Finite Group Databasebeta

← #13137941

Subgroups — #13137941

Order 1728.

491 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 491 conjugacy classes of subgroups — too many to draw. Only its 111 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
1C4⋊C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2#11721129C211yesmaximal, characteristic
3C2×((C3⋊D3)⋊C3)⋊C8C211yesmaximal, characteristic
4#11720529C211yesmaximal, characteristic, complemented
5#63634⋊C3C211yesmaximal, characteristic, complemented
6#11720315C211yesmaximal, characteristic, complemented
7#11720306C211yesmaximal, characteristic, complemented
8#10590573—31nomaximal
9D3×#807—31nomaximal
10C2×C4×(C3⋊D3)⋊C3—21no—
11C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
12C2×((C3⋊D3)⋊C3)⋊C4—21no—
13C4⋊C4×H(3,𝔽3)C2211yescharacteristic
14C4⋊#1084C2211yescharacteristic
15C2×#6180C2211yescharacteristic
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18#3750⋊C3—21no—
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20#88448—21no—
21((C3⋊D3)⋊C3)⋊C8C414yescomplemented
22C4⋊C4×C3⋊D3D311yescharacteristic
23C3×D3×C4⋊C4—31no—
24#63900—31no—
25D3×C4⋊Dic3—31no—
26C2×#3710—31no—
27C2×D3×C3⋊C8—31no—
28#63634D311yescharacteristic, complemented
29C3×#807—31no—
30C3×#807—31no—
31C3×#807—31no—
32C3×#807—61no—
33#63576—31no—
34#63575—31no—
35#63575—31no—
36#63575—31no—
37C22×(C3⋊D3)⋊C3D411yescharacteristic
38C2×C4×H(3,𝔽3)—21no—
39C2×C4×H(3,𝔽3)C2311yescharacteristic
40C2×#1084—21no—
41C2×(C3⋊Dic3)⋊C3—21no—
42C2×(C3⋊Dic3)⋊C3—21no—
43C2×(C3⋊Dic3)⋊C3D411yescharacteristic
44C4×(C3⋊D3)⋊C3—42no—
45C4×(C3⋊D3)⋊C3C2×C412yes—
46C4×(C3⋊D3)⋊C3Q812yes—
47((C3⋊D3)⋊C3)⋊C4—42no—
48#6180C2×C412yescomplemented
49(C3⋊(C3⋊C8))⋊C3C2×C412yescomplemented
50C2×#807—91no—
51D3×#107—91no—
52D3×#107—91no—
53C2×C4×C3⋊D3—21no—
54C2×C4×C3⋊D3C2×D311yescharacteristic
55C2×C3×C4×D3—31no—
56C2×C3×C4×D3—61no—
57C2×#624—61no—
58C2×D3×Dic3—61no—
59C32×C4⋊C4—21no—
60C32×C4⋊C4C2×D311yescharacteristic
61C32×C4⋊C4C2×D311yescharacteristic
62C4⋊(C3⋊Dic3)C2×D311yescharacteristic
63C4⋊(C3⋊Dic3)C2×D311yescharacteristic
64#3750—21no—
65C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
66C3×C4⋊Dic3—31no—
67C3×C4⋊Dic3—31no—
68C3×C4⋊Dic3—31no—
69C3×C4⋊Dic3—31no—
70C3×C4⋊Dic3—61no—
71C3×Dic3⋊C4—61no—
72C2×C3×C3⋊C8—31no—
73C2×C3×C3⋊C8—31no—
74C2×C3×C3⋊C8—31no—
75C2×C3×C3⋊C8—61no—
76#3724—61no—
77Dic3⋊Dic3—61no—
78#3710—34no—
79D3×C3⋊C8—34no—
80C22×H(3,𝔽3)C2×D411yescharacteristic
81C2×(C3⋊D3)⋊C3C4⋊C412yes—
82C2×(C3⋊D3)⋊C3D812yes—
83C2×(C3⋊D3)⋊C3Q1612yes—
84C4×H(3,𝔽3)—41no—
85C4×H(3,𝔽3)C22×C411yescharacteristic
86C4×H(3,𝔽3)C2×Q811yescharacteristic
87#1084—41no—
88(C3⋊Dic3)⋊C3—41no—
89(C3⋊Dic3)⋊C3—41no—
90(C3⋊Dic3)⋊C3C4⋊C412yes—
91C2×C3×C4⋊C4—91no—
92C2×C4⋊Dic3—91no—
93C22×C3⋊C8—91no—
94C2×D3×C8—91no—
95C2×D3×C8—91no—
96D3×C4⋊C4—31no—
97D3×C4⋊C4—91no—
98D3×C4⋊C4—91no—
99D3×C4⋊C4—91no—
100C3×#107—91no—
101C3×#107—91no—
102#818—91no—
103#818—91no—
104#807—31no—
105#807—31no—
106#807—31no—
107#807—61no—
108#807—91no—
109#807—91no—
110#807—91no—
111#807—91no—
112#807—91no—
113C22×C3⋊D3#6711yescharacteristic
114C22×C3×D3—31no—
115C2×C32×C4—21no—
116C2×C32×C4—21no—
117C2×C32×C4—21no—
118C2×C32×C4—41no—
119C2×C32×C4C22×D311yescharacteristic
120C2×C32×C4C22×D311yescharacteristic
121C2×C3⋊Dic3—21no—
122C2×C3⋊Dic3—21no—
123C2×C3⋊Dic3#6711yescharacteristic
124C4×C3⋊D3—42no—
125C4×C3⋊D3Dic612yes—
126C4×C3⋊D3C4×D312yes—
127C2×C3×Dic3—31no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—61no—
130C2×C3×Dic3—61no—
131C2×C3×Dic3—61no—
132C2×C3×Dic3—121no—
133C3×C4×D3—32no—
134C3×C4×D3—32no—
135C3×C4×D3—122no—
136#624—122no—
137D3×Dic3—122no—
138C3⋊(C3⋊C8)C4×D312yescomplemented
139C3×C3⋊C8—32no—
140C3×C3⋊C8—32no—
141C3×C3⋊C8—32no—
142C3×C3⋊C8—62no—
143C2×#107—271noSylow
144C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
145C2×H(3,𝔽3)C2×D811yescharacteristic
146C2×H(3,𝔽3)C2×Q1611yescharacteristic
147(C3⋊D3)⋊C3#10714yescomplemented
148C22×C3×C4—91no—
149C22×C3×C4—181no—
150C22×Dic3—181no—
151C2×C4×D3—31no—
152C2×C4×D3—61no—
153C2×C4×D3—91no—
154C2×C4×D3—181no—
155C2×C4×D3—181no—
156C2×C4×D3—181no—
157C2×C3×C8—91no—
158C2×C3×C8—91no—
159C3×C4⋊C4—31no—
160C3×C4⋊C4—31no—
161C3×C4⋊C4—61no—
162C3×C4⋊C4—91no—
163C3×C4⋊C4—91no—
164C3×C4⋊C4—91no—
165C3×C4⋊C4—181no—
166C3×C4⋊C4D3211yescharacteristic
167C4⋊Dic3—31no—
168C4⋊Dic3—31no—
169C4⋊Dic3—31no—
170C4⋊Dic3—31no—
171C4⋊Dic3—61no—
172C4⋊Dic3—91no—
173C4⋊Dic3—91no—
174C4⋊Dic3—181no—
175Dic3⋊C4—61no—
176Dic3⋊C4—181no—
177Dic3⋊C4—181no—
178Dic3⋊C4—181no—
179C2×C3⋊C8—31no—
180C2×C3⋊C8—31no—
181C2×C3⋊C8—31no—
182C2×C3⋊C8—61no—
183C2×C3⋊C8—91no—
184C2×C3⋊C8—94no—
185D3×C8—94no—
186D3×C8—94no—
187C22×C32—21no—
188C22×C32D3×D411yescharacteristic
189C22×C32C2×#6711yescharacteristic
190C2×C3⋊D3Dic3⋊C412yes—
191C2×C3⋊D3#21312yes—
192C2×C3⋊D3#21612yes—
193C2×C3×D3—32no—
194C2×C3×D3—32no—
195C2×C3×D3—32no—
196C32×C4—21no—
197C32×C4—21no—
198C32×C4—41no—
199C32×C4—41no—
200C32×C4—81no—
201C32×C4C2×Dic611yescharacteristic
202C32×C4C2×C4×D311yescharacteristic
203C32×C4C2×C4×D311yescharacteristic
204C32×C4D3×Q811yescharacteristic
205C3⋊Dic3—41no—
206C3⋊Dic3—41no—
207C3⋊Dic3Dic3⋊C412yes—
208C3×Dic3—32no—
209C3×Dic3—121no—
210C3×Dic3—121no—
211C3×Dic3—121no—
212C3×Dic3—121no—
213C3×Dic3—241no—
214C22×C8—271no—
215C2×C4⋊C4—91no—
216C2×C4⋊C4—271no—
217#107—91no—
218#107—91no—
219#107—271no—
220#107—271no—
221H(3,𝔽3)C2×#10711yescharacteristic, complemented, Sylow
222C23×C3—91no—
223C22×D3—31no—
224C22×D3—91no—
225C2×C3×C4—21no—
226C2×C3×C4—31no—
227C2×C3×C4—31no—
228C2×C3×C4—61no—
229C2×C3×C4—61no—
230C2×C3×C4—61no—
231C2×C3×C4—92no—
232C2×C3×C4—92no—
233C2×C3×C4—91no—
234C2×C3×C4—121no—
235C2×C3×C4—181no—
236C2×C3×C4—181no—
237C2×C3×C4—181no—
238C2×C3×C4—362no—
239C2×C3×C4C2×D3211yescharacteristic
240C2×Dic3—31no—
241C2×Dic3—61no—
242C2×Dic3—61no—
243C2×Dic3—61no—
244C2×Dic3—61no—
245C2×Dic3—91no—
246C2×Dic3—121no—
247C2×Dic3—181no—
248C2×Dic3—181no—
249C2×Dic3—362no—
250C4×D3—32no—
251C4×D3—32no—
252C4×D3—92no—
253C4×D3—92no—
254C4×D3—122no—
255C4×D3—362no—
256C4×D3—362no—
257C4×D3—362no—
258C3×C8—92no—
259C3×C8—92no—
260C3⋊C8—32no—
261C3⋊C8—32no—
262C3⋊C8—32no—
263C3⋊C8—62no—
264C3⋊C8—92no—
265C2×C32—21no—
266C2×C32—21no—
267C2×C32—21no—
268C2×C32D3×C4⋊C411yescharacteristic
269C2×C32D3×D811yescharacteristic
270C2×C32D3×Q1611yescharacteristic
271C2×C32C2×Dic3⋊C411yescharacteristic
272C2×C32C2×#21311yescharacteristic
273C2×C32C2×#21611yescharacteristic
274C3⋊D3#80714yescomplemented
275C3×D3—34no—
276C22×C4—91no—
277C22×C4—181no—
278C22×C4—541no—
279C2×C8—91no—
280C2×C8—91no—
281C2×C8—274no—
282C4⋊C4—91no—
283C4⋊C4—91no—
284C4⋊C4—91no—
285C4⋊C4—181no—
286C4⋊C4—541no—
287C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
288C22×C3—31no—
289C22×C3—31no—
290C22×C3—61no—
291C22×C3—92no—
292C22×C3—92no—
293C22×C3—92no—
294C22×C3D3×#6711yescharacteristic
295C2×D3—32no—
296C2×D3—32no—
297C2×D3—32no—
298C2×D3—92no—
299C2×D3—92no—
300C2×D3—92no—
301C3×C4—31no—
302C3×C4—31no—
303C3×C4—31no—
304C3×C4—31no—
305C3×C4—41no—
306C3×C4—61no—
307C3×C4—61no—
308C3×C4—92no—
309C3×C4—121no—
310C3×C4—121no—
311C3×C4—241no—
312C3×C4—361no—
313C3×C4—361no—
314C3×C4—361no—
315C3×C4D3×Dic611yescharacteristic
316C3×C4C4×D3211yescharacteristic
317Dic3—32no—
318Dic3—92no—
319Dic3—121no—
320Dic3—121no—
321Dic3—121no—
322Dic3—121no—
323Dic3—241no—
324Dic3—361no—
325Dic3—361no—
326C32—21no—
327C32D3×#10711yescharacteristic, complemented
328C32C2×#80711yescharacteristic, complemented
329C23—91no—
330C2×C4—21no—
331C2×C4—91no—
332C2×C4—92no—
333C2×C4—92no—
334C2×C4—181no—
335C2×C4—181no—
336C2×C4—181no—
337C2×C4—362no—
338C2×C4—1082no—
339C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
340C8—92no—
341C8—92no—
342C2×C3—31no—
343C2×C3—31no—
344C2×C3—31no—
345C2×C3—31no—
346C2×C3—31no—
347C2×C3—31no—
348C2×C3—61no—
349C2×C3—61no—
350C2×C3—61no—
351C2×C3—94no—
352C2×C3D3×Dic3⋊C411yescharacteristic
353C2×C3D3×#21311yescharacteristic
354C2×C3D3×#21611yescharacteristic
355D3—34no—
356D3—94no—
357C22—92no—
358C22—92no—
359C22—92no—
360C22#8867611yescharacteristic
361C4—41no—
362C4—92no—
363C4—361no—
364C4—361no—
365C4—361no—
366C4((C3⋊D3)⋊C3)⋊Q811yescharacteristic
367C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
368C3—31no—
369C3—31no—
370C3—61no—
371C3D3×#80711yescharacteristic
372C2—94no—
373C2#1172111911yescharacteristic
374C2#1172119411yescharacteristic
375C2#1172123611yescharacteristic
subgroupG/H
centerC22#88676
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radicalC22×(C3⋊D3)⋊C3D4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#107
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
4. #11720529C2semidirect
5. #63634⋊C3C2semidirect
6. #11720315C2semidirect
7. #11720306C2semidirect
21. ((C3⋊D3)⋊C3)⋊C8C4semidirect
28. #63634D3semidirect
48. #6180C2×C4semidirect
49. (C3⋊(C3⋊C8))⋊C3C2×C4semidirect
138. C3⋊(C3⋊C8)C4×D3semidirect
147. (C3⋊D3)⋊C3#107semidirect
221. H(3,𝔽3)C2×#107semidirect
274. C3⋊D3#807semidirect
327. C32D3×#107semidirect
328. C32C2×#807semidirect

Maximal subgroups

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

subgroupindex
C4⋊C4×(C3⋊D3)⋊C32
#117211292
C2×((C3⋊D3)⋊C3)⋊C82
#117205292
#63634⋊C32
#117203152
#117203062
#105905733
D3×#8073