Finite Group Databasebeta

← #13138917

Subgroups — #13138917

Order 1728.

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

normal subgroups only

This group has 386 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#11721094C211yesmaximal, characteristic
2#11721089C211yesmaximal, characteristic
3#11720553C211yesmaximal, characteristic
4#63644⋊C3C211yesmaximal, characteristic, complemented
5#63643⋊C3C211yesmaximal, characteristic, complemented
6#11720315C211yesmaximal, characteristic, complemented
7#11720313C211yesmaximal, characteristic, complemented
8#10590711—31nomaximal
9#10590709—31nomaximal
10C2×C4×#272C2211yescharacteristic
11C2×#634⋊C3C2211yescharacteristic
12C2×C8×H(3,𝔽3)C2211yescharacteristic
13C2×#6180C2211yescharacteristic
14C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
15C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17#88451—21no—
18#88451—21no—
19#88448—21no—
20#88447—21no—
21#63857—31no—
22#63856—31no—
23#63841—31no—
24#63841—31no—
25#63644D311yescharacteristic, complemented
26#63643D311yescharacteristic, complemented
27C3×(C2×D3)⋊C8—31no—
28C3×(C2×D3)⋊C8—31no—
29C3×(C2×D3)⋊C8—61no—
30C3×#817—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—
47C8×H(3,𝔽3)—21no—
48#6180—21no—
49#4868—91no—
50#4815—91no—
51#4723—91no—
52C2×#634C2×D311yescharacteristic
53C2×C3×C4×D3—31no—
54C2×C3×C4×D3—31no—
55C2×C3×C4×D3—61no—
56C2×C3×Dic6—31no—
57C2×C32×C8—21no—
58C2×C32×C8C2×D311yescharacteristic
59C2×C32×C8C2×D311yescharacteristic
60C4⋊(C3⋊Dic3)C2×D311yescharacteristic
61C4⋊(C3⋊Dic3)C2×D311yescharacteristic
62C4⋊(C3⋊Dic3)C2×D311yescharacteristic
63C3×C4⋊Dic3—31no—
64C3×C4⋊Dic3—31no—
65C3×C4⋊Dic3—31no—
66C2×C3×C3⋊C8—31no—
67C2×C3×C3⋊C8—31no—
68C2×C3×C3⋊C8—61no—
69#3724—61no—
70Dic3⋊Dic3—61no—
71(C2×D3)⋊Dic3—61no—
72(C2×D3)⋊Dic3—61no—
73(C2×D3)⋊Dic3—61no—
74(C2×D3)⋊Dic3—61no—
75Dic32—61no—
76Dic32—61no—
77C22×H(3,𝔽3)C2×D411yescharacteristic
78C2×#272—21no—
79C2×#272—21no—
80C2×#272—21no—
81C4×H(3,𝔽3)#4111yescharacteristic
82C4×H(3,𝔽3)#4111yescharacteristic
83#1084—21no—
84(C3⋊Dic3)⋊C3—41no—
85(C3⋊Dic3)⋊C3—41no—
86(C3⋊Dic3)⋊C3—41no—
87(C3⋊Dic3)⋊C3—41no—
88#946—91no—
89#926—91no—
90#924—91no—
91#897—91no—
92#892—91no—
93#892—91no—
94C3×#106—91no—
95C3×#103—91no—
96C3×C22⋊C8—91no—
97#835—91no—
98(C2×D3)⋊C8—31no—
99(C2×D3)⋊C8—31no—
100(C2×D3)⋊C8—61no—
101#818—91no—
102#817—31no—
103#817—91no—
104#816—31no—
105#816—91no—
106#816—91no—
107#810—91no—
108#807—91no—
109#807—91no—
110C22×C3×D3—31no—
111C22×C3×D3—31no—
112C22×C3×D3—61no—
113C2×C32×C4—21no—
114C2×C32×C4C22×D311yescharacteristic
115C2×C32×C4C22×D311yescharacteristic
116C2×C3⋊Dic3—21no—
117C2×C3⋊Dic3—21no—
118C2×C3⋊Dic3—21no—
119C2×C3⋊Dic3—21no—
120#634—21no—
121#634—21no—
122C2×C3×Dic3—31no—
123C2×C3×Dic3—31no—
124C2×C3×Dic3—61no—
125C2×C3×Dic3—61no—
126C2×C3×Dic3—61no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—61no—
129C3×C4×D3—61no—
130C3×C4×D3—61no—
131C3×C4×D3—61no—
132C3×C4×D3—61no—
133C3×C4×D3—121no—
134C3×C4×D3—121no—
135C3×Dic6—61no—
136C3×Dic6—61no—
137C32×C8—21no—
138C32×C8—21no—
139C32×C8—41no—
140C3×C3⋊C8—61no—
141C3×C3⋊C8—61no—
142C3×C3⋊C8—121no—
143#485—271noSylow
144C2×H(3,𝔽3)#12311yescharacteristic
145C2×H(3,𝔽3)#13511yescharacteristic
146C2×H(3,𝔽3)#13711yescharacteristic
147#272—41no—
148C2×C3×Q8—91no—
149C22×C3×C4—91no—
150C2×C4×D3—31no—
151C2×C4×D3—31no—
152C2×C4×D3—61no—
153C2×Dic6—31no—
154C2×Dic6—91no—
155C2×C3×C8—31no—
156C2×C3×C8—31no—
157C2×C3×C8—61no—
158C2×C3×C8—91no—
159C2×C3×C8D3211yescharacteristic
160C3×C4⋊C4—91no—
161C3×C4⋊C4—91no—
162C3×C4⋊C4—91no—
163#217—181no—
164#217—181no—
165(C2×D3)⋊C4—181no—
166(C2×D3)⋊C4—181no—
167(C2×D3)⋊C4—181no—
168(C2×D3)⋊C4—181no—
169C4⋊Dic3—31no—
170C4⋊Dic3—31no—
171C4⋊Dic3—31no—
172C4⋊Dic3—91no—
173C4⋊Dic3—91no—
174C4⋊Dic3—91no—
175C4⋊Dic3—181no—
176Dic3⋊C4—181no—
177Dic3⋊C4—181no—
178C4×Dic3—181no—
179C4×Dic3—181no—
180C4×Dic3—181no—
181C2×C3⋊C8—31no—
182C2×C3⋊C8—31no—
183C2×C3⋊C8—61no—
184C22×C32—21no—
185C22×C32D3×D411yescharacteristic
186C22×C32D3×D411yescharacteristic
187C2×C3×D3—62no—
188C2×C3×D3—61no—
189C2×C3×D3—61no—
190C2×C3×D3—61no—
191C2×C3×D3—61no—
192C2×C3×D3—61no—
193C2×C3×D3—61no—
194C2×C3×D3—121no—
195C2×C3×D3—121no—
196C32×C4—21no—
197C32×C4—21no—
198C32×C4#23511yescharacteristic
199C32×C4#23511yescharacteristic
200C32×C4#23711yescharacteristic
201C32×C4#23911yescharacteristic
202C3⋊Dic3—41no—
203C3⋊Dic3—41no—
204C3⋊Dic3—41no—
205C3⋊Dic3—41no—
206C3×Dic3—61no—
207C3×Dic3—61no—
208C3×Dic3—121no—
209C3×Dic3—121no—
210C3×Dic3—121no—
211C3×Dic3—121no—
212C3×Dic3—121no—
213C22⋊Q8—271no—
214#117—271no—
215#107—271no—
216#106—91no—
217#103—91no—
218#103—271no—
219C22⋊C8—91no—
220H(3,𝔽3)#48511yescharacteristic, complemented, Sylow
221C23×C3—91no—
222C22×D3—31no—
223C22×D3—31no—
224C22×D3—61no—
225C3×Q8—181no—
226C3×Q8—181no—
227C2×C3×C4—31no—
228C2×C3×C4—31no—
229C2×C3×C4—61no—
230C2×C3×C4—91no—
231C2×C3×C4—181no—
232C2×C3×C4—181no—
233C2×C3×C4—181no—
234C2×C3×C4—181no—
235C2×C3×C4—181no—
236C2×C3×C4—181no—
237C2×C3×C4C2×D3211yescharacteristic
238C2×Dic3—31no—
239C2×Dic3—31no—
240C2×Dic3—61no—
241C2×Dic3—61no—
242C2×Dic3—61no—
243C2×Dic3—61no—
244C2×Dic3—61no—
245C2×Dic3—181no—
246C2×Dic3—181no—
247C2×Dic3—181no—
248C2×Dic3—181no—
249C4×D3—61no—
250C4×D3—61no—
251C4×D3—61no—
252C4×D3—61no—
253C4×D3—121no—
254C4×D3—121no—
255Dic6—61no—
256Dic6—61no—
257Dic6—181no—
258Dic6—181no—
259C3×C8—21no—
260C3×C8—61no—
261C3×C8—61no—
262C3×C8—121no—
263C3×C8—181no—
264C3⋊C8—61no—
265C3⋊C8—61no—
266C3⋊C8—121no—
267C2×C32—21no—
268C2×C32—21no—
269C2×C32—21no—
270C2×C32#88311yescharacteristic
271C2×C32#89411yescharacteristic
272C2×C32#91511yescharacteristic
273C2×C32#91611yescharacteristic
274C2×C32#91611yescharacteristic
275C2×C32#91811yescharacteristic
276C3×D3—122no—
277C3×D3—121no—
278C3×D3—121no—
279C2×Q8—91no—
280C22×C4—91no—
281C2×C8—91no—
282C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
283C4⋊C4—91no—
284C4⋊C4—91no—
285C4⋊C4—91no—
286C4⋊C4—541no—
287C22⋊C4—541no—
288C22⋊C4—541no—
289C42—541no—
290C22×C3—31no—
291C22×C3—31no—
292C22×C3—61no—
293C22×C3—181no—
294C22×C3—181no—
295C22×C3—181no—
296C22×C3#380211yescharacteristic
297C2×D3—61no—
298C2×D3—61no—
299C2×D3—61no—
300C2×D3—61no—
301C2×D3—61no—
302C2×D3—61no—
303C2×D3—62no—
304C2×D3—121no—
305C2×D3—121no—
306C3×C4—31no—
307C3×C4—31no—
308C3×C4—31no—
309C3×C4—31no—
310C3×C4—61no—
311C3×C4—61no—
312C3×C4—181no—
313C3×C4—361no—
314C3×C4—361no—
315C3×C4—361no—
316C3×C4—361no—
317C3×C4#379911yescharacteristic
318C3×C4#380411yescharacteristic
319Dic3—61no—
320Dic3—61no—
321Dic3—121no—
322Dic3—121no—
323Dic3—121no—
324Dic3—121no—
325Dic3—121no—
326Dic3—361no—
327Dic3—361no—
328Dic3—361no—
329Dic3—361no—
330C32—21no—
331C32#481511yescharacteristic, complemented
332C32#486811yescharacteristic, complemented
333C23—91no—
334Q8—181no—
335Q8—181no—
336C2×C4—91no—
337C2×C4—181no—
338C2×C4—181no—
339C2×C4—181no—
340C2×C4—181no—
341C2×C4—181no—
342C2×C4—181no—
343C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
344C8—21no—
345C8—181no—
346C2×C3—31no—
347C2×C3—31no—
348C2×C3—31no—
349C2×C3—31no—
350C2×C3—31no—
351C2×C3—31no—
352C2×C3—61no—
353C2×C3—61no—
354C2×C3—61no—
355C2×C3—361no—
356C2×C3#6380911yescharacteristic
357C2×C3#6381111yescharacteristic
358C2×C3#6385411yescharacteristic
359D3—121no—
360D3—121no—
361D3—122no—
362C22—181no—
363C22—181no—
364C22—181no—
365C22#8865511yescharacteristic
366C4—181no—
367C4—361no—
368C4—361no—
369C4—361no—
370C4—361no—
371C4#8865211yescharacteristic
372C4#8867111yescharacteristic
373C3—31no—
374C3—31no—
375C3—61no—
376C3#1059071311yescharacteristic
377C2—361no—
378C2#1172084311yescharacteristic
379C2#1172084511yescharacteristic
380C2#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)#485
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. #63644⋊C3C2semidirect
5. #63643⋊C3C2semidirect
6. #11720315C2semidirect
7. #11720313C2semidirect
25. #63644D3semidirect
26. #63643D3semidirect
220. H(3,𝔽3)#485semidirect
331. C32#4815semidirect
332. C32#4868semidirect

Maximal subgroups

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

subgroupindex
#117210942
#117210892
#117205532
#63644⋊C32
#63643⋊C32
#117203152
#117203132
#105907113
#105907093