Finite Group Databasebeta

← #13140541

Subgroups — #13140541

Order 1728.

407 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 407 conjugacy classes of subgroups — too many to draw. Only its 79 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×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic
2#11721093C211yesmaximal, characteristic, complemented
3#11720578C211yesmaximal, characteristic
4#63659⋊C3C211yesmaximal, characteristic
5#11720314C211yesmaximal, characteristic, complemented
6#11720311C211yesmaximal, characteristic, complemented
7#11720306C211yesmaximal, characteristic, complemented
8#10591479—31nomaximal
9#10591409—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C22×(C3⋊Dic3)⋊C3—21no—
12C4⋊#1084C2211yescharacteristic
13C2×#6180C2211yescharacteristic
14#3757⋊C3—21no—
15C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#88449—21no—
20#88448—21no—
21D4×C3⋊Dic3D311yescharacteristic
22C3×D4×Dic3—31no—
23#63872—31no—
24#63869—31no—
25#63659D311yescharacteristic
26C3×#832—31no—
27C3×#832—31no—
28C3×#832—31no—
29C3×#832—61no—
30#63576—31no—
31#63575—31no—
32#63573—31no—
33#63573—31no—
34#63572—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×(C3⋊Dic3)⋊C3—21no—
42C2×(C3⋊Dic3)⋊C3—21no—
43C2×(C3⋊Dic3)⋊C3—42no—
44C2×(C3⋊Dic3)⋊C3D411yescharacteristic
45C2×(C3⋊Dic3)⋊C3D411yescharacteristic
46#6180—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—21no—
55C22×C3×Dic3—61no—
56#3757—21no—
57C4⋊(C3⋊Dic3)C2×D311yescharacteristic
58C4⋊(C3⋊Dic3)C2×D311yescharacteristic
59C4×C3⋊Dic3C2×D311yescharacteristic
60C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
61C3×#217—61no—
62C3×C4⋊Dic3—31no—
63C3×C4⋊Dic3—31no—
64C3×C4⋊Dic3—31no—
65C3×C4⋊Dic3—31no—
66C3×C4⋊Dic3—61no—
67C3×C4×Dic3—31no—
68C2×C3×C3⋊C8—31no—
69C2×C3×C3⋊C8—31no—
70C2×C3×C3⋊C8—31no—
71C2×C3×C3⋊C8—61no—
72#3724—61no—
73Dic3⋊Dic3—61no—
74Dic3⋊Dic3—61no—
75Dic3⋊Dic3—61no—
76C22×H(3,𝔽3)—22no—
77C22×H(3,𝔽3)—41no—
78C22×H(3,𝔽3)—41no—
79C22×H(3,𝔽3)C2×D411yescharacteristic
80C4×H(3,𝔽3)C2×Q811yescharacteristic
81C4×H(3,𝔽3)#4111yescharacteristic
82#1084—41no—
83(C3⋊Dic3)⋊C3—21no—
84(C3⋊Dic3)⋊C3—21no—
85(C3⋊Dic3)⋊C3—41no—
86(C3⋊Dic3)⋊C3—41no—
87C3×C4×D4—91no—
88D4×Dic3—31no—
89D4×Dic3—91no—
90#890—91no—
91#890—91no—
92#870—91no—
93C3×#102—91no—
94C3×#102—91no—
95#832—31no—
96#832—31no—
97#832—31no—
98#832—61no—
99#832—91no—
100#818—91no—
101#817—91no—
102Dic3⋊C8—91no—
103Dic3⋊C8—91no—
104#808—91no—
105#808—91no—
106#807—91no—
107#807—91no—
108C4⋊(C3⋊C8)—91no—
109C23×C32—21no—
110C23×C32—21no—
111C23×C32—41no—
112C32×D4—21no—
113C32×D4—21no—
114C32×D4—22no—
115C32×D4—41no—
116C32×D4Dic612yes—
117C32×D4Dic612yes—
118C2×C32×C4—21no—
119C2×C32×C4C22×D311yescharacteristic
120C2×C32×C4C22×D311yescharacteristic
121C2×C3⋊Dic3—21no—
122C2×C3⋊Dic3—21no—
123C2×C3⋊Dic3—42no—
124C2×C3⋊Dic3#6711yescharacteristic
125C2×C3⋊Dic3#6711yescharacteristic
126C2×C3×Dic3—31no—
127C2×C3×Dic3—31no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—61no—
130C2×C3×Dic3—61no—
131C2×C3×Dic3—61no—
132C2×C3×Dic3—121no—
133C2×C3×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)—42no—
141C2×H(3,𝔽3)C22⋊Q811yescharacteristic
142C2×H(3,𝔽3)#13511yescharacteristic
143C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
144C2×C3×D4—31no—
145C2×C3×D4—31no—
146C2×C3×D4—61no—
147C2×C3×D4D3211yescharacteristic
148C22×C3×C4—181no—
149C22×Dic3—61no—
150C22×Dic3—181no—
151C2×C3×C8—91no—
152C2×C3×C8—91no—
153C3×C4⋊C4—91no—
154C3×C4⋊C4—91no—
155C3×C4⋊C4—91no—
156C3×C22⋊C4—181no—
157C3×C42—91no—
158#217—61no—
159#217—181no—
160C4⋊Dic3—31no—
161C4⋊Dic3—31no—
162C4⋊Dic3—31no—
163C4⋊Dic3—31no—
164C4⋊Dic3—61no—
165C4⋊Dic3—91no—
166C4⋊Dic3—91no—
167C4⋊Dic3—181no—
168C4⋊Dic3—181no—
169Dic3⋊C4—181no—
170Dic3⋊C4—181no—
171Dic3⋊C4—181no—
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—22no—
181C22×C32—22no—
182C22×C32—21no—
183C22×C32—42no—
184C22×C32—41no—
185C22×C32—41no—
186C22×C32—41no—
187C22×C32—41no—
188C22×C32—81no—
189C22×C32—81no—
190C22×C32D3×D411yescharacteristic
191C22×C32C2×#6711yescharacteristic
192C32×C4—21no—
193C32×C4—21no—
194C32×C4C2×Dic611yescharacteristic
195C32×C4C2×Dic611yescharacteristic
196C32×C4#23511yescharacteristic
197C32×C4#23711yescharacteristic
198C3⋊Dic3—21no—
199C3⋊Dic3—21no—
200C3⋊Dic3—41no—
201C3⋊Dic3—41no—
202C3×Dic3—61no—
203C3×Dic3—61no—
204C3×Dic3—121no—
205C3×Dic3—121no—
206C3×Dic3—121no—
207C3×Dic3—121no—
208C3×Dic3—241no—
209#125—271no—
210C4×D4—91no—
211#107—271no—
212#106—271no—
213C4⋊C8—271no—
214#102—91no—
215#102—91no—
216H(3,𝔽3)#47811yescharacteristic, complemented, Sylow
217C23×C3—21no—
218C23×C3—61no—
219C23×C3—61no—
220C23×C3—121no—
221C3×D4—21no—
222C3×D4—32no—
223C3×D4—32no—
224C3×D4—61no—
225C3×D4—61no—
226C3×D4—62no—
227C3×D4—121no—
228C3×D4#62712yes—
229C2×C3×C4—31no—
230C2×C3×C4—31no—
231C2×C3×C4—61no—
232C2×C3×C4—91no—
233C2×C3×C4—91no—
234C2×C3×C4—181no—
235C2×C3×C4—181no—
236C2×C3×C4—181no—
237C2×C3×C4—362no—
238C2×C3×C4C2×D3211yescharacteristic
239C2×Dic3—31no—
240C2×Dic3—31no—
241C2×Dic3—61no—
242C2×Dic3—61no—
243C2×Dic3—61no—
244C2×Dic3—61no—
245C2×Dic3—91no—
246C2×Dic3—91no—
247C2×Dic3—121no—
248C2×Dic3—122no—
249C2×Dic3—181no—
250C2×Dic3—181no—
251C2×Dic3—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—42no—
263C2×C32—42no—
264C2×C32—82no—
265C2×C32#87811yescharacteristic
266C2×C32#91211yescharacteristic
267C2×C32#91411yescharacteristic
268C2×C32#92411yescharacteristic
269C2×C32#93211yescharacteristic
270C2×C32#95011yescharacteristic
271C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
272C22×C4—181no—
273C2×C8—91no—
274C2×C8—91no—
275C4⋊C4—91no—
276C4⋊C4—91no—
277C4⋊C4—91no—
278C4⋊C4—541no—
279C4⋊C4—541no—
280C22⋊C4—181no—
281C42—91no—
282C22×C3—22no—
283C22×C3—31no—
284C22×C3—31no—
285C22×C3—41no—
286C22×C3—41no—
287C22×C3—62no—
288C22×C3—62no—
289C22×C3—61no—
290C22×C3—122no—
291C22×C3—121no—
292C22×C3—121no—
293C22×C3—121no—
294C22×C3—121no—
295C22×C3—241no—
296C22×C3—241no—
297C22×C3D3×#6711yescharacteristic
298C3×C4—31no—
299C3×C4—31no—
300C3×C4—31no—
301C3×C4—31no—
302C3×C4—61no—
303C3×C4—61no—
304C3×C4—181no—
305C3×C4—181no—
306C3×C4—361no—
307C3×C4—361no—
308C3×C4—361no—
309C3×C4#380411yescharacteristic
310C3×C4C2×#62711yescharacteristic
311Dic3—61no—
312Dic3—61no—
313Dic3—121no—
314Dic3—121no—
315Dic3—121no—
316Dic3—121no—
317Dic3—181no—
318Dic3—181no—
319Dic3—241no—
320Dic3—361no—
321Dic3—361no—
322C32—21no—
323C32#476611yescharacteristic, complemented
324C32#500911yescharacteristic, complemented
325C23—21no—
326D4—21no—
327D4#619612yes—
328C2×C4—91no—
329C2×C4—91no—
330C2×C4—181no—
331C2×C4—181no—
332C2×C4—181no—
333C2×C4—362no—
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—42no—
344C2×C3—61no—
345C2×C3—61no—
346C2×C3—61no—
347C2×C3—122no—
348C2×C3—122no—
349C2×C3—242no—
350C2×C3#6393311yescharacteristic
351C2×C3#6393411yescharacteristic
352C2×C3#6396411yescharacteristic
353C22—22no—
354C22—41no—
355C22—41no—
356C22#8867611yescharacteristic
357C4—181no—
358C4—181no—
359C4—361no—
360C4—361no—
361C4—361no—
362C4C2×#619611yescharacteristic
363C4#8867111yescharacteristic
364C3—31no—
365C3—31no—
366C3—61no—
367C3#1059141111yescharacteristic
368C2—42no—
369C2#1172119611yescharacteristic
370C2#1172120111yescharacteristic
371C2#1172129911yescharacteristic
subgroupG/H
centerC22#88676
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
Fitting, prime radicalC2×D4×H(3,𝔽3)C22
socleC22×C3D3×#67
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. #11721093C2semidirect
5. #11720314C2semidirect
6. #11720311C2semidirect
7. #11720306C2semidirect
216. H(3,𝔽3)#478semidirect
323. C32#4766semidirect
324. C32#5009semidirect

Maximal subgroups

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

subgroupindex
D4×(C3⋊Dic3)⋊C32
#117210932
#117205782
#63659⋊C32
#117203142
#117203112
#117203062
#105914793
#105914093