Finite Group Databasebeta

← #13138854

Subgroups — #13138854

Order 1728.

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

normal subgroups only

This group has 407 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#11721141C211yesmaximal, characteristic
2#11721081C211yesmaximal, characteristic, complemented
3#11720550C211yesmaximal, characteristic, complemented
4#63648⋊C3C211yesmaximal, characteristic, complemented
5#63647⋊C3C211yesmaximal, characteristic, complemented
6#11720308C211yesmaximal, characteristic, complemented
7#11720306C211yesmaximal, characteristic, complemented
8#10590704—31nomaximal
9#10590652—31nomaximal
10C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
11C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
12C2×#6198—21no—
13C2×C8×H(3,𝔽3)C2211yescharacteristic
14C4⋊#1084C2211yescharacteristic
15C4⋊#1084C2211yescharacteristic
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
18#88450—21no—
19#88450—21no—
20#88447—21no—
21#63916—31no—
22#63909—31no—
23#63849—31no—
24#63841—31no—
25#63648D311yescharacteristic, complemented
26#63647D311yescharacteristic, complemented
27C3×#821—31no—
28C3×(C2×D3)⋊C8—31no—
29C3×#817—31no—
30C3×#817—31no—
31C3×#817—61no—
32#63576—31no—
33#63575—31no—
34#63564—31no—
35#63562—31no—
36C22×(C3⋊D3)⋊C3—21no—
37C22×(C3⋊D3)⋊C3D411yescharacteristic
38C2×C4×H(3,𝔽3)C2311yescharacteristic
39C2×#1084—21no—
40C2×#1084—21no—
41C2×(C3⋊Dic3)⋊C3—21no—
42C2×(C3⋊Dic3)⋊C3D411yescharacteristic
43C4⋊((C3⋊D3)⋊C3)—21no—
44C4⋊((C3⋊D3)⋊C3)—21no—
45C4×(C3⋊D3)⋊C3—21no—
46C4×(C3⋊D3)⋊C3—21no—
47#6198—42no—
48C8×H(3,𝔽3)—21no—
49(C3⋊(C3⋊C8))⋊C3—21no—
50#4866—91no—
51#4779—91no—
52#4737—91no—
53C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
54C2×C4×C3⋊D3C2×D311yescharacteristic
55C2×C3×D12—31no—
56C2×C3×C4×D3—31no—
57C2×#626—61no—
58C2×#626—61no—
59C2×C32×C8—21no—
60C2×C32×C8C2×D311yescharacteristic
61C2×C32×C8C2×D311yescharacteristic
62C4⋊(C3⋊Dic3)C2×D311yescharacteristic
63C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
64C3×C4⋊Dic3—31no—
65C3×C4⋊Dic3—31no—
66C3×C4⋊Dic3—31no—
67C3×C4⋊Dic3—31no—
68C3×C4⋊Dic3—31no—
69C3×C4⋊Dic3—61no—
70C3×C4⋊Dic3—61no—
71C2×C3×C3⋊C8—31no—
72#3722—61no—
73#3722—61no—
74(C2×D3)⋊Dic3—61no—
75(C2×D3)⋊Dic3—61no—
76Dic32—61no—
77Dic32—61no—
78C22×H(3,𝔽3)C2×D411yescharacteristic
79C2×(C3⋊D3)⋊C3—21no—
80C2×(C3⋊D3)⋊C3—21no—
81C2×(C3⋊D3)⋊C3—21no—
82C2×(C3⋊D3)⋊C3—41no—
83C2×(C3⋊D3)⋊C3—41no—
84C2×(C3⋊D3)⋊C3—41no—
85C4×H(3,𝔽3)#4111yescharacteristic
86C4×H(3,𝔽3)#4111yescharacteristic
87#1084—41no—
88#1084—41no—
89(C3⋊Dic3)⋊C3—21no—
90(C3⋊Dic3)⋊C3—41no—
91#938—91no—
92#930—91no—
93#926—91no—
94#895—91no—
95#892—91no—
96#892—91no—
97C3×#106—91no—
98C3×#102—91no—
99C3×C22⋊C8—91no—
100#832—91no—
101#821—31no—
102#821—91no—
103#821—91no—
104(C2×D3)⋊C8—31no—
105(C2×D3)⋊C8—91no—
106#818—91no—
107#817—31no—
108#817—31no—
109#817—61no—
110#809—91no—
111#807—91no—
112#807—91no—
113C22×C3⋊D3—21no—
114C22×C3⋊D3D1211yescharacteristic
115C22×C3×D3—31no—
116C22×C3×D3—61no—
117C2×C32×C4—21no—
118C2×C32×C4C22×D311yescharacteristic
119C2×C32×C4C22×D311yescharacteristic
120C2×C3⋊Dic3—21no—
121C2×C3⋊Dic3D1211yescharacteristic
122C4⋊(C3⋊D3)—21no—
123C4⋊(C3⋊D3)—21no—
124C4×C3⋊D3—21no—
125C4×C3⋊D3—21no—
126C2×C3×Dic3—31no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—61no—
130C2×C3×Dic3—61no—
131C2×C3×Dic3—61no—
132C2×C3×Dic3—121no—
133C2×C3×Dic3—121no—
134C3×D12—61no—
135C3×D12—61no—
136C3×C4×D3—61no—
137C3×C4×D3—61no—
138#626—122no—
139#626—122no—
140C32×C8—22no—
141C32×C8—21no—
142C32×C8—21no—
143C3⋊(C3⋊C8)—21no—
144C3×C3⋊C8—61no—
145#482—271noSylow
146C2×H(3,𝔽3)#12311yescharacteristic
147C2×H(3,𝔽3)#13511yescharacteristic
148C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
149(C3⋊D3)⋊C3—41no—
150(C3⋊D3)⋊C3—81no—
151C2×C3×D4—91no—
152C22×C3×C4—91no—
153C2×#67—181no—
154C2×#67—181no—
155C2×D12—31no—
156C2×D12—91no—
157C2×D12—181no—
158C2×C4×D3—31no—
159C2×C4×D3—91no—
160C2×C3×C8—31no—
161C2×C3×C8—31no—
162C2×C3×C8—61no—
163C2×C3×C8—91no—
164C2×C3×C8D3211yescharacteristic
165C3×C4⋊C4—91no—
166C3×C4⋊C4—91no—
167C3×C4⋊C4—91no—
168#217—181no—
169#217—181no—
170(C2×D3)⋊C4—181no—
171(C2×D3)⋊C4—181no—
172(C2×D3)⋊C4—181no—
173(C2×D3)⋊C4—181no—
174C4⋊Dic3—31no—
175C4⋊Dic3—31no—
176C4⋊Dic3—31no—
177C4⋊Dic3—31no—
178C4⋊Dic3—31no—
179C4⋊Dic3—61no—
180C4⋊Dic3—61no—
181C4⋊Dic3—91no—
182C4×Dic3—181no—
183C4×Dic3—181no—
184C4×Dic3—181no—
185C2×C3⋊C8—31no—
186C2×C3⋊C8—91no—
187C22×C32—21no—
188C22×C32C2×D1211yescharacteristic
189C22×C32D3×D411yescharacteristic
190C2×C3⋊D3—21no—
191C2×C3⋊D3—21no—
192C2×C3⋊D3—21no—
193C2×C3⋊D3—41no—
194C2×C3⋊D3—41no—
195C2×C3⋊D3—41no—
196C2×C3×D3—61no—
197C2×C3×D3—61no—
198C2×C3×D3—61no—
199C2×C3×D3—121no—
200C2×C3×D3—121no—
201C2×C3×D3—121no—
202C32×C4—21no—
203C32×C4—21no—
204C32×C4#23511yescharacteristic
205C32×C4#23711yescharacteristic
206C32×C4#23711yescharacteristic
207C32×C4#23711yescharacteristic
208C3⋊Dic3—21no—
209C3⋊Dic3—41no—
210C3×Dic3—61no—
211C3×Dic3—121no—
212C3×Dic3—121no—
213C3×Dic3—121no—
214C3×Dic3—121no—
215C3×Dic3—121no—
216C3×Dic3—241no—
217C3×Dic3—241no—
218#121—271no—
219#117—271no—
220#107—271no—
221#106—91no—
222#102—91no—
223#102—271no—
224C22⋊C8—91no—
225H(3,𝔽3)#48211yescharacteristic, complemented, Sylow
226C23×C3—91no—
227C23×C3—181no—
228C22×D3—31no—
229C22×D3—61no—
230C22×D3—91no—
231C22×D3—181no—
232C3×D4—181no—
233C3×D4—181no—
234C2×C3×C4—31no—
235C2×C3×C4—31no—
236C2×C3×C4—61no—
237C2×C3×C4—91no—
238C2×C3×C4—181no—
239C2×C3×C4—181no—
240C2×C3×C4—181no—
241C2×C3×C4—181no—
242C2×C3×C4—181no—
243C2×C3×C4C2×D3211yescharacteristic
244#67—362no—
245#67—362no—
246C2×Dic3—31no—
247C2×Dic3—61no—
248C2×Dic3—61no—
249C2×Dic3—61no—
250C2×Dic3—61no—
251C2×Dic3—61no—
252C2×Dic3—91no—
253C2×Dic3—121no—
254C2×Dic3—121no—
255C2×Dic3—181no—
256D12—61no—
257D12—61no—
258D12—181no—
259D12—181no—
260D12—362no—
261C4×D3—61no—
262C4×D3—61no—
263C4×D3—181no—
264C4×D3—181no—
265C3×C8—21no—
266C3×C8—61no—
267C3×C8—61no—
268C3×C8—62no—
269C3×C8—181no—
270C3⋊C8—61no—
271C3⋊C8—181no—
272C2×C32—21no—
273C2×C32—21no—
274C2×C32—21no—
275C2×C32#88311yescharacteristic
276C2×C32#88611yescharacteristic
277C2×C32#90411yescharacteristic
278C2×C32#90811yescharacteristic
279C2×C32#91111yescharacteristic
280C2×C32#91611yescharacteristic
281C3⋊D3—41no—
282C3⋊D3—81no—
283C3×D3—121no—
284C3×D3—241no—
285C2×D4—91no—
286C2×D4—541no—
287C22×C4—91no—
288C2×C8—91no—
289C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
290C4⋊C4—91no—
291C4⋊C4—91no—
292C4⋊C4—91no—
293C22⋊C4—541no—
294C22⋊C4—541no—
295C42—541no—
296C22×C3—31no—
297C22×C3—31no—
298C22×C3—61no—
299C22×C3—181no—
300C22×C3—181no—
301C22×C3—181no—
302C22×C3—361no—
303C22×C3—361no—
304C22×C3—361no—
305C22×C3D3×D1211yescharacteristic
306C2×D3—61no—
307C2×D3—61no—
308C2×D3—61no—
309C2×D3—121no—
310C2×D3—121no—
311C2×D3—121no—
312C2×D3—181no—
313C2×D3—181no—
314C2×D3—181no—
315C2×D3—361no—
316C2×D3—361no—
317C2×D3—361no—
318C3×C4—31no—
319C3×C4—31no—
320C3×C4—31no—
321C3×C4—31no—
322C3×C4—61no—
323C3×C4—61no—
324C3×C4—181no—
325C3×C4—361no—
326C3×C4—361no—
327C3×C4—361no—
328C3×C4#379511yescharacteristic
329C3×C4#380511yescharacteristic
330Dic3—61no—
331Dic3—121no—
332Dic3—121no—
333Dic3—121no—
334Dic3—121no—
335Dic3—121no—
336Dic3—181no—
337Dic3—241no—
338Dic3—241no—
339Dic3—361no—
340C32—21no—
341C32#473711yescharacteristic, complemented
342C32#477911yescharacteristic, complemented
343C23—91no—
344C23—181no—
345D4—181no—
346D4—181no—
347D4—1082no—
348C2×C4—91no—
349C2×C4—181no—
350C2×C4—181no—
351C2×C4—181no—
352C2×C4—181no—
353C2×C4—181no—
354C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
355C8—21no—
356C8—181no—
357C2×C3—31no—
358C2×C3—31no—
359C2×C3—31no—
360C2×C3—31no—
361C2×C3—31no—
362C2×C3—31no—
363C2×C3—61no—
364C2×C3—61no—
365C2×C3—61no—
366C2×C3—361no—
367C2×C3—721no—
368C2×C3#6379511yescharacteristic
369C2×C3#6380611yescharacteristic
370C2×C3#6389111yescharacteristic
371D3—121no—
372D3—241no—
373D3—361no—
374D3—721no—
375C22—181no—
376C22—181no—
377C22—181no—
378C22—361no—
379C22—361no—
380C22—361no—
381C22#8865411yescharacteristic
382C4—181no—
383C4—361no—
384C4—361no—
385C4—361no—
386C4#8864811yescharacteristic
387C4#8867211yescharacteristic
388C3—31no—
389C3—31no—
390C3—61no—
391C3#1059083711yescharacteristic
392C2—361no—
393C2—721no—
394C2#1172084911yescharacteristic
395C2#1172085411yescharacteristic
396C2#1172113111yescharacteristic
subgroupG/H
centerC22#88654
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3D3×D12
prime radical#11721141C2
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#482
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
2. #11721081C2semidirect
3. #11720550C2semidirect
4. #63648⋊C3C2semidirect
5. #63647⋊C3C2semidirect
6. #11720308C2semidirect
7. #11720306C2semidirect
25. #63648D3semidirect
26. #63647D3semidirect
225. H(3,𝔽3)#482semidirect
341. C32#4737semidirect
342. C32#4779semidirect

Maximal subgroups

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

subgroupindex
#117211412
#117210812
#117205502
#63648⋊C32
#63647⋊C32
#117203082
#117203062
#105907043
#105906523