Finite Group Databasebeta

← #13140955

Subgroups — #13140955

Order 1728.

495 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 495 conjugacy classes of subgroups — too many to draw. Only its 91 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
1C2×#88549C211yesmaximal, characteristic
2Q8×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
3#11721149C211yesmaximal, characteristic
4#63662⋊C3C211yesmaximal, characteristic, complemented
5#11720324C211yesmaximal, characteristic
6#11720314C211yesmaximal, characteristic, complemented
7#11720310C211yesmaximal, characteristic, complemented
8#10591571—31nomaximal
9Dic3×#215—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×C4⋊#272C2211yescharacteristic
12C2×#6199—21no—
13#88549C414yescomplemented
14C2×#6180C2211yescharacteristic
15C4⋊((C3⋊Dic3)⋊C3)—21no—
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18C4×(C3⋊Dic3)⋊C3—21no—
19C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#88451—21no—
22Q8×C3⋊Dic3D311yescharacteristic, complemented
23C3×Q8×Dic3—31no—
24C2×C3×#215—31no—
25C2×C3×#215—31no—
26C2×C3×#215—61no—
27#63898—31no—
28Dic3×D12—31no—
29#63662D311yescharacteristic, complemented
30C3×#835—31no—
31#63573—31no—
32#63573—31no—
33#63563—31no—
34#63562—31no—
35#63553—31no—
36Dic3×C3⋊C8—31no—
37C22×#272—21no—
38Q8×H(3,𝔽3)—21no—
39Q8×H(3,𝔽3)C2×C412yescomplemented
40C2×C4×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42C4⋊#272—21no—
43C4⋊#272C2×C412yes—
44C2×(C3⋊Dic3)⋊C3—21no—
45C2×(C3⋊Dic3)⋊C3—21no—
46C2×(C3⋊Dic3)⋊C3D411yescharacteristic
47C2×(C3⋊Dic3)⋊C3D411yescharacteristic
48#6199—42no—
49#6180C2×C412yescomplemented
50(C3⋊(C3⋊C8))⋊C3—21no—
51Dic3×SD16—91no—
52C4×#215—91no—
53#4788—91no—
54C2×C32×Q8—21no—
55C2×C32×Q8C2×D311yescharacteristic
56C2×C32×Q8C2×D311yescharacteristic
57C2×C3×D12—31no—
58C2×C3×D12—31no—
59C2×C3×D12—61no—
60C2×D3×Dic3—61no—
61C2×D3×Dic3—61no—
62C4⋊(C3⋊Dic3)—21no—
63C4⋊(C3⋊Dic3)C2×D311yescharacteristic
64C4⋊(C3⋊Dic3)C2×D311yescharacteristic
65C4×C3⋊Dic3—21no—
66C4×C3⋊Dic3C2×D311yescharacteristic
67C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
68C3×#215—34no—
69C3×#215—34no—
70C3×#215—64no—
71C3×C4⋊Dic3—31no—
72C3×C4⋊Dic3—31no—
73C3×C4⋊Dic3—61no—
74C3×C4×Dic3—31no—
75C3×C4×Dic3—61no—
76C2×C3×C3⋊C8—31no—
77C2×C3×C3⋊C8—31no—
78C2×C3×C3⋊C8—31no—
79C2×C3×C3⋊C8—61no—
80(C2×D3)⋊Dic3—61no—
81(C2×D3)⋊Dic3—61no—
82C22×H(3,𝔽3)C2×D411yescharacteristic
83C2×#272—22no—
84C2×#272—41no—
85C2×#272—41no—
86C4×H(3,𝔽3)—22no—
87C4×H(3,𝔽3)C22×C411yescharacteristic
88C4×H(3,𝔽3)#4111yescharacteristic
89(C3⋊Dic3)⋊C3—21no—
90(C3⋊Dic3)⋊C3—41no—
91(C3⋊Dic3)⋊C3—41no—
92(C3⋊Dic3)⋊C3SD1612yescomplemented
93C2×C3×SD16—91no—
94C3×C4×Q8—91no—
95Q8×Dic3—31no—
96Q8×Dic3—91no—
97C2×#215—31no—
98C2×#215—31no—
99C2×#215—61no—
100D4×Dic3—91no—
101#893—91no—
102C4×D12—91no—
103C3×#103—91no—
104#835—31no—
105#835—91no—
106#832—91no—
107#821—91no—
108#817—91no—
109C8×Dic3—91no—
110C8×Dic3—91no—
111#809—91no—
112#808—91no—
113#808—91no—
114C4×C3⋊C8—91no—
115C22×C3×D3—61no—
116C22×C3×D3—61no—
117C22×C3×D3—121no—
118C32×Q8—21no—
119C32×Q8—21no—
120C32×Q8—22no—
121C32×Q8—41no—
122C32×Q8C4×D312yescomplemented
123C32×Q8C4×D312yescomplemented
124C2×C32×C4—21no—
125C2×C32×C4—21no—
126C2×C32×C4—21no—
127C2×C32×C4—41no—
128C2×C32×C4C22×D311yescharacteristic
129C2×C32×C4C22×D311yescharacteristic
130C2×C3⋊Dic3—21no—
131C2×C3⋊Dic3—21no—
132C2×C3⋊Dic3#6711yescharacteristic
133C2×C3⋊Dic3#6711yescharacteristic
134C2×C3×Dic3—31no—
135C2×C3×Dic3—31no—
136C2×C3×Dic3—61no—
137C2×C3×Dic3—61no—
138C3×D12—32no—
139C3×D12—32no—
140C3×D12—62no—
141C3×D12—61no—
142C3×D12—61no—
143C3×D12—121no—
144D3×Dic3—122no—
145D3×Dic3—122no—
146C3⋊(C3⋊C8)—21no—
147C3×C3⋊C8—32no—
148C3×C3⋊C8—32no—
149C3×C3⋊C8—62no—
150C3×C3⋊C8—61no—
151C4×SD16—271noSylow
152C2×H(3,𝔽3)C4×D411yescharacteristic
153C2×H(3,𝔽3)C2×SD1611yescharacteristic
154C2×H(3,𝔽3)#13511yescharacteristic
155#272—42no—
156C2×C3×Q8—31no—
157C2×C3×Q8—31no—
158C2×C3×Q8—61no—
159C2×C3×Q8D3211yescharacteristic
160C2×C3×D4—91no—
161C22×Dic3—181no—
162C2×D12—31no—
163C2×D12—31no—
164C2×D12—61no—
165C2×C4×D3—181no—
166C2×C4×D3—181no—
167C3×SD16—94no—
168C2×C3×C8—91no—
169C2×C3×C8—91no—
170C3×C4⋊C4—91no—
171C3×C4⋊C4—91no—
172C3×C4⋊C4—181no—
173C3×C42—91no—
174C3×C42—181no—
175#217—181no—
176#215—34no—
177#215—34no—
178#215—64no—
179(C2×D3)⋊C4—181no—
180(C2×D3)⋊C4—181no—
181C4⋊Dic3—31no—
182C4⋊Dic3—31no—
183C4⋊Dic3—61no—
184C4⋊Dic3—91no—
185C4⋊Dic3—91no—
186C4⋊Dic3—181no—
187C4×Dic3—31no—
188C4×Dic3—61no—
189C4×Dic3—91no—
190C4×Dic3—181no—
191C2×C3⋊C8—31no—
192C2×C3⋊C8—31no—
193C2×C3⋊C8—31no—
194C2×C3⋊C8—61no—
195C2×C3⋊C8—91no—
196C22×C32—21no—
197C22×C32D3×D411yescharacteristic
198C22×C32C2×#6711yescharacteristic
199C2×C3×D3—62no—
200C2×C3×D3—62no—
201C2×C3×D3—122no—
202C2×C3×D3—121no—
203C2×C3×D3—121no—
204C2×C3×D3—121no—
205C2×C3×D3—121no—
206C2×C3×D3—241no—
207C2×C3×D3—241no—
208C32×C4—22no—
209C32×C4—22no—
210C32×C4—21no—
211C32×C4—21no—
212C32×C4—42no—
213C32×C4C2×C4×D311yescharacteristic
214C32×C4C2×C4×D311yescharacteristic
215C32×C4#23511yescharacteristic
216C32×C4#23711yescharacteristic
217C3⋊Dic3—21no—
218C3⋊Dic3—41no—
219C3⋊Dic3—41no—
220C3⋊Dic3#21512yescomplemented
221C3×Dic3—32no—
222C3×Dic3—61no—
223C3×Dic3—121no—
224C3×Dic3—121no—
225C2×SD16—91no—
226C4×Q8—91no—
227C4×D4—271no—
228#106—271no—
229#103—91no—
230#102—271no—
231C4×C8—271no—
232H(3,𝔽3)C4×SD1611yescharacteristic, complemented, Sylow
233C23×C3—181no—
234C22×D3—61no—
235C22×D3—61no—
236C22×D3—121no—
237C3×Q8—21no—
238C3×Q8—32no—
239C3×Q8—32no—
240C3×Q8—61no—
241C3×Q8—61no—
242C3×Q8—62no—
243C3×Q8—121no—
244C3×Q8#62412yes—
245C3×D4—92no—
246C3×D4—181no—
247C2×C3×C4—21no—
248C2×C3×C4—31no—
249C2×C3×C4—31no—
250C2×C3×C4—61no—
251C2×C3×C4—61no—
252C2×C3×C4—61no—
253C2×C3×C4—91no—
254C2×C3×C4—91no—
255C2×C3×C4—121no—
256C2×C3×C4—181no—
257C2×C3×C4—181no—
258C2×C3×C4C2×D3211yescharacteristic
259C2×Dic3—31no—
260C2×Dic3—31no—
261C2×Dic3—61no—
262C2×Dic3—61no—
263C2×Dic3—91no—
264C2×Dic3—91no—
265C2×Dic3—181no—
266C2×Dic3—181no—
267C2×Dic3—362no—
268D12—32no—
269D12—32no—
270D12—62no—
271D12—61no—
272D12—61no—
273D12—121no—
274C4×D3—362no—
275C4×D3—362no—
276C3×C8—92no—
277C3×C8—181no—
278C3⋊C8—32no—
279C3⋊C8—32no—
280C3⋊C8—62no—
281C3⋊C8—61no—
282C3⋊C8—181no—
283C2×C32—21no—
284C2×C32—21no—
285C2×C32—21no—
286C2×C32#67⋊C411yescharacteristic
287C2×C32D3×SD1611yescharacteristic
288C2×C32#91911yescharacteristic
289C2×C32C4×#6711yescharacteristic
290C2×C32C2×#21511yescharacteristic
291C2×C32#95011yescharacteristic
292C3×D3—122no—
293C3×D3—122no—
294C3×D3—242no—
295C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
296C2×D4—91no—
297C22×C4—541no—
298SD16—94no—
299C2×C8—91no—
300C2×C8—91no—
301C4⋊C4—91no—
302C4⋊C4—91no—
303C4⋊C4—181no—
304C22⋊C4—541no—
305C42—91no—
306C42—181no—
307C22×C3—31no—
308C22×C3—31no—
309C22×C3—61no—
310C22×C3—182no—
311C22×C3—361no—
312C22×C3—361no—
313C22×C3D3×#6711yescharacteristic
314C2×D3—62no—
315C2×D3—62no—
316C2×D3—122no—
317C2×D3—121no—
318C2×D3—121no—
319C2×D3—121no—
320C2×D3—121no—
321C2×D3—241no—
322C2×D3—241no—
323C3×C4—22no—
324C3×C4—31no—
325C3×C4—31no—
326C3×C4—31no—
327C3×C4—31no—
328C3×C4—62no—
329C3×C4—62no—
330C3×C4—61no—
331C3×C4—61no—
332C3×C4—92no—
333C3×C4—122no—
334C3×C4—181no—
335C3×C4—361no—
336C3×C4—361no—
337C3×C4#380411yescharacteristic
338C3×C4C2×#62411yescharacteristic
339Dic3—32no—
340Dic3—61no—
341Dic3—92no—
342Dic3—121no—
343Dic3—121no—
344Dic3—181no—
345Dic3—361no—
346Dic3—361no—
347C32—21no—
348C32#478811yescharacteristic, complemented
349C32C4×#21511yescharacteristic, complemented
350C23—181no—
351Q8—21no—
352Q8#619912yescomplemented
353D4—92no—
354D4—181no—
355C2×C4—21no—
356C2×C4—91no—
357C2×C4—91no—
358C2×C4—181no—
359C2×C4—181no—
360C2×C4—1082no—
361C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
362C8—92no—
363C8—181no—
364C2×C3—31no—
365C2×C3—31no—
366C2×C3—31no—
367C2×C3—31no—
368C2×C3—31no—
369C2×C3—31no—
370C2×C3—61no—
371C2×C3—61no—
372C2×C3—61no—
373C2×C3—362no—
374C2×C3D3×#21511yescharacteristic
375C2×C3#6394911yescharacteristic
376C2×C3#6395211yescharacteristic
377D3—122no—
378D3—122no—
379D3—242no—
380C22—182no—
381C22—361no—
382C22—361no—
383C22#8867611yescharacteristic
384C4—22no—
385C4—92no—
386C4—181no—
387C4—361no—
388C4—361no—
389C4#8867111yescharacteristic
390C4C2×#619911yescharacteristic
391C3—31no—
392C3—31no—
393C3—61no—
394C3#1059153611yescharacteristic
395C2—362no—
396C2#1172123211yescharacteristic
397C2#1172123511yescharacteristic
398C2#1172129311yescharacteristic
subgroupG/H
centerC22#88676
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×Q8×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radicalC2×C4⋊#272C22
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×Q8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C4×SD16
LCSG ▷ 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
2. Q8×(C3⋊Dic3)⋊C3C2semidirect
4. #63662⋊C3C2semidirect
6. #11720314C2semidirect
7. #11720310C2semidirect
13. #88549C4semidirect
22. Q8×C3⋊Dic3D3semidirect
29. #63662D3semidirect
39. Q8×H(3,𝔽3)C2×C4semidirect
49. #6180C2×C4semidirect
92. (C3⋊Dic3)⋊C3SD16semidirect
122. C32×Q8C4×D3semidirect
123. C32×Q8C4×D3semidirect
220. C3⋊Dic3#215semidirect
232. H(3,𝔽3)C4×SD16semidirect
348. C32#4788semidirect
349. C32C4×#215semidirect
352. Q8#6199semidirect

Maximal subgroups

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

subgroupindex
C2×#885492
Q8×(C3⋊Dic3)⋊C32
#117211492
#63662⋊C32
#117203242
#117203142
#117203102
#105915713
Dic3×#2153