Finite Group Databasebeta

← #13137882

Subgroups — #13137882

Order 1728.

446 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 73 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#11721756C211yesmaximal, characteristic, complemented
2#11721136C211yesmaximal, characteristic
3#63636⋊C3C211yesmaximal, characteristic
4#63635⋊C3C211yesmaximal, characteristic, complemented
5#11720308C211yesmaximal, characteristic
6#11720307C211yesmaximal, characteristic, complemented
7#11720304C211yesmaximal, characteristic, complemented
8#10590466—31nomaximal
9#10590465—31nomaximal
10C2×#6229—21no—
11C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
12C2×((C3⋊D3)⋊C3)⋊C4—21no—
13C4⋊C4×H(3,𝔽3)C2211yescharacteristic
14C4⋊#1084C2211yescharacteristic
15#88540—21no—
16C4×#1084C2211yescharacteristic
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20#88450—21no—
21C3×#888—31no—
22C3×#888—31no—
23C3×#888—61no—
24#63894—31no—
25Dic3×D12—31no—
26#63636D311yescharacteristic
27#63635D311yescharacteristic, complemented
28C3×#809—31no—
29C3×#808—31no—
30#63574—31no—
31#63573—31no—
32Dic3⋊(C3⋊C8)—31no—
33Dic3⋊(C3⋊C8)—31no—
34#63564—31no—
35#63562—31no—
36C22×(C3⋊D3)⋊C3—21no—
37C2×C4×H(3,𝔽3)—21no—
38C2×C4×H(3,𝔽3)C2311yescharacteristic
39C2×#1084—21no—
40C2×#1084D411yescharacteristic
41C2×#1084D411yescharacteristic
42#6229—42no—
43C2×(C3⋊Dic3)⋊C3—21no—
44C4⋊((C3⋊D3)⋊C3)—21no—
45C4⋊((C3⋊D3)⋊C3)Q812yescomplemented
46((C3⋊D3)⋊C3)⋊C4—42no—
47(C3⋊(C3⋊C8))⋊C3—21no—
48(C3⋊(C3⋊C8))⋊C3—21no—
49#5084—91no—
50#4870—91no—
51#4761—91no—
52C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
53C2×C3×D12—31no—
54C2×C3×Dic6—61no—
55C2×C3×Dic6—61no—
56C2×C3×Dic6—62no—
57C2×#624—61no—
58C2×D3×Dic3—61no—
59C32×C4⋊C4—21no—
60C32×C4⋊C4C2×D311yescharacteristic
61C32×C4⋊C4C2×D311yescharacteristic
62C4⋊(C3⋊Dic3)C2×D311yescharacteristic
63C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
64C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
65C3×C4⋊Dic3—31no—
66C3×C4⋊Dic3—31no—
67C3×C4⋊Dic3—31no—
68C3×C4⋊Dic3—61no—
69C3×Dic3⋊C4—61no—
70C3×Dic3⋊C4—61no—
71C3×Dic3⋊C4—62no—
72C3×C4×Dic3—31no—
73C3×C4×Dic3—31no—
74C3×C4×Dic3—61no—
75C2×C3×C3⋊C8—31no—
76C2×C3×C3⋊C8—31no—
77#3722—61no—
78(C2×D3)⋊Dic3—61no—
79C22×H(3,𝔽3)C2×D411yescharacteristic
80C2×(C3⋊D3)⋊C3—22no—
81C2×(C3⋊D3)⋊C3—41no—
82C2×(C3⋊D3)⋊C3—41no—
83C4×H(3,𝔽3)—41no—
84C4×H(3,𝔽3)C2×Q811yescharacteristic
85C4×H(3,𝔽3)#4111yescharacteristic
86#1084—21no—
87#1084—41no—
88#1084SD1612yes—
89(C3⋊Dic3)⋊C3—41no—
90C3×C4⋊Q8—91no—
91D4×Dic3—91no—
92#893—91no—
93#888—31no—
94#888—31no—
95#888—61no—
96C4×D12—91no—
97C3×#106—91no—
98C3×#102—91no—
99#832—91no—
100#821—91no—
101#817—91no—
102Dic3⋊C8—91no—
103Dic3⋊C8—91no—
104#809—31no—
105#809—91no—
106#809—91no—
107#808—31no—
108#808—91no—
109#808—91no—
110#808—91no—
111C4⋊(C3⋊C8)—91no—
112C22×C3⋊D3—21no—
113C22×C3×D3—61no—
114C2×C32×C4—22no—
115C2×C32×C4—21no—
116C2×C32×C4—21no—
117C2×C32×C4—21no—
118C2×C32×C4C22×D311yescharacteristic
119C2×C32×C4C22×D311yescharacteristic
120C2×C3⋊Dic3—21no—
121C4⋊(C3⋊D3)—21no—
122C4⋊(C3⋊D3)Dic612yescomplemented
123C2×C3×Dic3—31no—
124C2×C3×Dic3—31no—
125C2×C3×Dic3—31no—
126C2×C3×Dic3—31no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—62no—
130C2×C3×Dic3—61no—
131C2×C3×Dic3—61no—
132C2×C3×Dic3—61no—
133C3×D12—32no—
134C3×D12—61no—
135C3×Dic6—124no—
136C3×Dic6—122no—
137C3×Dic6—122no—
138#624—122no—
139D3×Dic3—122no—
140C3⋊(C3⋊C8)—21no—
141C3⋊(C3⋊C8)—21no—
142C3×C3⋊C8—61no—
143C3×C3⋊C8—61no—
144#476—271noSylow
145C2×H(3,𝔽3)C22⋊Q811yescharacteristic
146C2×H(3,𝔽3)C2×SD1611yescharacteristic
147C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
148(C3⋊D3)⋊C3—42no—
149C2×C3×Q8—181no—
150C2×C3×D4—91no—
151C22×Dic3—181no—
152C2×D12—31no—
153C2×D12—91no—
154C2×C4×D3—181no—
155C2×C4×D3—181no—
156C2×Dic6—61no—
157C2×Dic6—61no—
158C2×Dic6—62no—
159C2×C3×C8—91no—
160C2×C3×C8—91no—
161C3×C4⋊C4—31no—
162C3×C4⋊C4—31no—
163C3×C4⋊C4—61no—
164C3×C4⋊C4—91no—
165C3×C4⋊C4—91no—
166C3×C4⋊C4—181no—
167C3×C4⋊C4D3211yescharacteristic
168C3×C42—91no—
169#217—181no—
170(C2×D3)⋊C4—181no—
171(C2×D3)⋊C4—181no—
172C4⋊Dic3—31no—
173C4⋊Dic3—31no—
174C4⋊Dic3—31no—
175C4⋊Dic3—61no—
176C4⋊Dic3—91no—
177Dic3⋊C4—61no—
178Dic3⋊C4—61no—
179Dic3⋊C4—62no—
180C4×Dic3—31no—
181C4×Dic3—31no—
182C4×Dic3—61no—
183C2×C3⋊C8—31no—
184C2×C3⋊C8—31no—
185C2×C3⋊C8—91no—
186C2×C3⋊C8—91no—
187C22×C32—21no—
188C22×C32D3×D411yescharacteristic
189C22×C32D3×D411yescharacteristic
190C2×C3⋊D3—22no—
191C2×C3⋊D3—41no—
192C2×C3⋊D3—41no—
193C2×C3×D3—62no—
194C2×C3×D3—121no—
195C2×C3×D3—121no—
196C32×C4—21no—
197C32×C4—21no—
198C32×C4—42no—
199C32×C4—41no—
200C32×C4—41no—
201C32×C4C2×Dic611yescharacteristic
202C32×C4#23511yescharacteristic
203C32×C4#23711yescharacteristic
204C32×C4D3×Q811yescharacteristic
205C3⋊Dic3—41no—
206C3×Dic3—32no—
207C3×Dic3—32no—
208C3×Dic3—62no—
209C3×Dic3—62no—
210C3×Dic3—61no—
211C3×Dic3—61no—
212C3×Dic3—122no—
213C3×Dic3—121no—
214C3×Dic3—121no—
215C3×Dic3—121no—
216C4⋊Q8—91no—
217C4×D4—271no—
218#106—91no—
219#106—271no—
220C4⋊C8—271no—
221#102—91no—
222#102—271no—
223H(3,𝔽3)#47611yescharacteristic, complemented, Sylow
224C23×C3—181no—
225C22×D3—61no—
226C22×D3—181no—
227C3×Q8—362no—
228C3×D4—92no—
229C3×D4—181no—
230C2×C3×C4—21no—
231C2×C3×C4—31no—
232C2×C3×C4—31no—
233C2×C3×C4—61no—
234C2×C3×C4—61no—
235C2×C3×C4—62no—
236C2×C3×C4—61no—
237C2×C3×C4—91no—
238C2×C3×C4—91no—
239C2×C3×C4—181no—
240C2×C3×C4—181no—
241C2×C3×C4C2×D3211yescharacteristic
242C2×Dic3—31no—
243C2×Dic3—31no—
244C2×Dic3—31no—
245C2×Dic3—31no—
246C2×Dic3—61no—
247C2×Dic3—61no—
248C2×Dic3—61no—
249C2×Dic3—61no—
250C2×Dic3—62no—
251C2×Dic3—61no—
252C2×Dic3—181no—
253C2×Dic3—362no—
254D12—32no—
255D12—61no—
256D12—92no—
257D12—181no—
258C4×D3—362no—
259C4×D3—362no—
260Dic6—122no—
261Dic6—122no—
262Dic6—124no—
263C3×C8—181no—
264C3×C8—181no—
265C3⋊C8—61no—
266C3⋊C8—61no—
267C3⋊C8—181no—
268C3⋊C8—181no—
269C2×C32—21no—
270C2×C32—21no—
271C2×C32—21no—
272C2×C32#87811yescharacteristic
273C2×C32#89611yescharacteristic
274C2×C32#91111yescharacteristic
275C2×C32D3×SD1611yescharacteristic
276C2×C32D3×SD1611yescharacteristic
277C2×C32#91411yescharacteristic
278C3⋊D3—42no—
279C3×D3—122no—
280C2×Q8—181no—
281C2×D4—91no—
282C22×C4—541no—
283C2×C8—91no—
284C2×C8—91no—
285C4⋊C4—91no—
286C4⋊C4—91no—
287C4⋊C4—181no—
288C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
289C22⋊C4—541no—
290C42—91no—
291C22×C3—31no—
292C22×C3—31no—
293C22×C3—61no—
294C22×C3—182no—
295C22×C3—361no—
296C22×C3—361no—
297C22×C3#381111yescharacteristic
298C2×D3—62no—
299C2×D3—121no—
300C2×D3—121no—
301C2×D3—182no—
302C2×D3—361no—
303C2×D3—361no—
304C3×C4—31no—
305C3×C4—31no—
306C3×C4—31no—
307C3×C4—31no—
308C3×C4—41no—
309C3×C4—61no—
310C3×C4—61no—
311C3×C4—92no—
312C3×C4—121no—
313C3×C4—121no—
314C3×C4—122no—
315C3×C4—181no—
316C3×C4—361no—
317C3×C4—361no—
318C3×C4D3×Dic611yescharacteristic
319C3×C4#379511yescharacteristic
320Dic3—32no—
321Dic3—32no—
322Dic3—62no—
323Dic3—61no—
324Dic3—61no—
325Dic3—62no—
326Dic3—121no—
327Dic3—121no—
328Dic3—122no—
329Dic3—121no—
330Dic3—361no—
331C32—21no—
332C32#476111yescharacteristic, complemented
333C32#487011yescharacteristic, complemented
334C23—181no—
335Q8—362no—
336D4—92no—
337D4—181no—
338C2×C4—21no—
339C2×C4—91no—
340C2×C4—91no—
341C2×C4—181no—
342C2×C4—181no—
343C2×C4—1082no—
344C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
345C8—181no—
346C8—181no—
347C2×C3—31no—
348C2×C3—31no—
349C2×C3—31no—
350C2×C3—31no—
351C2×C3—31no—
352C2×C3—31no—
353C2×C3—61no—
354C2×C3—61no—
355C2×C3—61no—
356C2×C3—362no—
357C2×C3#6389611yescharacteristic
358C2×C3#6393711yescharacteristic
359C2×C3#6394011yescharacteristic
360D3—122no—
361D3—362no—
362C22—182no—
363C22—361no—
364C22—361no—
365C22#8867711yescharacteristic
366C4—41no—
367C4—92no—
368C4—181no—
369C4—361no—
370C4—361no—
371C4((C3⋊D3)⋊C3)⋊Q811yescharacteristic
372C4#8864811yescharacteristic
373C3—31no—
374C3—31no—
375C3—61no—
376C3#1059047211yescharacteristic
377C2—362no—
378C2#1172113511yescharacteristic
379C2#1172119311yescharacteristic
380C2#1172122811yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3#3811
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C22
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#476
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. #11721756C2semidirect
4. #63635⋊C3C2semidirect
6. #11720307C2semidirect
7. #11720304C2semidirect
27. #63635D3semidirect
45. C4⋊((C3⋊D3)⋊C3)Q8semidirect
122. C4⋊(C3⋊D3)Dic6semidirect
223. H(3,𝔽3)#476semidirect
332. C32#4761semidirect
333. C32#4870semidirect

Maximal subgroups

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

subgroupindex
#117217562
#117211362
#63636⋊C32
#63635⋊C32
#117203082
#117203072
#117203042
#105904663
#105904653