Finite Group Databasebeta

← #13140909

Subgroups — #13140909

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#64155⋊C3C211yesmaximal, characteristic
2#11721081C211yesmaximal, characteristic
3#11720583C211yesmaximal, characteristic, complemented
4#63662⋊C3C211yesmaximal, characteristic, complemented
5#11720317C211yesmaximal, characteristic
6#11720314C211yesmaximal, characteristic, complemented
7#11720306C211yesmaximal, characteristic, complemented
8#10591561—31nomaximal
9#10591559—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
12C4⋊#1084C2211yescharacteristic
13C2×#6180C2211yescharacteristic
14#3752⋊C3—21no—
15C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17#3750⋊C3—21no—
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#88450—21no—
20#88447—21no—
21#64155D311yescharacteristic
22C3×#946—31no—
23#63849—31no—
24#63841—31no—
25#63662D311yescharacteristic, complemented
26C3×#835—31no—
27C3×#835—31no—
28C3×#835—31no—
29C3×#835—61no—
30#63576—31no—
31#63575—31no—
32#63573—31no—
33#63573—31no—
34#63557—31no—
35(C2×D3)⋊(C3⋊C8)—31no—
36C22×(C3⋊D3)⋊C3D411yescharacteristic
37Q8×H(3,𝔽3)—21no—
38Q8×H(3,𝔽3)—21no—
39C2×C4×H(3,𝔽3)—21no—
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×#1084—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—
47#6180—21no—
48(C3⋊(C3⋊C8))⋊C3—21no—
49#5045—91no—
50#4815—91no—
51#4814—91no—
52C2×C32×Q8—21no—
53C2×C32×Q8C2×D311yescharacteristic
54C2×C32×Q8C2×D311yescharacteristic
55C2×C4×C3⋊D3C2×D311yescharacteristic
56C2×C3×C4×D3—31no—
57#3752—21no—
58C4⋊(C3⋊Dic3)C2×D311yescharacteristic
59C4⋊(C3⋊Dic3)C2×D311yescharacteristic
60#3750—21no—
61C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
62C3×(C2×D3)⋊C4—61no—
63C3×C4⋊Dic3—31no—
64C3×C4⋊Dic3—31no—
65C3×C4⋊Dic3—31no—
66C3×C4⋊Dic3—31no—
67C3×C4⋊Dic3—61no—
68C3×Dic3⋊C4—61no—
69C2×C3×C3⋊C8—31no—
70C2×C3×C3⋊C8—31no—
71C2×C3×C3⋊C8—31no—
72C2×C3×C3⋊C8—61no—
73#3722—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)—41no—
82C4×H(3,𝔽3)#4111yescharacteristic
83C4×H(3,𝔽3)#4111yescharacteristic
84#1084—41no—
85(C3⋊Dic3)⋊C3—21no—
86(C3⋊Dic3)⋊C3—41no—
87(C3⋊Dic3)⋊C3—41no—
88C3×C22⋊Q8—91no—
89#946—31no—
90#946—91no—
91#926—91no—
92#892—91no—
93#892—91no—
94C3×#103—91no—
95C3×#103—91no—
96#835—31no—
97#835—31no—
98#835—31no—
99#835—61no—
100#835—91no—
101#830—91no—
102(C2×D3)⋊C8—91no—
103(C2×D3)⋊C8—91no—
104#818—91no—
105#817—91no—
106#808—91no—
107#808—91no—
108#807—91no—
109#807—91no—
110C22×C3⋊D3#6711yescharacteristic
111C22×C3×D3—31no—
112C32×Q8—22no—
113C32×Q8—21no—
114C32×Q8—21no—
115C32×Q8—21no—
116C32×Q8—21no—
117C32×Q8—41no—
118C2×C32×C4—21no—
119C2×C32×C4—21no—
120C2×C32×C4—21no—
121C2×C32×C4—41no—
122C2×C32×C4C22×D311yescharacteristic
123C2×C32×C4C22×D311yescharacteristic
124C2×C3⋊Dic3—21no—
125C2×C3⋊Dic3—21no—
126C2×C3⋊Dic3#6711yescharacteristic
127C4×C3⋊D3—21no—
128C4×C3⋊D3—21no—
129C2×C3×Dic3—31no—
130C2×C3×Dic3—61no—
131C2×C3×Dic3—61no—
132C2×C3×Dic3—61no—
133C2×C3×Dic3—61no—
134C2×C3×Dic3—121no—
135C3×C4×D3—61no—
136C3×C4×D3—61no—
137C3⋊(C3⋊C8)—21no—
138C3×C3⋊C8—61no—
139C3×C3⋊C8—61no—
140C3×C3⋊C8—61no—
141C3×C3⋊C8—121no—
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—31no—
148C2×C3×Q8—31no—
149C2×C3×Q8—61no—
150C2×C3×Q8D3211yescharacteristic
151C22×C3×C4—91no—
152C2×C4×D3—31no—
153C2×C4×D3—91no—
154C2×C3×C8—91no—
155C2×C3×C8—91no—
156C3×C4⋊C4—91no—
157C3×C4⋊C4—91no—
158C3×C4⋊C4—91no—
159C3×C4⋊C4—181no—
160C3×C22⋊C4—181no—
161#217—181no—
162(C2×D3)⋊C4—61no—
163(C2×D3)⋊C4—181no—
164(C2×D3)⋊C4—181no—
165(C2×D3)⋊C4—181no—
166C4⋊Dic3—31no—
167C4⋊Dic3—31no—
168C4⋊Dic3—31no—
169C4⋊Dic3—31no—
170C4⋊Dic3—61no—
171C4⋊Dic3—91no—
172C4⋊Dic3—91no—
173Dic3⋊C4—61no—
174Dic3⋊C4—181no—
175C4×Dic3—181no—
176C4×Dic3—181no—
177C4×Dic3—181no—
178C2×C3⋊C8—31no—
179C2×C3⋊C8—31no—
180C2×C3⋊C8—31no—
181C2×C3⋊C8—61no—
182C2×C3⋊C8—91no—
183C22×C32—21no—
184C22×C32D3×D411yescharacteristic
185C22×C32C2×#6711yescharacteristic
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—42no—
195C32×C4—41no—
196C32×C4—41no—
197C32×C4#23511yescharacteristic
198C32×C4#23711yescharacteristic
199C32×C4#23711yescharacteristic
200C32×C4#23711yescharacteristic
201C3⋊Dic3—21no—
202C3⋊Dic3—41no—
203C3⋊Dic3—41no—
204C3×Dic3—61no—
205C3×Dic3—121no—
206C3×Dic3—121no—
207C3×Dic3—121no—
208C3×Dic3—121no—
209C3×Dic3—241no—
210C22⋊Q8—91no—
211#117—271no—
212#107—271no—
213#106—271no—
214#103—91no—
215#103—91no—
216C22⋊C8—271no—
217H(3,𝔽3)#48511yescharacteristic, complemented, Sylow
218C23×C3—91no—
219C22×D3—31no—
220C22×D3—91no—
221C3×Q8—21no—
222C3×Q8—21no—
223C3×Q8—61no—
224C3×Q8—61no—
225C3×Q8—61no—
226C3×Q8—61no—
227C3×Q8—62no—
228C3×Q8—121no—
229C2×C3×C4—21no—
230C2×C3×C4—31no—
231C2×C3×C4—31no—
232C2×C3×C4—61no—
233C2×C3×C4—61no—
234C2×C3×C4—61no—
235C2×C3×C4—91no—
236C2×C3×C4—121no—
237C2×C3×C4—181no—
238C2×C3×C4—181no—
239C2×C3×C4—181no—
240C2×C3×C4—181no—
241C2×C3×C4—181no—
242C2×C3×C4C2×D3211yescharacteristic
243C2×Dic3—31no—
244C2×Dic3—61no—
245C2×Dic3—61no—
246C2×Dic3—61no—
247C2×Dic3—61no—
248C2×Dic3—91no—
249C2×Dic3—121no—
250C2×Dic3—181no—
251C2×Dic3—181no—
252C4×D3—61no—
253C4×D3—61no—
254C4×D3—181no—
255C4×D3—181no—
256C3×C8—181no—
257C3×C8—181no—
258C3⋊C8—61no—
259C3⋊C8—61no—
260C3⋊C8—61no—
261C3⋊C8—121no—
262C3⋊C8—181no—
263C2×C32—21no—
264C2×C32—21no—
265C2×C32—21no—
266C2×C32#88311yescharacteristic
267C2×C32#91511yescharacteristic
268C2×C32#91911yescharacteristic
269C2×C32#93511yescharacteristic
270C2×C32#94211yescharacteristic
271C2×C32#95011yescharacteristic
272C3⋊D3—41no—
273C3×D3—121no—
274C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
275C22×C4—91no—
276C2×C8—91no—
277C2×C8—91no—
278C4⋊C4—91no—
279C4⋊C4—91no—
280C4⋊C4—91no—
281C4⋊C4—181no—
282C22⋊C4—181no—
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×#6711yescharacteristic
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—41no—
303C3×C4—61no—
304C3×C4—61no—
305C3×C4—121no—
306C3×C4—121no—
307C3×C4—122no—
308C3×C4—181no—
309C3×C4—361no—
310C3×C4—361no—
311C3×C4—361no—
312C3×C4#380411yescharacteristic
313C3×C4#380511yescharacteristic
314Dic3—61no—
315Dic3—121no—
316Dic3—121no—
317Dic3—121no—
318Dic3—121no—
319Dic3—181no—
320Dic3—241no—
321Dic3—361no—
322Dic3—361no—
323C32—21no—
324C32#481411yescharacteristic, complemented
325C32#504511yescharacteristic, complemented
326C23—91no—
327Q8—21no—
328Q8—21no—
329C2×C4—21no—
330C2×C4—91no—
331C2×C4—181no—
332C2×C4—181no—
333C2×C4—181no—
334C2×C4—181no—
335C2×C4—181no—
336C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
337C8—181no—
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#6394611yescharacteristic
350C2×C3#6394911yescharacteristic
351C2×C3#6396911yescharacteristic
352D3—121no—
353D3—361no—
354C22—181no—
355C22—181no—
356C22—181no—
357C22#8867611yescharacteristic
358C4—41no—
359C4—181no—
360C4—361no—
361C4—361no—
362C4—361no—
363C4#8867111yescharacteristic
364C4#8867211yescharacteristic
365C3—31no—
366C3—31no—
367C3—61no—
368C3#1059155811yescharacteristic
369C2—361no—
370C2#1172123511yescharacteristic
371C2#1172123811yescharacteristic
372C2#1172131111yescharacteristic
subgroupG/H
centerC22#88676
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×Q8×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radicalC22×(C3⋊D3)⋊C3D4
hypercenterC2×Q8((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×Q8

Complements

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

NH ≅ G/Ntype
3. #11720583C2semidirect
4. #63662⋊C3C2semidirect
6. #11720314C2semidirect
7. #11720306C2semidirect
25. #63662D3semidirect
217. H(3,𝔽3)#485semidirect
324. C32#4814semidirect
325. C32#5045semidirect

Maximal subgroups

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

subgroupindex
#64155⋊C32
#117210812
#117205832
#63662⋊C32
#117203172
#117203142
#117203062
#105915613
#105915593