Finite Group Databasebeta

← #13138971

Subgroups — #13138971

Order 1728.

449 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 449 conjugacy classes of subgroups — too many to draw. Only its 73 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#11721156C211yesmaximal, characteristic
2#11721129C211yesmaximal, characteristic, complemented
3#11720554C211yesmaximal, characteristic, complemented
4#63647⋊C3C211yesmaximal, characteristic, complemented
5#63644⋊C3C211yesmaximal, characteristic, complemented
6#11720324C211yesmaximal, characteristic, complemented
7#11720307C211yesmaximal, characteristic, complemented
8#10590676—31nomaximal
9#10590649—31nomaximal
10C2×C4⋊#272C2211yescharacteristic
11C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
12C2×#6200—21no—
13C2×((C3⋊D3)⋊C3)⋊C4—21no—
14C2×C8×H(3,𝔽3)C2211yescharacteristic
15C4⋊#1084C2211yescharacteristic
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#88451—21no—
20#88448—21no—
21#63916—31no—
22#63908—31no—
23#63900—31no—
24D3×C4⋊Dic3—31no—
25#63647D311yescharacteristic, complemented
26#63644D311yescharacteristic, complemented
27C3×#821—31no—
28C3×#821—31no—
29C3×#821—61no—
30C3×(C2×D3)⋊C8—31no—
31C3×#817—31no—
32#63574—31no—
33#63573—31no—
34#63563—31no—
35#63562—31no—
36C22×#272—21no—
37C22×(C3⋊D3)⋊C3D411yescharacteristic
38C2×C4×H(3,𝔽3)C2311yescharacteristic
39C2×#1084—21no—
40C4⋊#272—21no—
41C4⋊#272—21no—
42C2×(C3⋊Dic3)⋊C3—21no—
43C2×(C3⋊Dic3)⋊C3—21no—
44C2×(C3⋊Dic3)⋊C3D411yescharacteristic
45C4×(C3⋊D3)⋊C3—21no—
46C4×(C3⋊D3)⋊C3—21no—
47#6200—42no—
48((C3⋊D3)⋊C3)⋊C4—42no—
49C8×H(3,𝔽3)—21no—
50(C3⋊(C3⋊C8))⋊C3—21no—
51#4863—91no—
52#4777—91no—
53#4735—91no—
54C2×C4×C3⋊D3C2×D311yescharacteristic
55C2×C3×D12—31no—
56C2×C3×D12—31no—
57C2×C3×D12—61no—
58C2×C3×C4×D3—31no—
59C2×#626—61no—
60C2×#625—61no—
61C2×#624—61no—
62C2×D3×Dic3—61no—
63C2×C32×C8—21no—
64C2×C32×C8C2×D311yescharacteristic
65C2×C32×C8C2×D311yescharacteristic
66C4⋊(C3⋊Dic3)C2×D311yescharacteristic
67C4⋊(C3⋊Dic3)C2×D311yescharacteristic
68C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
69C3×C4⋊Dic3—31no—
70C3×C4⋊Dic3—31no—
71C3×C4⋊Dic3—31no—
72C3×C4⋊Dic3—31no—
73C3×C4⋊Dic3—61no—
74C2×C3×C3⋊C8—31no—
75#3724—61no—
76Dic3⋊Dic3—61no—
77(C2×D3)⋊Dic3—61no—
78(C2×D3)⋊Dic3—61no—
79C22×H(3,𝔽3)C2×D411yescharacteristic
80C2×#272—41no—
81C2×#272—41no—
82C2×#272—41no—
83C2×(C3⋊D3)⋊C3—21no—
84C2×(C3⋊D3)⋊C3—21no—
85C2×(C3⋊D3)⋊C3SD1612yes—
86C4×H(3,𝔽3)#4111yescharacteristic
87C4×H(3,𝔽3)#4111yescharacteristic
88#1084—41no—
89(C3⋊Dic3)⋊C3—21no—
90(C3⋊Dic3)⋊C3—41no—
91(C3⋊Dic3)⋊C3—41no—
92#938—91no—
93#930—91no—
94C2×C4⋊Dic3—91no—
95#895—91no—
96D3×C4⋊C4—91no—
97D3×C4⋊C4—91no—
98C3×#106—91no—
99C3×#102—91no—
100C3×C22⋊C8—91no—
101#832—91no—
102#821—31no—
103#821—31no—
104#821—61no—
105#821—91no—
106(C2×D3)⋊C8—31no—
107(C2×D3)⋊C8—91no—
108#817—31no—
109#817—91no—
110#817—91no—
111#809—91no—
112#808—91no—
113#808—91no—
114C22×C3⋊D3D1211yescharacteristic
115C22×C3×D3—31no—
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⋊Dic3—21no—
124C2×C3⋊Dic3D1211yescharacteristic
125C4×C3⋊D3—21no—
126C4×C3⋊D3—21no—
127C2×C3×Dic3—31no—
128C2×C3×Dic3—61no—
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—121no—
139C3×C4×D3—61no—
140C3×C4×D3—61no—
141#626—122no—
142#625—122no—
143#624—122no—
144D3×Dic3—122no—
145C32×C8—21no—
146C32×C8—21no—
147C32×C8—41no—
148C3⋊(C3⋊C8)—21no—
149C3×C3⋊C8—61no—
150#481—271noSylow
151C2×H(3,𝔽3)#12311yescharacteristic
152C2×H(3,𝔽3)C2×SD1611yescharacteristic
153C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
154#272—81no—
155(C3⋊D3)⋊C3—22no—
156C2×C3×D4—91no—
157C22×C3×C4—91no—
158C2×#67—181no—
159C2×#67—181no—
160C22×Dic3—181no—
161C2×D12—31no—
162C2×D12—31no—
163C2×D12—61no—
164C2×D12—181no—
165C2×C4×D3—31no—
166C2×C4×D3—91no—
167C2×C4×D3—181no—
168C2×C4×D3—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×C4⋊C4—91no—
177#217—181no—
178(C2×D3)⋊C4—181no—
179(C2×D3)⋊C4—181no—
180C4⋊Dic3—31no—
181C4⋊Dic3—31no—
182C4⋊Dic3—31no—
183C4⋊Dic3—31no—
184C4⋊Dic3—61no—
185C4⋊Dic3—91no—
186C4⋊Dic3—91no—
187C4⋊Dic3—181no—
188Dic3⋊C4—181no—
189Dic3⋊C4—181no—
190C2×C3⋊C8—31no—
191C2×C3⋊C8—91no—
192C22×C32—21no—
193C22×C32C2×D1211yescharacteristic
194C22×C32D3×D411yescharacteristic
195C2×C3⋊D3—21no—
196C2×C3⋊D3—21no—
197C2×C3⋊D3#20412yes—
198C2×C3×D3—32no—
199C2×C3×D3—61no—
200C2×C3×D3—61no—
201C2×C3×D3—122no—
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#23711yescharacteristic
214C32×C4#23711yescharacteristic
215C32×C4#23911yescharacteristic
216C3⋊Dic3—21no—
217C3⋊Dic3—41no—
218C3⋊Dic3—41no—
219C3×Dic3—61no—
220C3×Dic3—121no—
221C3×Dic3—121no—
222C3×Dic3—121no—
223C3×Dic3—121no—
224C3×Dic3—241no—
225#121—271no—
226C2×C4⋊C4—271no—
227#106—91no—
228#106—271no—
229#102—91no—
230#102—271no—
231C22⋊C8—91no—
232H(3,𝔽3)#48111yescharacteristic, complemented, Sylow
233C23×C3—91no—
234C23×C3—181no—
235C22×D3—31no—
236C22×D3—61no—
237C22×D3—61no—
238C22×D3—91no—
239C22×D3—121no—
240C3×D4—181no—
241C3×D4—181no—
242C2×C3×C4—31no—
243C2×C3×C4—31no—
244C2×C3×C4—61no—
245C2×C3×C4—91no—
246C2×C3×C4—181no—
247C2×C3×C4—181no—
248C2×C3×C4—181no—
249C2×C3×C4—181no—
250C2×C3×C4—181no—
251C2×C3×C4C2×D3211yescharacteristic
252#67—362no—
253#67—362no—
254C2×Dic3—31no—
255C2×Dic3—61no—
256C2×Dic3—61no—
257C2×Dic3—61no—
258C2×Dic3—61no—
259C2×Dic3—91no—
260C2×Dic3—121no—
261C2×Dic3—181no—
262C2×Dic3—181no—
263C2×Dic3—362no—
264D12—61no—
265D12—61no—
266D12—61no—
267D12—61no—
268D12—62no—
269D12—121no—
270D12—362no—
271C4×D3—61no—
272C4×D3—61no—
273C4×D3—181no—
274C4×D3—181no—
275C4×D3—362no—
276C4×D3—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#88611yescharacteristic
288C2×C32#89411yescharacteristic
289C2×C32C2×#20411yescharacteristic
290C2×C32#90811yescharacteristic
291C2×C32D3×SD1611yescharacteristic
292C2×C32#91411yescharacteristic
293C3⋊D3—22no—
294C3×D3—62no—
295C3×D3—242no—
296C3×D3—241no—
297C3×D3—241no—
298C2×D4—91no—
299C2×D4—541no—
300C22×C4—91no—
301C22×C4—541no—
302C2×C8—91no—
303C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
304C4⋊C4—91no—
305C4⋊C4—91no—
306C4⋊C4—91no—
307C4⋊C4—541no—
308C22⋊C4—541no—
309C22×C3—31no—
310C22×C3—31no—
311C22×C3—61no—
312C22×C3—92no—
313C22×C3—181no—
314C22×C3—181no—
315C22×C3—361no—
316C22×C3—361no—
317C22×C3—361no—
318C22×C3D3×D1211yescharacteristic
319C2×D3—32no—
320C2×D3—61no—
321C2×D3—61no—
322C2×D3—92no—
323C2×D3—121no—
324C2×D3—121no—
325C2×D3—121no—
326C2×D3—121no—
327C2×D3—121no—
328C2×D3—121no—
329C2×D3—122no—
330C2×D3—181no—
331C2×D3—181no—
332C2×D3—241no—
333C2×D3—241no—
334C3×C4—31no—
335C3×C4—31no—
336C3×C4—31no—
337C3×C4—31no—
338C3×C4—61no—
339C3×C4—61no—
340C3×C4—181no—
341C3×C4—361no—
342C3×C4—361no—
343C3×C4—361no—
344C3×C4#379611yescharacteristic
345C3×C4#380411yescharacteristic
346Dic3—61no—
347Dic3—121no—
348Dic3—121no—
349Dic3—121no—
350Dic3—121no—
351Dic3—181no—
352Dic3—241no—
353Dic3—361no—
354Dic3—361no—
355C32—21no—
356C32#473511yescharacteristic, complemented
357C32#486311yescharacteristic, complemented
358C23—91no—
359C23—181no—
360D4—181no—
361D4—181no—
362D4—1082no—
363C2×C4—91no—
364C2×C4—181no—
365C2×C4—181no—
366C2×C4—181no—
367C2×C4—181no—
368C2×C4—181no—
369C2×C4—1082no—
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—182no—
383C2×C3—721no—
384C2×C3D3×#20411yescharacteristic
385C2×C3#6379411yescharacteristic
386C2×C3#6389011yescharacteristic
387D3—62no—
388D3—182no—
389D3—241no—
390D3—241no—
391D3—242no—
392C22—92no—
393C22—181no—
394C22—181no—
395C22—361no—
396C22—361no—
397C22—361no—
398C22#8865411yescharacteristic
399C4—181no—
400C4—361no—
401C4—361no—
402C4—361no—
403C4#8864911yescharacteristic
404C4#8867111yescharacteristic
405C3—31no—
406C3—31no—
407C3—61no—
408C3#1059067711yescharacteristic
409C2—182no—
410C2—721no—
411C2#1172084611yescharacteristic
412C2#1172084811yescharacteristic
413C2#1172113011yescharacteristic
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#11721156C2
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#481
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. #11721129C2semidirect
3. #11720554C2semidirect
4. #63647⋊C3C2semidirect
5. #63644⋊C3C2semidirect
6. #11720324C2semidirect
7. #11720307C2semidirect
25. #63647D3semidirect
26. #63644D3semidirect
232. H(3,𝔽3)#481semidirect
356. C32#4735semidirect
357. C32#4863semidirect

Maximal subgroups

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

subgroupindex
#117211562
#117211292
#117205542
#63647⋊C32
#63644⋊C32
#117203242
#117203072
#105906763
#105906493