Finite Group Databasebeta

← #13138984

Subgroups — #13138984

Order 1728.

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

normal subgroups only

This group has 435 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#11721158C211yesmaximal, characteristic
2#11721093C211yesmaximal, characteristic, complemented
3#11720554C211yesmaximal, characteristic, complemented
4#63648⋊C3C211yesmaximal, characteristic, complemented
5#63642⋊C3C211yesmaximal, characteristic, complemented
6#11720324C211yesmaximal, characteristic, complemented
7#11720308C211yesmaximal, characteristic, complemented
8#10590888—31nomaximal
9#10590882—31nomaximal
10C2×C4⋊#272C2211yescharacteristic
11C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
12C2×#6200—21no—
13C2×#6200—21no—
14C2×C8×H(3,𝔽3)C2211yescharacteristic
15C4⋊#1084C2211yescharacteristic
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#88449—21no—
20#88448—21no—
21#63914—31no—
22#63911—31no—
23#63872—31no—
24#63869—31no—
25#63648D311yescharacteristic, complemented
26#63642D311yescharacteristic, complemented
27C3×#821—31no—
28C3×#821—31no—
29C3×#821—31no—
30C3×#821—61no—
31C3×#815—31no—
32#63564—31no—
33#63563—31no—
34#63562—31no—
35#63562—31no—
36C22×#272—21no—
37C22×(C3⋊D3)⋊C3—21no—
38C2×C4×H(3,𝔽3)C2311yescharacteristic
39C2×#1084—21no—
40C4⋊#272—21no—
41C4⋊#272—21no—
42C2×(C3⋊Dic3)⋊C3—21no—
43C2×(C3⋊Dic3)⋊C3D411yescharacteristic
44C2×(C3⋊Dic3)⋊C3D411yescharacteristic
45C4⋊((C3⋊D3)⋊C3)—21no—
46C4⋊((C3⋊D3)⋊C3)—21no—
47#6200—42no—
48#6200—42no—
49C8×H(3,𝔽3)—21no—
50(C3⋊(C3⋊C8))⋊C3—21no—
51#4764—91no—
52#4764—91no—
53#4713—91no—
54C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
55C2×C3×D12—31no—
56C2×C3×D12—31no—
57C2×C3×D12—31no—
58C2×C3×D12—61no—
59C2×#626—61no—
60C2×#626—61no—
61C2×#625—61no—
62C2×#625—61no—
63C2×C32×C8—21no—
64C2×C32×C8C2×D311yescharacteristic
65C2×C32×C8C2×D311yescharacteristic
66C4⋊(C3⋊Dic3)C2×D311yescharacteristic
67C4×C3⋊Dic3C2×D311yescharacteristic
68C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
69C3×C4⋊Dic3—31no—
70C3×C4⋊Dic3—31no—
71C3×C4⋊Dic3—31no—
72C3×C4⋊Dic3—61no—
73C3×C4×Dic3—31no—
74C2×C3×C3⋊C8—31no—
75#3724—61no—
76Dic3⋊Dic3—61no—
77Dic3⋊Dic3—61no—
78Dic3⋊Dic3—61no—
79C22×H(3,𝔽3)C2×D411yescharacteristic
80C2×#272—41no—
81C2×#272—41no—
82C2×#272—41no—
83C2×(C3⋊D3)⋊C3—41no—
84C2×(C3⋊D3)⋊C3—41no—
85C2×(C3⋊D3)⋊C3—41no—
86C4×H(3,𝔽3)#4111yescharacteristic
87C4×H(3,𝔽3)#4111yescharacteristic
88#1084—41no—
89(C3⋊Dic3)⋊C3—21no—
90(C3⋊Dic3)⋊C3—21no—
91(C3⋊Dic3)⋊C3—41no—
92#940—91no—
93#940—91no—
94#890—91no—
95#890—91no—
96#874—91no—
97#870—91no—
98C3×#102—91no—
99C3×#102—91no—
100C3×#97—91no—
101#832—91no—
102#832—91no—
103#821—31no—
104#821—31no—
105#821—31no—
106#821—61no—
107#821—91no—
108#821—91no—
109#821—91no—
110#815—31no—
111#815—91no—
112#809—91no—
113#809—91no—
114C22×C3⋊D3—21no—
115C22×C3×D3—61no—
116C22×C3×D3—61no—
117C22×C3×D3—61no—
118C22×C3×D3—121no—
119C2×C32×C4—21no—
120C2×C32×C4C22×D311yescharacteristic
121C2×C32×C4C22×D311yescharacteristic
122C2×C3⋊Dic3—21no—
123C2×C3⋊Dic3D1211yescharacteristic
124C2×C3⋊Dic3D1211yescharacteristic
125C4⋊(C3⋊D3)—21no—
126C4⋊(C3⋊D3)—21no—
127C2×C3×Dic3—31no—
128C2×C3×Dic3—31no—
129C2×C3×Dic3—61no—
130C2×C3×Dic3—61no—
131C2×C3×Dic3—61no—
132C2×C3×Dic3—121no—
133C3×D12—62no—
134C3×D12—61no—
135C3×D12—61no—
136C3×D12—61no—
137C3×D12—61no—
138C3×D12—61no—
139C3×D12—61no—
140C3×D12—121no—
141#626—122no—
142#626—122no—
143#625—122no—
144#625—122no—
145C32×C8—21no—
146C32×C8—21no—
147C32×C8—41no—
148C3⋊(C3⋊C8)—21no—
149C3×C3⋊C8—61no—
150#490—271noSylow
151C2×H(3,𝔽3)#12411yescharacteristic
152C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
153C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
154#272—81no—
155(C3⋊D3)⋊C3—81no—
156C2×C3×D4—91no—
157C2×C3×D4—91no—
158C2×#67—181no—
159C2×#67—181no—
160C2×#67—181no—
161C2×#67—181no—
162C2×D12—31no—
163C2×D12—31no—
164C2×D12—31no—
165C2×D12—61no—
166C2×D12—91no—
167C2×D12—181no—
168C2×D12—181no—
169C2×C3×C8—31no—
170C2×C3×C8—31no—
171C2×C3×C8—61no—
172C2×C3×C8—91no—
173C2×C3×C8D3211yescharacteristic
174C3×C4⋊C4—91no—
175C3×C4⋊C4—91no—
176C3×C42—91no—
177C4⋊Dic3—31no—
178C4⋊Dic3—31no—
179C4⋊Dic3—31no—
180C4⋊Dic3—61no—
181C4⋊Dic3—91no—
182C4⋊Dic3—181no—
183C4⋊Dic3—181no—
184Dic3⋊C4—181no—
185Dic3⋊C4—181no—
186Dic3⋊C4—181no—
187Dic3⋊C4—181no—
188C4×Dic3—31no—
189C4×Dic3—91no—
190C2×C3⋊C8—31no—
191C2×C3⋊C8—91no—
192C22×C32—21no—
193C22×C32C2×D1211yescharacteristic
194C22×C32D3×D411yescharacteristic
195C2×C3⋊D3—41no—
196C2×C3⋊D3—41no—
197C2×C3⋊D3—41no—
198C2×C3×D3—122no—
199C2×C3×D3—121no—
200C2×C3×D3—121no—
201C2×C3×D3—121no—
202C2×C3×D3—121no—
203C2×C3×D3—121no—
204C2×C3×D3—121no—
205C2×C3×D3—121no—
206C2×C3×D3—121no—
207C2×C3×D3—121no—
208C2×C3×D3—241no—
209C2×C3×D3—241no—
210C32×C4—21no—
211C32×C4—21no—
212C32×C4#23511yescharacteristic
213C32×C4#23511yescharacteristic
214C32×C4#23511yescharacteristic
215C32×C4#23711yescharacteristic
216C3⋊Dic3—21no—
217C3⋊Dic3—21no—
218C3⋊Dic3—41no—
219C3×Dic3—61no—
220C3×Dic3—61no—
221C3×Dic3—121no—
222C3×Dic3—121no—
223C3×Dic3—121no—
224C3×Dic3—241no—
225#127—271no—
226#125—271no—
227#102—91no—
228#102—91no—
229#102—271no—
230#102—271no—
231#97—91no—
232H(3,𝔽3)#49011yescharacteristic, complemented, Sylow
233C23×C3—181no—
234C23×C3—181no—
235C22×D3—61no—
236C22×D3—61no—
237C22×D3—61no—
238C22×D3—121no—
239C22×D3—181no—
240C3×D4—181no—
241C3×D4—181no—
242C3×D4—181no—
243C3×D4—181no—
244C2×C3×C4—31no—
245C2×C3×C4—31no—
246C2×C3×C4—61no—
247C2×C3×C4—91no—
248C2×C3×C4—91no—
249C2×C3×C4—181no—
250C2×C3×C4—181no—
251C2×C3×C4C2×D3211yescharacteristic
252#67—362no—
253#67—362no—
254#67—362no—
255#67—362no—
256C2×Dic3—31no—
257C2×Dic3—31no—
258C2×Dic3—61no—
259C2×Dic3—61no—
260C2×Dic3—61no—
261C2×Dic3—91no—
262C2×Dic3—91no—
263C2×Dic3—121no—
264C2×Dic3—181no—
265D12—61no—
266D12—61no—
267D12—61no—
268D12—61no—
269D12—62no—
270D12—61no—
271D12—61no—
272D12—121no—
273D12—181no—
274D12—181no—
275D12—362no—
276D12—362no—
277C3×C8—21no—
278C3×C8—61no—
279C3×C8—61no—
280C3×C8—121no—
281C3×C8—181no—
282C3⋊C8—61no—
283C3⋊C8—181no—
284C2×C32—21no—
285C2×C32—21no—
286C2×C32—21no—
287C2×C32#87511yescharacteristic
288C2×C32#88511yescharacteristic
289C2×C32#90811yescharacteristic
290C2×C32#90811yescharacteristic
291C2×C32#91111yescharacteristic
292C2×C32#91411yescharacteristic
293C3⋊D3—81no—
294C3×D3—242no—
295C3×D3—241no—
296C3×D3—241no—
297C3×D3—241no—
298C2×D4—91no—
299C2×D4—91no—
300C2×D4—541no—
301C2×D4—541no—
302C2×C8—91no—
303C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
304C4⋊C4—91no—
305C4⋊C4—91no—
306C4⋊C4—541no—
307C4⋊C4—541no—
308C42—91no—
309C22×C3—31no—
310C22×C3—31no—
311C22×C3—61no—
312C22×C3—361no—
313C22×C3—361no—
314C22×C3—361no—
315C22×C3—361no—
316C22×C3—361no—
317C22×C3—361no—
318C22×C3D3×D1211yescharacteristic
319C2×D3—121no—
320C2×D3—121no—
321C2×D3—121no—
322C2×D3—121no—
323C2×D3—121no—
324C2×D3—121no—
325C2×D3—122no—
326C2×D3—121no—
327C2×D3—121no—
328C2×D3—121no—
329C2×D3—241no—
330C2×D3—241no—
331C2×D3—361no—
332C2×D3—361no—
333C2×D3—361no—
334C3×C4—31no—
335C3×C4—31no—
336C3×C4—31no—
337C3×C4—31no—
338C3×C4—61no—
339C3×C4—61no—
340C3×C4—181no—
341C3×C4—181no—
342C3×C4—361no—
343C3×C4—361no—
344C3×C4#379811yescharacteristic
345C3×C4#380411yescharacteristic
346Dic3—61no—
347Dic3—61no—
348Dic3—121no—
349Dic3—121no—
350Dic3—121no—
351Dic3—181no—
352Dic3—181no—
353Dic3—241no—
354Dic3—361no—
355C32—21no—
356C32#471311yescharacteristic, complemented
357C32#476411yescharacteristic, complemented
358C23—181no—
359C23—181no—
360D4—181no—
361D4—181no—
362D4—181no—
363D4—181no—
364D4—1082no—
365D4—1082no—
366C2×C4—91no—
367C2×C4—91no—
368C2×C4—181no—
369C2×C4—181no—
370C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
371C8—21no—
372C8—181no—
373C2×C3—31no—
374C2×C3—31no—
375C2×C3—31no—
376C2×C3—31no—
377C2×C3—31no—
378C2×C3—31no—
379C2×C3—61no—
380C2×C3—61no—
381C2×C3—61no—
382C2×C3—721no—
383C2×C3—721no—
384C2×C3#6379411yescharacteristic
385C2×C3#6379511yescharacteristic
386C2×C3#6385211yescharacteristic
387D3—241no—
388D3—241no—
389D3—242no—
390D3—241no—
391D3—721no—
392C22—361no—
393C22—361no—
394C22—361no—
395C22—361no—
396C22—361no—
397C22—361no—
398C22#8865411yescharacteristic
399C4—181no—
400C4—181no—
401C4—361no—
402C4—361no—
403C4#8865011yescharacteristic
404C4#8867111yescharacteristic
405C3—31no—
406C3—31no—
407C3—61no—
408C3#1059088211yescharacteristic
409C2—721no—
410C2—721no—
411C2#1172084811yescharacteristic
412C2#1172084911yescharacteristic
413C2#1172110311yescharacteristic
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#11721158C2
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#490
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. #11721093C2semidirect
3. #11720554C2semidirect
4. #63648⋊C3C2semidirect
5. #63642⋊C3C2semidirect
6. #11720324C2semidirect
7. #11720308C2semidirect
25. #63648D3semidirect
26. #63642D3semidirect
232. H(3,𝔽3)#490semidirect
356. C32#4713semidirect
357. C32#4764semidirect

Maximal subgroups

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

subgroupindex
#117211582
#117210932
#117205542
#63648⋊C32
#63642⋊C32
#117203242
#117203082
#105908883
#105908823