—362no—
369C2#1172114811yescharacteristic
370C2#1172119711yescharacteristic
371C2#1172123311yescharacteristic
subgroupG/H
centerC22#88676
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radicalC2×C4⋊#272C22
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#478
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #64099⋊C3C2semidirect
4. #63635⋊C3C2semidirect
6. #11720315C2semidirect
7. #11720311C2semidirect
21. #64099D3semidirect
25. #63635D3semidirect
216. H(3,𝔽3)#478semidirect
323. C32#4871semidirect
324. C32#5066semidirect

Maximal subgroups

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

subgroupindex
#64099⋊C32
#117211492
#117205312
#63635⋊C32
#117203242
#117203152
#117203112
#105904743
#105904693