Finite Group Databasebeta

← #13140943

Subgroups — #13140943

Order 1728.

410 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 67 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#11721863C211yesmaximal, characteristic
2#11721089C211yesmaximal, characteristic
3#63662⋊C3C211yesmaximal, characteristic, complemented
4#63662⋊C3C211yesmaximal, characteristic, complemented
5#11720323C211yesmaximal, characteristic
6#11720307C211yesmaximal, characteristic, complemented
7#11720306C211yesmaximal, characteristic, complemented
8#10591559—31nomaximal
9#10591558—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×C4×#272C2211yescharacteristic
12#88542—21no—
13C4⋊#1084C2211yescharacteristic
14#88540—21no—
15C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#88451—21no—
20#88447—21no—
21C3×#946—31no—
22C3×#946—31no—
23C3×#946—61no—
24#63841—31no—
25#63841—31no—
26#63662D311yescharacteristic, complemented
27#63662D311yescharacteristic, complemented
28C3×#835—31no—
29C3×#835—31no—
30#63576—31no—
31#63575—31no—
32#63574—31no—
33#63573—31no—
34(C2×D3)⋊(C3⋊C8)—31no—
35(C2×D3)⋊(C3⋊C8)—31no—
36C22×#272D411yescharacteristic
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×#1084D411yescharacteristic
43C4×#272—21no—
44C4×#272—21no—
45C2×(C3⋊Dic3)⋊C3—21no—
46C2×(C3⋊Dic3)⋊C3—21no—
47(C3⋊(C3⋊C8))⋊C3—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×C3×C4×D3—31no—
56C2×C3×C4×D3—31no—
57C2×C3×C4×D3—61no—
58C4⋊(C3⋊Dic3)C2×D311yescharacteristic
59C4⋊(C3⋊Dic3)C2×D311yescharacteristic
60C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
61C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
62C3×(C2×D3)⋊C4—61no—
63C3×(C2×D3)⋊C4—61no—
64C3×(C2×D3)⋊C4—62no—
65C3×C4⋊Dic3—31no—
66C3×C4⋊Dic3—31no—
67C3×C4⋊Dic3—31no—
68C3×C4⋊Dic3—31no—
69C3×C4⋊Dic3—61no—
70C3×Dic3⋊C4—61no—
71C3×Dic3⋊C4—61no—
72C3×Dic3⋊C4—62no—
73C2×C3×C3⋊C8—31no—
74C2×C3×C3⋊C8—31no—
75(C2×D3)⋊Dic3—61no—
76(C2×D3)⋊Dic3—61no—
77Dic32—61no—
78Dic32—61no—
79C22×H(3,𝔽3)C2×D411yescharacteristic
80C2×#272—21no—
81C2×#272—21no—
82C2×#272—21no—
83C4×H(3,𝔽3)—41no—
84C4×H(3,𝔽3)#4111yescharacteristic
85C4×H(3,𝔽3)#4111yescharacteristic
86#1084—21no—
87#1084—41no—
88(C3⋊Dic3)⋊C3—41no—
89(C3⋊Dic3)⋊C3—41no—
90C3×C22⋊Q8—91no—
91#946—31no—
92#946—31no—
93#946—61no—
94#926—91no—
95#892—91no—
96#892—91no—
97C3×#103—91no—
98C3×#103—91no—
99#835—31no—
100#835—31no—
101#835—91no—
102#835—91no—
103#830—91no—
104(C2×D3)⋊C8—91no—
105(C2×D3)⋊C8—91no—
106#818—91no—
107#817—91no—
108#808—91no—
109#808—91no—
110#807—91no—
111#807—91no—
112C22×C3×D3—31no—
113C22×C3×D3—31no—
114C22×C3×D3—61no—
115C32×Q8—21no—
116C32×Q8—21no—
117C32×Q8—21no—
118C32×Q8—21no—
119C32×Q8—41no—
120C32×Q8—41no—
121C2×C32×C4—22no—
122C2×C32×C4—21no—
123C2×C32×C4—21no—
124C2×C32×C4—21no—
125C2×C32×C4C22×D311yescharacteristic
126C2×C32×C4C22×D311yescharacteristic
127C2×C3⋊Dic3—21no—
128C2×C3⋊Dic3—21no—
129C2×C3×Dic3—31no—
130C2×C3×Dic3—31no—
131C2×C3×Dic3—61no—
132C2×C3×Dic3—62no—
133C2×C3×Dic3—61no—
134C2×C3×Dic3—61no—
135C2×C3×Dic3—61no—
136C2×C3×Dic3—61no—
137C3×C4×D3—61no—
138C3×C4×D3—61no—
139C3×C4×D3—61no—
140C3×C4×D3—61no—
141C3×C4×D3—121no—
142C3×C4×D3—121no—
143C3⋊(C3⋊C8)—21no—
144C3⋊(C3⋊C8)—21no—
145C3×C3⋊C8—61no—
146C3×C3⋊C8—61no—
147#485—271noSylow
148C2×H(3,𝔽3)#12311yescharacteristic
149C2×H(3,𝔽3)#13511yescharacteristic
150C2×H(3,𝔽3)#13711yescharacteristic
151#272—41no—
152C2×C3×Q8—31no—
153C2×C3×Q8—31no—
154C2×C3×Q8—61no—
155C2×C3×Q8D3211yescharacteristic
156C22×C3×C4—91no—
157C2×C4×D3—31no—
158C2×C4×D3—31no—
159C2×C4×D3—61no—
160C2×C3×C8—91no—
161C2×C3×C8—91no—
162C3×C4⋊C4—91no—
163C3×C4⋊C4—91no—
164C3×C4⋊C4—91no—
165C3×C4⋊C4—181no—
166C3×C22⋊C4—181no—
167#217—181no—
168(C2×D3)⋊C4—61no—
169(C2×D3)⋊C4—61no—
170(C2×D3)⋊C4—62no—
171(C2×D3)⋊C4—181no—
172(C2×D3)⋊C4—181no—
173C4⋊Dic3—31no—
174C4⋊Dic3—31no—
175C4⋊Dic3—31no—
176C4⋊Dic3—31no—
177C4⋊Dic3—61no—
178C4⋊Dic3—91no—
179C4⋊Dic3—91no—
180Dic3⋊C4—61no—
181Dic3⋊C4—61no—
182Dic3⋊C4—62no—
183C4×Dic3—181no—
184C4×Dic3—181no—
185C4×Dic3—181no—
186C2×C3⋊C8—31no—
187C2×C3⋊C8—31no—
188C2×C3⋊C8—91no—
189C2×C3⋊C8—91no—
190C22×C32—21no—
191C22×C32D3×D411yescharacteristic
192C22×C32D3×D411yescharacteristic
193C2×C3×D3—62no—
194C2×C3×D3—61no—
195C2×C3×D3—61no—
196C2×C3×D3—61no—
197C2×C3×D3—61no—
198C2×C3×D3—61no—
199C2×C3×D3—61no—
200C2×C3×D3—121no—
201C2×C3×D3—121no—
202C32×C4—21no—
203C32×C4—21no—
204C32×C4—42no—
205C32×C4—41no—
206C32×C4—41no—
207C32×C4#23511yescharacteristic
208C32×C4#23511yescharacteristic
209C32×C4#23711yescharacteristic
210C32×C4#23711yescharacteristic
211C3⋊Dic3—41no—
212C3⋊Dic3—41no—
213C3×Dic3—61no—
214C3×Dic3—61no—
215C3×Dic3—121no—
216C3×Dic3—122no—
217C3×Dic3—121no—
218C3×Dic3—121no—
219C3×Dic3—121no—
220C3×Dic3—121no—
221C22⋊Q8—91no—
222#117—271no—
223#107—271no—
224#106—271no—
225#103—91no—
226#103—91no—
227C22⋊C8—271no—
228H(3,𝔽3)#48511yescharacteristic, complemented, Sylow
229C23×C3—91no—
230C22×D3—31no—
231C22×D3—31no—
232C22×D3—61no—
233C3×Q8—21no—
234C3×Q8—21no—
235C3×Q8—61no—
236C3×Q8—61no—
237C3×Q8—61no—
238C3×Q8—61no—
239C3×Q8—121no—
240C3×Q8—121no—
241C2×C3×C4—21no—
242C2×C3×C4—31no—
243C2×C3×C4—31no—
244C2×C3×C4—61no—
245C2×C3×C4—61no—
246C2×C3×C4—62no—
247C2×C3×C4—61no—
248C2×C3×C4—91no—
249C2×C3×C4—181no—
250C2×C3×C4—181no—
251C2×C3×C4—181no—
252C2×C3×C4—181no—
253C2×C3×C4—181no—
254C2×C3×C4C2×D3211yescharacteristic
255C2×Dic3—31no—
256C2×Dic3—31no—
257C2×Dic3—61no—
258C2×Dic3—61no—
259C2×Dic3—61no—
260C2×Dic3—62no—
261C2×Dic3—61no—
262C2×Dic3—61no—
263C2×Dic3—181no—
264C2×Dic3—181no—
265C4×D3—61no—
266C4×D3—61no—
267C4×D3—61no—
268C4×D3—61no—
269C4×D3—121no—
270C4×D3—121no—
271C3×C8—181no—
272C3×C8—181no—
273C3⋊C8—61no—
274C3⋊C8—61no—
275C3⋊C8—181no—
276C3⋊C8—181no—
277C2×C32—21no—
278C2×C32—21no—
279C2×C32—21no—
280C2×C32#88311yescharacteristic
281C2×C32#88311yescharacteristic
282C2×C32#91511yescharacteristic
283C2×C32#91611yescharacteristic
284C2×C32#91811yescharacteristic
285C2×C32#91911yescharacteristic
286C3×D3—122no—
287C3×D3—121no—
288C3×D3—121no—
289C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
290C22×C4—91no—
291C2×C8—91no—
292C2×C8—91no—
293C4⋊C4—91no—
294C4⋊C4—91no—
295C4⋊C4—91no—
296C4⋊C4—181no—
297C22⋊C4—181no—
298C22⋊C4—541no—
299C42—541no—
300C22×C3—31no—
301C22×C3—31no—
302C22×C3—61no—
303C22×C3—181no—
304C22×C3—181no—
305C22×C3—181no—
306C22×C3#381111yescharacteristic
307C2×D3—61no—
308C2×D3—61no—
309C2×D3—61no—
310C2×D3—61no—
311C2×D3—61no—
312C2×D3—61no—
313C2×D3—62no—
314C2×D3—121no—
315C2×D3—121no—
316C3×C4—31no—
317C3×C4—31no—
318C3×C4—31no—
319C3×C4—31no—
320C3×C4—41no—
321C3×C4—61no—
322C3×C4—61no—
323C3×C4—121no—
324C3×C4—121no—
325C3×C4—122no—
326C3×C4—181no—
327C3×C4—361no—
328C3×C4—361no—
329C3×C4—361no—
330C3×C4#380411yescharacteristic
331C3×C4#380411yescharacteristic
332Dic3—61no—
333Dic3—61no—
334Dic3—121no—
335Dic3—121no—
336Dic3—121no—
337Dic3—122no—
338Dic3—121no—
339Dic3—121no—
340Dic3—361no—
341Dic3—361no—
342C32—21no—
343C32#481411yescharacteristic, complemented
344C32#481511yescharacteristic, complemented
345C23—91no—
346Q8—21no—
347Q8—21no—
348C2×C4—21no—
349C2×C4—91no—
350C2×C4—181no—
351C2×C4—181no—
352C2×C4—181no—
353C2×C4—181no—
354C2×C4—181no—
355C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
356C8—181no—
357C8—181no—
358C2×C3—31no—
359C2×C3—31no—
360C2×C3—31no—
361C2×C3—31no—
362C2×C3—31no—
363C2×C3—31no—
364C2×C3—61no—
365C2×C3—61no—
366C2×C3—61no—
367C2×C3—361no—
368C2×C3#6395011yescharacteristic
369C2×C3#6395111yescharacteristic
370C2×C3#6397511yescharacteristic
371D3—121no—
372D3—121no—
373D3—122no—
374C22—181no—
375C22—181no—
376C22—181no—
377C22#8867711yescharacteristic
378C4—41no—
379C4—181no—
380C4—361no—
381C4—361no—
382C4—361no—
383C4#8867111yescharacteristic
384C4#8867111yescharacteristic
385C3—31no—
386C3—31no—
387C3—61no—
388C3#1059156111yescharacteristic
389C2—361no—
390C2#1172123011yescharacteristic
391C2#1172123111yescharacteristic
392C2#1172131911yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×Q8×H(3,𝔽3)C22
socleC22×C3#3811
prime radicalC22×#272D4
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. #63662⋊C3C2semidirect
4. #63662⋊C3C2semidirect
6. #11720307C2semidirect
7. #11720306C2semidirect
26. #63662D3semidirect
27. #63662D3semidirect
228. H(3,𝔽3)#485semidirect
343. C32#4814semidirect
344. C32#4815semidirect

Maximal subgroups

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

subgroupindex
#117218632
#117210892
#63662⋊C32
#117203232
#117203072
#117203062
#105915593
#105915583