Finite Group Databasebeta

← #13137958

Subgroups — #13137958

Order 1728.

475 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 475 conjugacy classes of subgroups — too many to draw. Only its 87 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#11721760C211yesmaximal, characteristic
2#11721089C211yesmaximal, characteristic
3C2×#88434C211yesmaximal, characteristic
4#63635⋊C3C211yesmaximal, characteristic, complemented
5#63634⋊C3C211yesmaximal, characteristic, complemented
6#11720307C211yesmaximal, characteristic, complemented
7#11720306C211yesmaximal, characteristic, complemented
8#10590572—31nomaximal
9#10590571—31nomaximal
10C2×C4×#272C2211yescharacteristic
11C4⋊C4×H(3,𝔽3)C2211yescharacteristic
12#88542—21no—
13C4⋊#1084C2211yescharacteristic
14C4×#1084—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—
21#88434C414yescomplemented
22C3×#892—31no—
23C3×#892—31no—
24C3×#892—61no—
25#63841—31no—
26#63841—31no—
27C2×D3×C3⋊C8—31no—
28C2×D3×C3⋊C8—31no—
29#63635D311yescharacteristic, complemented
30#63634D311yescharacteristic, complemented
31C3×#808—31no—
32C3×#807—31no—
33#63576—31no—
34#63575—31no—
35#63574—31no—
36#63573—31no—
37C22×#272D411yescharacteristic
38C2×C4×H(3,𝔽3)—21no—
39C2×C4×H(3,𝔽3)C2311yescharacteristic
40C2×#1084—21no—
41C2×#1084D411yescharacteristic
42C4×#272C2×C412yes—
43C4×#272Q812yes—
44C2×(C3⋊Dic3)⋊C3—21no—
45C2×(C3⋊Dic3)⋊C3—21no—
46(C3⋊(C3⋊C8))⋊C3C2×C412yes—
47(C3⋊(C3⋊C8))⋊C3C2×C412yes—
48#4998—91no—
49#4880—91no—
50#4860—91no—
51C2×C3×C4×D3—31no—
52C2×C3×C4×D3—31no—
53C2×C3×C4×D3—61no—
54C32×C4⋊C4—21no—
55C32×C4⋊C4C2×D311yescharacteristic
56C32×C4⋊C4C2×D311yescharacteristic
57C4⋊(C3⋊Dic3)C2×D311yescharacteristic
58C4⋊(C3⋊Dic3)C2×D311yescharacteristic
59C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
60C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
61C3×(C2×D3)⋊C4—61no—
62C3×(C2×D3)⋊C4—61no—
63C3×(C2×D3)⋊C4—62no—
64C3×C4⋊Dic3—31no—
65C3×C4⋊Dic3—31no—
66C3×C4⋊Dic3—31no—
67C3×C4⋊Dic3—31no—
68C3×C4⋊Dic3—61no—
69C3×C4×Dic3—61no—
70C3×C4×Dic3—61no—
71C3×C4×Dic3—62no—
72C2×C3×C3⋊C8—31no—
73C2×C3×C3⋊C8—31no—
74(C2×D3)⋊Dic3—61no—
75(C2×D3)⋊Dic3—61no—
76Dic32—61no—
77Dic32—61no—
78D3×C3⋊C8—34no—
79D3×C3⋊C8—34no—
80C22×H(3,𝔽3)C2×D411yescharacteristic
81C2×#272—21no—
82C2×#272—21no—
83C2×#272C4⋊C412yes—
84C4×H(3,𝔽3)—41no—
85C4×H(3,𝔽3)C22×C411yescharacteristic
86C4×H(3,𝔽3)C2×Q811yescharacteristic
87#1084—41no—
88#1084C4⋊C412yes—
89(C3⋊Dic3)⋊C3—41no—
90(C3⋊Dic3)⋊C3—41no—
91C3×#117—91no—
92#926—91no—
93C22×C3⋊C8—91no—
94C2×D3×C8—91no—
95C2×D3×C8—91no—
96#892—31no—
97#892—31no—
98#892—61no—
99#892—91no—
100#892—91no—
101C3×#107—91no—
102C3×#106—91no—
103#818—91no—
104#817—91no—
105#808—31no—
106#808—91no—
107#808—91no—
108#808—91no—
109#807—31no—
110#807—91no—
111#807—91no—
112#807—91no—
113C22×C3×D3—31no—
114C22×C3×D3—31no—
115C22×C3×D3—61no—
116C2×C32×C4—22no—
117C2×C32×C4—21no—
118C2×C32×C4—21no—
119C2×C32×C4—21no—
120C2×C32×C4C22×D311yescharacteristic
121C2×C32×C4C22×D311yescharacteristic
122C2×C3⋊Dic3—21no—
123C2×C3⋊Dic3—21no—
124C2×C3×Dic3—31no—
125C2×C3×Dic3—31no—
126C2×C3×Dic3—61no—
127C2×C3×Dic3—62no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—61no—
130C2×C3×Dic3—61no—
131C2×C3×Dic3—61no—
132C3×C4×D3—32no—
133C3×C4×D3—32no—
134C3×C4×D3—32no—
135C3×C4×D3—32no—
136C3×C4×D3—62no—
137C3×C4×D3—62no—
138C3⋊(C3⋊C8)C4×D312yes—
139C3⋊(C3⋊C8)C4×D312yes—
140C3×C3⋊C8—32no—
141C3×C3⋊C8—32no—
142#427—271noSylow
143C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
144C2×H(3,𝔽3)#13511yescharacteristic
145C2×H(3,𝔽3)#13511yescharacteristic
146#272—22no—
147C22×C3×C4—91no—
148C2×C4×D3—31no—
149C2×C4×D3—31no—
150C2×C4×D3—61no—
151C2×C3×C8—91no—
152C2×C3×C8—91no—
153C3×C4⋊C4—31no—
154C3×C4⋊C4—31no—
155C3×C4⋊C4—61no—
156C3×C4⋊C4—91no—
157C3×C4⋊C4—91no—
158C3×C4⋊C4—91no—
159C3×C4⋊C4D3211yescharacteristic
160C3×C22⋊C4—181no—
161C3×C42—181no—
162#217—181no—
163(C2×D3)⋊C4—61no—
164(C2×D3)⋊C4—61no—
165(C2×D3)⋊C4—62no—
166(C2×D3)⋊C4—181no—
167(C2×D3)⋊C4—181no—
168C4⋊Dic3—31no—
169C4⋊Dic3—31no—
170C4⋊Dic3—31no—
171C4⋊Dic3—31no—
172C4⋊Dic3—61no—
173C4⋊Dic3—91no—
174C4⋊Dic3—91no—
175C4×Dic3—61no—
176C4×Dic3—61no—
177C4×Dic3—62no—
178C4×Dic3—181no—
179C4×Dic3—181no—
180C4×Dic3—181no—
181C2×C3⋊C8—31no—
182C2×C3⋊C8—31no—
183C2×C3⋊C8—91no—
184C2×C3⋊C8—91no—
185C2×C3⋊C8—94no—
186D3×C8—94no—
187D3×C8—94no—
188C22×C32—21no—
189C22×C32D3×D411yescharacteristic
190C22×C32D3×D411yescharacteristic
191C2×C3×D3—32no—
192C2×C3×D3—32no—
193C2×C3×D3—62no—
194C2×C3×D3—61no—
195C2×C3×D3—61no—
196C2×C3×D3—61no—
197C2×C3×D3—61no—
198C2×C3×D3—121no—
199C2×C3×D3—121no—
200C32×C4—21no—
201C32×C4—21no—
202C32×C4—42no—
203C32×C4—41no—
204C32×C4—41no—
205C32×C4C2×C4×D311yescharacteristic
206C32×C4C2×C4×D311yescharacteristic
207C32×C4D3×Q811yescharacteristic
208C32×C4D3×Q811yescharacteristic
209C3⋊Dic3—41no—
210C3⋊Dic3—41no—
211C3×Dic3—32no—
212C3×Dic3—32no—
213C3×Dic3—62no—
214C3×Dic3—122no—
215C3×Dic3—121no—
216C3×Dic3—121no—
217C3×Dic3—121no—
218C3×Dic3—121no—
219C22×C8—271no—
220#117—91no—
221#117—271no—
222#107—91no—
223#107—271no—
224#106—91no—
225#106—271no—
226H(3,𝔽3)#42711yescharacteristic, complemented, Sylow
227C23×C3—91no—
228C22×D3—31no—
229C22×D3—31no—
230C22×D3—61no—
231C2×C3×C4—21no—
232C2×C3×C4—31no—
233C2×C3×C4—31no—
234C2×C3×C4—61no—
235C2×C3×C4—61no—
236C2×C3×C4—62no—
237C2×C3×C4—61no—
238C2×C3×C4—91no—
239C2×C3×C4—92no—
240C2×C3×C4—92no—
241C2×C3×C4—181no—
242C2×C3×C4—181no—
243C2×C3×C4—181no—
244C2×C3×C4C2×D3211yescharacteristic
245C2×Dic3—31no—
246C2×Dic3—31no—
247C2×Dic3—61no—
248C2×Dic3—61no—
249C2×Dic3—61no—
250C2×Dic3—62no—
251C2×Dic3—61no—
252C2×Dic3—61no—
253C2×Dic3—181no—
254C2×Dic3—181no—
255C4×D3—32no—
256C4×D3—32no—
257C4×D3—32no—
258C4×D3—32no—
259C4×D3—62no—
260C4×D3—62no—
261C3×C8—92no—
262C3×C8—92no—
263C3⋊C8—32no—
264C3⋊C8—32no—
265C3⋊C8—92no—
266C3⋊C8—92no—
267C2×C32—21no—
268C2×C32—21no—
269C2×C32—21no—
270C2×C32D3×C4⋊C411yescharacteristic
271C2×C32D3×C4⋊C411yescharacteristic
272C2×C32#91211yescharacteristic
273C2×C32#91611yescharacteristic
274C2×C32#91611yescharacteristic
275C2×C32#91911yescharacteristic
276C3×D3—62no—
277C3×D3—62no—
278C3×D3—122no—
279C22×C4—91no—
280C2×C8—91no—
281C2×C8—91no—
282C2×C8—274no—
283C4⋊C4—91no—
284C4⋊C4—91no—
285C4⋊C4—91no—
286C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
287C22⋊C4—181no—
288C22⋊C4—541no—
289C42—181no—
290C42—541no—
291C22×C3—31no—
292C22×C3—31no—
293C22×C3—61no—
294C22×C3—92no—
295C22×C3—181no—
296C22×C3—181no—
297C22×C3#381111yescharacteristic
298C2×D3—32no—
299C2×D3—32no—
300C2×D3—62no—
301C2×D3—61no—
302C2×D3—61no—
303C2×D3—61no—
304C2×D3—61no—
305C2×D3—121no—
306C2×D3—121no—
307C3×C4—31no—
308C3×C4—31no—
309C3×C4—31no—
310C3×C4—31no—
311C3×C4—41no—
312C3×C4—61no—
313C3×C4—61no—
314C3×C4—92no—
315C3×C4—121no—
316C3×C4—121no—
317C3×C4—122no—
318C3×C4—361no—
319C3×C4—361no—
320C3×C4—361no—
321C3×C4#379711yescharacteristic
322C3×C4C4×D3211yescharacteristic
323Dic3—32no—
324Dic3—32no—
325Dic3—62no—
326Dic3—121no—
327Dic3—121no—
328Dic3—122no—
329Dic3—121no—
330Dic3—121no—
331Dic3—361no—
332Dic3—361no—
333C32—21no—
334C32#486011yescharacteristic, complemented
335C32#488011yescharacteristic, complemented
336C23—91no—
337C2×C4—21no—
338C2×C4—91no—
339C2×C4—92no—
340C2×C4—92no—
341C2×C4—181no—
342C2×C4—181no—
343C2×C4—181no—
344C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
345C8—92no—
346C8—92no—
347C2×C3—31no—
348C2×C3—31no—
349C2×C3—31no—
350C2×C3—31no—
351C2×C3—31no—
352C2×C3—31no—
353C2×C3—61no—
354C2×C3—61no—
355C2×C3—61no—
356C2×C3—182no—
357C2×C3#6388311yescharacteristic
358C2×C3#6393611yescharacteristic
359C2×C3#6395011yescharacteristic
360D3—62no—
361D3—62no—
362D3—122no—
363C22—92no—
364C22—181no—
365C22—181no—
366C22#8867711yescharacteristic
367C4—41no—
368C4—92no—
369C4—361no—
370C4—361no—
371C4—361no—
372C4#8865111yescharacteristic
373C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
374C3—31no—
375C3—31no—
376C3—61no—
377C3#1059057411yescharacteristic
378C2—182no—
379C2#1172114711yescharacteristic
380C2#1172119211yescharacteristic
381C2#1172123011yescharacteristic
subgroupG/H
centerC22#88677
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3#3811
prime radicalC22×#272D4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#427
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. #63635⋊C3C2semidirect
5. #63634⋊C3C2semidirect
6. #11720307C2semidirect
7. #11720306C2semidirect
21. #88434C4semidirect
29. #63635D3semidirect
30. #63634D3semidirect
226. H(3,𝔽3)#427semidirect
334. C32#4860semidirect
335. C32#4880semidirect

Maximal subgroups

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

subgroupindex
#117217602
#117210892
C2×#884342
#63635⋊C32
#63634⋊C32
#117203072
#117203062
#105905723
#105905713