Finite Group Databasebeta

← #13138907

Subgroups — #13138907

Order 1728.

414 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 414 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#11721152C211yesmaximal, characteristic
2#11721094C211yesmaximal, characteristic
3#11720553C211yesmaximal, characteristic
4#63645⋊C3C211yesmaximal, characteristic, complemented
5#63643⋊C3C211yesmaximal, characteristic, complemented
6#11720315C211yesmaximal, characteristic, complemented
7#11720313C211yesmaximal, characteristic, complemented
8#10590684—31nomaximal
9#10590682—31nomaximal
10C2×C4×#272C2211yescharacteristic
11C2×#634⋊C3C2211yescharacteristic
12C2×#6199—21no—
13C2×C8×H(3,𝔽3)C2211yescharacteristic
14C2×#6180C2211yescharacteristic
15C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18#88451—21no—
19#88449—21no—
20#88448—21no—
21D3×C4⋊Dic3—31no—
22D3×C4⋊Dic3—31no—
23#63857—31no—
24#63856—31no—
25#63645D311yescharacteristic, complemented
26#63643D311yescharacteristic, complemented
27C3×(C2×D3)⋊C8—31no—
28C3×(C2×D3)⋊C8—31no—
29C3×(C2×D3)⋊C8—61no—
30C3×#818—31no—
31C3×#816—31no—
32#63575—31no—
33#63575—31no—
34#63567—31no—
35#63566—31no—
36C22×#272D411yescharacteristic
37C2×C4×H(3,𝔽3)C2311yescharacteristic
38C2×#1084D411yescharacteristic
39C4×#272—21no—
40C4×#272—21no—
41C2×(C3⋊Dic3)⋊C3—21no—
42C2×(C3⋊Dic3)⋊C3—21no—
43C2×(C3⋊Dic3)⋊C3—21no—
44C2×(C3⋊Dic3)⋊C3—21no—
45#634⋊C3—21no—
46#634⋊C3—21no—
47#6199—42no—
48C8×H(3,𝔽3)—21no—
49#6180—21no—
50#4884—91no—
51#4811—91no—
52#4722—91no—
53C2×#634C2×D311yescharacteristic
54C2×C3×C4×D3—31no—
55C2×C3×C4×D3—31no—
56C2×C3×C4×D3—61no—
57C2×C3×Dic6—31no—
58C2×D3×Dic3—61no—
59C2×D3×Dic3—61no—
60C2×C32×C8—21no—
61C2×C32×C8C2×D311yescharacteristic
62C2×C32×C8C2×D311yescharacteristic
63C4⋊(C3⋊Dic3)C2×D311yescharacteristic
64C4⋊(C3⋊Dic3)C2×D311yescharacteristic
65C4⋊(C3⋊Dic3)C2×D311yescharacteristic
66C3×C4⋊Dic3—31no—
67C3×C4⋊Dic3—31no—
68C3×C4⋊Dic3—31no—
69C2×C3×C3⋊C8—31no—
70C2×C3×C3⋊C8—31no—
71C2×C3×C3⋊C8—61no—
72#3724—61no—
73Dic3⋊Dic3—61no—
74Dic3⋊Dic3—61no—
75Dic3⋊Dic3—61no—
76(C2×D3)⋊Dic3—61no—
77(C2×D3)⋊Dic3—61no—
78C22×H(3,𝔽3)C2×D411yescharacteristic
79C2×#272—21no—
80C2×#272—21no—
81C2×#272Q1612yes—
82C4×H(3,𝔽3)#4111yescharacteristic
83C4×H(3,𝔽3)#4111yescharacteristic
84#1084—21no—
85(C3⋊Dic3)⋊C3—41no—
86(C3⋊Dic3)⋊C3—41no—
87(C3⋊Dic3)⋊C3—41no—
88(C3⋊Dic3)⋊C3—41no—
89#946—91no—
90C2×C4⋊Dic3—91no—
91#924—91no—
92#897—91no—
93D3×C4⋊C4—91no—
94D3×C4⋊C4—91no—
95C3×#107—91no—
96C3×#103—91no—
97C3×C22⋊C8—91no—
98#835—91no—
99(C2×D3)⋊C8—31no—
100(C2×D3)⋊C8—31no—
101(C2×D3)⋊C8—61no—
102#818—31no—
103#818—91no—
104#818—91no—
105#816—31no—
106#816—91no—
107#816—91no—
108#810—91no—
109#807—91no—
110#807—91no—
111C22×C3×D3—31no—
112C22×C3×D3—31no—
113C22×C3×D3—61no—
114C2×C32×C4—21no—
115C2×C32×C4C22×D311yescharacteristic
116C2×C32×C4C22×D311yescharacteristic
117C2×C3⋊Dic3—21no—
118C2×C3⋊Dic3—21no—
119C2×C3⋊Dic3—21no—
120C2×C3⋊Dic3—21no—
121#634—21no—
122#634—21no—
123C2×C3×Dic3—31no—
124C2×C3×Dic3—31no—
125C2×C3×Dic3—61no—
126C2×C3×Dic3—61no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—61no—
130C3×C4×D3—61no—
131C3×C4×D3—61no—
132C3×C4×D3—61no—
133C3×C4×D3—61no—
134C3×C4×D3—121no—
135C3×C4×D3—121no—
136C3×Dic6—61no—
137C3×Dic6—61no—
138D3×Dic3—122no—
139D3×Dic3—122no—
140C32×C8—21no—
141C32×C8—21no—
142C32×C8—41no—
143C3×C3⋊C8—61no—
144C3×C3⋊C8—61no—
145C3×C3⋊C8—121no—
146#484—271noSylow
147C2×H(3,𝔽3)#12311yescharacteristic
148C2×H(3,𝔽3)C2×Q1611yescharacteristic
149C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
150#272—22no—
151C2×C3×Q8—91no—
152C22×C3×C4—91no—
153C22×Dic3—181no—
154C2×C4×D3—31no—
155C2×C4×D3—31no—
156C2×C4×D3—61no—
157C2×C4×D3—181no—
158C2×C4×D3—181no—
159C2×Dic6—31no—
160C2×Dic6—91no—
161C2×C3×C8—31no—
162C2×C3×C8—31no—
163C2×C3×C8—61no—
164C2×C3×C8—91no—
165C2×C3×C8D3211yescharacteristic
166C3×C4⋊C4—91no—
167C3×C4⋊C4—91no—
168C3×C4⋊C4—91no—
169#217—181no—
170(C2×D3)⋊C4—181no—
171(C2×D3)⋊C4—181no—
172C4⋊Dic3—31no—
173C4⋊Dic3—31no—
174C4⋊Dic3—31no—
175C4⋊Dic3—91no—
176C4⋊Dic3—91no—
177C4⋊Dic3—91no—
178C4⋊Dic3—181no—
179C4⋊Dic3—181no—
180Dic3⋊C4—181no—
181Dic3⋊C4—181no—
182Dic3⋊C4—181no—
183Dic3⋊C4—181no—
184C2×C3⋊C8—31no—
185C2×C3⋊C8—31no—
186C2×C3⋊C8—61no—
187C22×C32—21no—
188C22×C32D3×D411yescharacteristic
189C22×C32D3×D411yescharacteristic
190C2×C3×D3—32no—
191C2×C3×D3—32no—
192C2×C3×D3—62no—
193C2×C3×D3—61no—
194C2×C3×D3—61no—
195C2×C3×D3—61no—
196C2×C3×D3—61no—
197C2×C3×D3—121no—
198C2×C3×D3—121no—
199C32×C4—21no—
200C32×C4—21no—
201C32×C4#23511yescharacteristic
202C32×C4#23511yescharacteristic
203C32×C4#23711yescharacteristic
204C32×C4#23911yescharacteristic
205C3⋊Dic3—41no—
206C3⋊Dic3—41no—
207C3⋊Dic3—41no—
208C3⋊Dic3—41no—
209C3×Dic3—61no—
210C3×Dic3—61no—
211C3×Dic3—121no—
212C3×Dic3—121no—
213C3×Dic3—121no—
214C3×Dic3—121no—
215C3×Dic3—121no—
216C22⋊Q8—271no—
217C2×C4⋊C4—271no—
218#107—91no—
219#107—271no—
220#103—91no—
221#103—271no—
222C22⋊C8—91no—
223H(3,𝔽3)#48411yescharacteristic, complemented, Sylow
224C23×C3—91no—
225C22×D3—31no—
226C22×D3—31no—
227C22×D3—61no—
228C3×Q8—181no—
229C3×Q8—181no—
230C2×C3×C4—31no—
231C2×C3×C4—31no—
232C2×C3×C4—61no—
233C2×C3×C4—91no—
234C2×C3×C4—181no—
235C2×C3×C4—181no—
236C2×C3×C4—181no—
237C2×C3×C4—181no—
238C2×C3×C4—181no—
239C2×C3×C4—181no—
240C2×C3×C4C2×D3211yescharacteristic
241C2×Dic3—31no—
242C2×Dic3—31no—
243C2×Dic3—61no—
244C2×Dic3—61no—
245C2×Dic3—61no—
246C2×Dic3—61no—
247C2×Dic3—61no—
248C2×Dic3—181no—
249C2×Dic3—181no—
250C2×Dic3—181no—
251C2×Dic3—181no—
252C2×Dic3—362no—
253C4×D3—61no—
254C4×D3—61no—
255C4×D3—61no—
256C4×D3—61no—
257C4×D3—121no—
258C4×D3—121no—
259C4×D3—362no—
260C4×D3—362no—
261Dic6—61no—
262Dic6—61no—
263Dic6—181no—
264Dic6—181no—
265C3×C8—21no—
266C3×C8—61no—
267C3×C8—61no—
268C3×C8—121no—
269C3×C8—181no—
270C3⋊C8—61no—
271C3⋊C8—61no—
272C3⋊C8—121no—
273C2×C32—21no—
274C2×C32—21no—
275C2×C32—21no—
276C2×C32#88311yescharacteristic
277C2×C32#89411yescharacteristic
278C2×C32#91111yescharacteristic
279C2×C32#91411yescharacteristic
280C2×C32D3×Q1611yescharacteristic
281C2×C32D3×Q1611yescharacteristic
282C3×D3—62no—
283C3×D3—62no—
284C3×D3—122no—
285C2×Q8—91no—
286C22×C4—91no—
287C22×C4—541no—
288C2×C8—91no—
289C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
290C4⋊C4—91no—
291C4⋊C4—91no—
292C4⋊C4—91no—
293C4⋊C4—541no—
294C4⋊C4—541no—
295C22⋊C4—541no—
296C22×C3—31no—
297C22×C3—31no—
298C22×C3—61no—
299C22×C3—92no—
300C22×C3—181no—
301C22×C3—181no—
302C22×C3#380211yescharacteristic
303C2×D3—32no—
304C2×D3—32no—
305C2×D3—62no—
306C2×D3—61no—
307C2×D3—61no—
308C2×D3—61no—
309C2×D3—61no—
310C2×D3—121no—
311C2×D3—121no—
312C3×C4—31no—
313C3×C4—31no—
314C3×C4—31no—
315C3×C4—31no—
316C3×C4—61no—
317C3×C4—61no—
318C3×C4—181no—
319C3×C4—361no—
320C3×C4—361no—
321C3×C4—361no—
322C3×C4—361no—
323C3×C4#379911yescharacteristic
324C3×C4#380411yescharacteristic
325Dic3—61no—
326Dic3—61no—
327Dic3—121no—
328Dic3—121no—
329Dic3—121no—
330Dic3—121no—
331Dic3—121no—
332Dic3—361no—
333Dic3—361no—
334Dic3—361no—
335Dic3—361no—
336C32—21no—
337C32#481111yescharacteristic, complemented
338C32#488411yescharacteristic, complemented
339C23—91no—
340Q8—181no—
341Q8—181no—
342C2×C4—91no—
343C2×C4—181no—
344C2×C4—181no—
345C2×C4—181no—
346C2×C4—181no—
347C2×C4—181no—
348C2×C4—181no—
349C2×C4—1082no—
350C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
351C8—21no—
352C8—181no—
353C2×C3—31no—
354C2×C3—31no—
355C2×C3—31no—
356C2×C3—31no—
357C2×C3—31no—
358C2×C3—31no—
359C2×C3—61no—
360C2×C3—61no—
361C2×C3—61no—
362C2×C3—182no—
363C2×C3#6379811yescharacteristic
364C2×C3#6380211yescharacteristic
365C2×C3#6385411yescharacteristic
366D3—62no—
367D3—62no—
368D3—122no—
369C22—92no—
370C22—181no—
371C22—181no—
372C22#8865511yescharacteristic
373C4—181no—
374C4—361no—
375C4—361no—
376C4—361no—
377C4—361no—
378C4#8865211yescharacteristic
379C4#8867111yescharacteristic
380C3—31no—
381C3—31no—
382C3—61no—
383C3#1059068911yescharacteristic
384C2—182no—
385C2#1172083311yescharacteristic
386C2#1172083911yescharacteristic
387C2#1172110211yescharacteristic
subgroupG/H
centerC22#88655
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3#3802
prime radicalC22×#272D4
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#484
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
4. #63645⋊C3C2semidirect
5. #63643⋊C3C2semidirect
6. #11720315C2semidirect
7. #11720313C2semidirect
25. #63645D3semidirect
26. #63643D3semidirect
223. H(3,𝔽3)#484semidirect
337. C32#4811semidirect
338. C32#4884semidirect

Maximal subgroups

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

subgroupindex
#117211522
#117210942
#117205532
#63645⋊C32
#63643⋊C32
#117203152
#117203132
#105906843
#105906823