Finite Group Databasebeta

← #13140524

Subgroups — #13140524

Order 1728.

451 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 451 conjugacy classes of subgroups — too many to draw. Only its 77 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
1D4×#1084C211yesmaximal, characteristic
2#11721095C211yesmaximal, characteristic, complemented
3#63659⋊C3C211yesmaximal, characteristic
4#63659⋊C3C211yesmaximal, characteristic
5#11720307C211yesmaximal, characteristic, complemented
6#11720306C211yesmaximal, characteristic, complemented
7#11720304C211yesmaximal, characteristic, complemented
8#10591411—31nomaximal
9#10591409—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C22×#1084—21no—
12#88552—21no—
13C4⋊#1084C2211yescharacteristic
14C4×#1084C2211yescharacteristic
15C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#88448—21no—
20#88448—21no—
21C3×D4×Dic3—31no—
22C3×D4×Dic3—31no—
23C3×D4×Dic3—61no—
24#63869—31no—
25#63869—31no—
26#63659D311yescharacteristic
27#63659D311yescharacteristic
28C3×#832—31no—
29C3×#832—31no—
30#63576—31no—
31#63575—31no—
32#63574—31no—
33#63573—31no—
34Dic3⋊(C3⋊C8)—31no—
35Dic3⋊(C3⋊C8)—31no—
36C23×H(3,𝔽3)—21no—
37D4×H(3,𝔽3)—21no—
38D4×H(3,𝔽3)Q812yes—
39C2×C4×H(3,𝔽3)C2311yescharacteristic
40C2×#1084—21no—
41C2×#1084—42no—
42C2×#1084D411yescharacteristic
43C2×#1084D411yescharacteristic
44C2×(C3⋊Dic3)⋊C3—21no—
45C2×(C3⋊Dic3)⋊C3—21no—
46(C3⋊(C3⋊C8))⋊C3—21no—
47(C3⋊(C3⋊C8))⋊C3—21no—
48#5009—91no—
49#4766—91no—
50#4763—91no—
51C2×C32×D4—21no—
52C2×C32×D4C2×D311yescharacteristic
53C2×C32×D4C2×D311yescharacteristic
54C22×C3×Dic3—61no—
55C22×C3×Dic3—61no—
56C22×C3×Dic3—62no—
57C4⋊(C3⋊Dic3)C2×D311yescharacteristic
58C4⋊(C3⋊Dic3)C2×D311yescharacteristic
59C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
60C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
61C3×#217—61no—
62C3×#217—61no—
63C3×#217—62no—
64C3×C4⋊Dic3—31no—
65C3×C4⋊Dic3—31no—
66C3×C4⋊Dic3—31no—
67C3×C4⋊Dic3—31no—
68C3×C4⋊Dic3—61no—
69C3×C4×Dic3—31no—
70C3×C4×Dic3—31no—
71C3×C4×Dic3—61no—
72C2×C3×C3⋊C8—31no—
73C2×C3×C3⋊C8—31no—
74#3724—61no—
75#3724—61no—
76Dic3⋊Dic3—61no—
77Dic3⋊Dic3—61no—
78C22×H(3,𝔽3)—22no—
79C22×H(3,𝔽3)—41no—
80C22×H(3,𝔽3)—41no—
81C22×H(3,𝔽3)C2×D411yescharacteristic
82C4×H(3,𝔽3)C2×Q811yescharacteristic
83C4×H(3,𝔽3)#4111yescharacteristic
84#1084—21no—
85#1084—21no—
86#1084—41no—
87(C3⋊Dic3)⋊C3—41no—
88(C3⋊Dic3)⋊C3—41no—
89C3×C4×D4—91no—
90D4×Dic3—31no—
91D4×Dic3—31no—
92D4×Dic3—61no—
93#890—91no—
94#890—91no—
95#870—91no—
96C3×#102—91no—
97C3×#102—91no—
98#832—31no—
99#832—31no—
100#832—91no—
101#832—91no—
102#818—91no—
103#817—91no—
104Dic3⋊C8—91no—
105Dic3⋊C8—91no—
106#808—91no—
107#808—91no—
108#807—91no—
109#807—91no—
110C4⋊(C3⋊C8)—91no—
111C23×C32—22no—
112C23×C32—21no—
113C23×C32—21no—
114C32×D4—22no—
115C32×D4—21no—
116C32×D4—21no—
117C32×D4—22no—
118C32×D4Dic612yes—
119C32×D4Dic612yes—
120C2×C32×C4—21no—
121C2×C32×C4C22×D311yescharacteristic
122C2×C32×C4C22×D311yescharacteristic
123C2×C3⋊Dic3—21no—
124C2×C3⋊Dic3—21no—
125C2×C3×Dic3—31no—
126C2×C3×Dic3—31no—
127C2×C3×Dic3—31no—
128C2×C3×Dic3—31no—
129C2×C3×Dic3—61no—
130C2×C3×Dic3—61no—
131C2×C3×Dic3—62no—
132C2×C3×Dic3—61no—
133C2×C3×Dic3—61no—
134C2×C3×Dic3—61no—
135C2×C3×Dic3—61no—
136C2×C3×Dic3—124no—
137C2×C3×Dic3—122no—
138C2×C3×Dic3—122no—
139C3⋊(C3⋊C8)—21no—
140C3⋊(C3⋊C8)—21no—
141C3×C3⋊C8—61no—
142C3×C3⋊C8—61no—
143#478—271noSylow
144C2×H(3,𝔽3)—42no—
145C2×H(3,𝔽3)C22⋊Q811yescharacteristic
146C2×H(3,𝔽3)#13511yescharacteristic
147C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
148C2×C3×D4—31no—
149C2×C3×D4—31no—
150C2×C3×D4—61no—
151C2×C3×D4D3211yescharacteristic
152C22×C3×C4—181no—
153C22×Dic3—61no—
154C22×Dic3—61no—
155C22×Dic3—62no—
156C2×C3×C8—91no—
157C2×C3×C8—91no—
158C3×C4⋊C4—91no—
159C3×C4⋊C4—91no—
160C3×C4⋊C4—91no—
161C3×C22⋊C4—181no—
162C3×C42—91no—
163#217—61no—
164#217—61no—
165#217—62no—
166C4⋊Dic3—31no—
167C4⋊Dic3—31no—
168C4⋊Dic3—31no—
169C4⋊Dic3—31no—
170C4⋊Dic3—61no—
171C4⋊Dic3—91no—
172C4⋊Dic3—91no—
173C4⋊Dic3—181no—
174C4⋊Dic3—181no—
175Dic3⋊C4—181no—
176Dic3⋊C4—181no—
177Dic3⋊C4—181no—
178Dic3⋊C4—181no—
179C4×Dic3—31no—
180C4×Dic3—31no—
181C4×Dic3—61no—
182C2×C3⋊C8—31no—
183C2×C3⋊C8—31no—
184C2×C3⋊C8—91no—
185C2×C3⋊C8—91no—
186C22×C32—24no—
187C22×C32—22no—
188C22×C32—22no—
189C22×C32—21no—
190C22×C32—42no—
191C22×C32—42no—
192C22×C32—41no—
193C22×C32—41no—
194C22×C32—41no—
195C22×C32—41no—
196C22×C32D3×D411yescharacteristic
197C22×C32D3×D411yescharacteristic
198C32×C4—21no—
199C32×C4—21no—
200C32×C4C2×Dic611yescharacteristic
201C32×C4C2×Dic611yescharacteristic
202C32×C4#23711yescharacteristic
203C32×C4#23711yescharacteristic
204C3⋊Dic3—41no—
205C3⋊Dic3—41no—
206C3×Dic3—62no—
207C3×Dic3—61no—
208C3×Dic3—61no—
209C3×Dic3—61no—
210C3×Dic3—61no—
211C3×Dic3—121no—
212C3×Dic3—122no—
213C3×Dic3—121no—
214C3×Dic3—121no—
215C3×Dic3—121no—
216C3×Dic3—121no—
217#125—271no—
218C4×D4—91no—
219#107—271no—
220#106—271no—
221C4⋊C8—271no—
222#102—91no—
223#102—91no—
224H(3,𝔽3)#47811yescharacteristic, complemented, Sylow
225C23×C3—21no—
226C23×C3—61no—
227C23×C3—61no—
228C23×C3—62no—
229C3×D4—21no—
230C3×D4—32no—
231C3×D4—32no—
232C3×D4—61no—
233C3×D4—61no—
234C3×D4—62no—
235C3×D4—62no—
236C3×D4#62712yes—
237C2×C3×C4—31no—
238C2×C3×C4—31no—
239C2×C3×C4—61no—
240C2×C3×C4—91no—
241C2×C3×C4—91no—
242C2×C3×C4—181no—
243C2×C3×C4—181no—
244C2×C3×C4—181no—
245C2×C3×C4—362no—
246C2×C3×C4C2×D3211yescharacteristic
247C2×Dic3—31no—
248C2×Dic3—31no—
249C2×Dic3—31no—
250C2×Dic3—31no—
251C2×Dic3—61no—
252C2×Dic3—61no—
253C2×Dic3—61no—
254C2×Dic3—61no—
255C2×Dic3—62no—
256C2×Dic3—61no—
257C2×Dic3—61no—
258C2×Dic3—122no—
259C2×Dic3—122no—
260C2×Dic3—124no—
261C2×Dic3—181no—
262C2×Dic3—181no—
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—44no—
273C2×C32—42no—
274C2×C32—42no—
275C2×C32#87811yescharacteristic
276C2×C32#87811yescharacteristic
277C2×C32#91111yescharacteristic
278C2×C32#91211yescharacteristic
279C2×C32#91411yescharacteristic
280C2×C32#91611yescharacteristic
281C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
282C22×C4—181no—
283C2×C8—91no—
284C2×C8—91no—
285C4⋊C4—91no—
286C4⋊C4—91no—
287C4⋊C4—91no—
288C4⋊C4—541no—
289C4⋊C4—541no—
290C22⋊C4—181no—
291C42—91no—
292C22×C3—22no—
293C22×C3—31no—
294C22×C3—31no—
295C22×C3—41no—
296C22×C3—41no—
297C22×C3—62no—
298C22×C3—62no—
299C22×C3—64no—
300C22×C3—61no—
301C22×C3—121no—
302C22×C3—121no—
303C22×C3—121no—
304C22×C3—121no—
305C22×C3—122no—
306C22×C3—122no—
307C22×C3#381111yescharacteristic
308C3×C4—31no—
309C3×C4—31no—
310C3×C4—31no—
311C3×C4—31no—
312C3×C4—61no—
313C3×C4—61no—
314C3×C4—181no—
315C3×C4—181no—
316C3×C4—361no—
317C3×C4—361no—
318C3×C4—361no—
319C3×C4#380511yescharacteristic
320C3×C4C2×#62711yescharacteristic
321Dic3—61no—
322Dic3—61no—
323Dic3—61no—
324Dic3—61no—
325Dic3—62no—
326Dic3—121no—
327Dic3—121no—
328Dic3—121no—
329Dic3—122no—
330Dic3—121no—
331Dic3—121no—
332Dic3—361no—
333Dic3—361no—
334C32—21no—
335C32#476311yescharacteristic, complemented
336C32#476611yescharacteristic, complemented
337C23—21no—
338D4—21no—
339D4#619612yes—
340C2×C4—91no—
341C2×C4—91no—
342C2×C4—181no—
343C2×C4—181no—
344C2×C4—181no—
345C2×C4—362no—
346C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
347C8—181no—
348C8—181no—
349C2×C3—31no—
350C2×C3—31no—
351C2×C3—31no—
352C2×C3—31no—
353C2×C3—31no—
354C2×C3—31no—
355C2×C3—42no—
356C2×C3—61no—
357C2×C3—61no—
358C2×C3—61no—
359C2×C3—122no—
360C2×C3—122no—
361C2×C3—124no—
362C2×C3#6393611yescharacteristic
363C2×C3#6393711yescharacteristic
364C2×C3#6398211yescharacteristic
365C22—22no—
366C22—41no—
367C22—41no—
368C22#8867711yescharacteristic
369C4—181no—
370C4—181no—
371C4—361no—
372C4—361no—
373C4—361no—
374C4C2×#619611yescharacteristic
375C4#8867211yescharacteristic
376C3—31no—
377C3—31no—
378C3—61no—
379C3#1059147911yescharacteristic
380C2—42no—
381C2#1172119211yescharacteristic
382C2#1172119311yescharacteristic
383C2#1172130911yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
Fitting, prime radicalC2×D4×H(3,𝔽3)C22
socleC22×C3#3811
hypercenterC2×D4((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 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #11721095C2semidirect
5. #11720307C2semidirect
6. #11720306C2semidirect
7. #11720304C2semidirect
224. H(3,𝔽3)#478semidirect
335. C32#4763semidirect
336. C32#4766semidirect

Maximal subgroups

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

subgroupindex
D4×#10842
#117210952
#63659⋊C32
#117203072
#117203062
#117203042
#105914113
#105914093