Finite Group Databasebeta

← #13158924

Subgroups — #13158924

Order 1728.

1120 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C4⋊(C3⋊(C3⋊D3))C211yesmaximal, characteristic
2C2×#88788C211yesmaximal, characteristic, complemented
3C2×#88785C211yesmaximal, characteristic, complemented
4C3×#63630C211yesmaximal, characteristic, complemented
5C32×C4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
6#11722024C211yesmaximal, characteristic, complemented
7#11722024C211yesmaximal, characteristic, complemented
8#10592535—31nomaximal
9#10590328—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C411yescharacteristic
11C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
12C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
13C33×C42C2211yescharacteristic
14#88859C411yescharacteristic
15#88859C411yescharacteristic
16C4⋊(C3⋊(C3⋊Dic3))C411yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21#88788—22no—
22#88785—22no—
23C2×#3743—31no—
24C2×C8×C3⋊D3—31no—
25C4×C4⋊(C3⋊D3)—31no—
26C4×C4⋊(C3⋊D3)—34no—
27C4×C4⋊(C3⋊D3)—64no—
28C2×#3712—34no—
29C2×#3710—34no—
30C32×C4⋊C8—31no—
31#63647—31no—
32#63647—31no—
33#63630D311yescharacteristic, complemented
34C3×C4⋊(C3⋊C8)D314yescomplemented
35C3×#803—34no—
36#63557—34no—
37#63557—34no—
38C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
39C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
40C2×C33×C4C2×C411yescharacteristic
41C2×C33×C4C2×C411yescharacteristic
42C2×C33×C4C2311yescharacteristic
43C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
44C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
45C4⋊(C3⋊(C3⋊D3))—22no—
46C4×C3⋊(C3⋊D3)—21no—
47C4×C3⋊(C3⋊D3)—21no—
48C4×C3⋊(C3⋊D3)—21no—
49C4×C3⋊(C3⋊D3)—21no—
50C3×C3⋊(C3⋊C8)—21no—
51C3×C3⋊(C3⋊C8)D412yescomplemented
52C32×C3⋊C8—21no—
53C32×C3⋊C8—21no—
54#4836—94no—
55#4706—91no—
56C2×C4⋊(C3⋊D3)—31no—
57C2×C4⋊(C3⋊D3)—34no—
58C2×C4⋊(C3⋊D3)—64no—
59C2×C4×C3⋊D3—31no—
60C2×C4×C3⋊D3—31no—
61C2×C4×C3⋊D3—34no—
62C2×C4×C3⋊D3—34no—
63C2×C4×C3⋊D3—64no—
64C2×C4×C3⋊D3—64no—
65C2×C32×C8—31no—
66C2×C32×C8—31no—
67C32×C42—24no—
68C32×C42C2×D314yes—
69C32×C42C2×D311yescharacteristic
70#3752—31no—
71#3752—31no—
72#3752—34no—
73#3752—34no—
74#3752—64no—
75#3752—64no—
76C4⋊(C3⋊Dic3)—31no—
77C4⋊(C3⋊Dic3)—34no—
78C4⋊(C3⋊Dic3)—64no—
79C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
80C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
81#3743—62no—
82C8×C3⋊D3—62no—
83C2×C3×C3⋊C8—34no—
84C2×C3×C3⋊C8—34no—
85C2×C3×C3⋊C8C2×D314yes—
86C2×C3×C3⋊C8C2×D314yes—
87#3712—68no—
88#3710—68no—
89C22×C33C22×C411yescharacteristic
90C2×C3⋊(C3⋊D3)—21no—
91C2×C3⋊(C3⋊D3)—21no—
92C2×C3⋊(C3⋊D3)—21no—
93C2×C3⋊(C3⋊D3)—21no—
94C2×C3⋊(C3⋊D3)—21no—
95C2×C3⋊(C3⋊D3)M1612yescomplemented
96C33×C4—21no—
97C33×C4—21no—
98C33×C4C2×D411yescharacteristic
99C33×C4#4111yescharacteristic
100C3⋊(C3⋊Dic3)—21no—
101C3⋊(C3⋊Dic3)—21no—
102C2×ΓL(1,𝔽25)—94no—
103C2×ΓL(1,𝔽25)—91no—
104C2×D3×C8—94no—
105C2×D3×C8—91no—
106C4×D12—94no—
107C4×D12—91no—
108C4×D12—184no—
109C3×C4⋊C8—34no—
110C3×#97—91no—
111(C2×D3)⋊C8—94no—
112(C2×D3)⋊C8—94no—
113(C2×D3)⋊C8—91no—
114(C2×D3)⋊C8—91no—
115C4⋊(C3⋊C8)C3⋊D311yescharacteristic, complemented
116#803—34no—
117C22×C3⋊D3—31no—
118C22×C3⋊D3—31no—
119C22×C3⋊D3—34no—
120C22×C3⋊D3—34no—
121C22×C3⋊D3—64no—
122C22×C3⋊D3—64no—
123C2×C32×C4—24no—
124C2×C32×C4—24no—
125C2×C32×C4—24no—
126C2×C32×C4C4×D314yes—
127C2×C32×C4C4×D314yes—
128C2×C32×C4C4×D311yescharacteristic
129C2×C32×C4C4×D311yescharacteristic
130C2×C32×C4C22×D314yes—
131C2×C32×C4C22×D311yescharacteristic
132C2×C3⋊Dic3—31no—
133C2×C3⋊Dic3—31no—
134C2×C3⋊Dic3—34no—
135C2×C3⋊Dic3—34no—
136C2×C3⋊Dic3—64no—
137C2×C3⋊Dic3—64no—
138C4⋊(C3⋊D3)—62no—
139C4⋊(C3⋊D3)—68no—
140C4⋊(C3⋊D3)—616no—
141C4×C3⋊D3—61no—
142C4×C3⋊D3—61no—
143C4×C3⋊D3—61no—
144C4×C3⋊D3—61no—
145C4×C3⋊D3—64no—
146C4×C3⋊D3—64no—
147C4×C3⋊D3—64no—
148C4×C3⋊D3—64no—
149C4×C3⋊D3—124no—
150C4×C3⋊D3—124no—
151C4×C3⋊D3—124no—
152C4×C3⋊D3—124no—
153C32×C8—61no—
154C32×C8—61no—
155C3⋊(C3⋊C8)—21no—
156C3⋊(C3⋊C8)D1212yescomplemented
157C3×C3⋊C8—24no—
158C3×C3⋊C8—24no—
159C3×C3⋊C8—38no—
160C3×C3⋊C8—64no—
161#435—271noSylow
162C2×C33C4×D411yescharacteristic
163C2×C33C2×M1611yescharacteristic
164C2×C33#13111yescharacteristic
165C3⋊(C3⋊D3)—22no—
166C3⋊(C3⋊D3)—41no—
167C2×D12—94no—
168C2×D12—91no—
169C2×D12—184no—
170C2×C4×D3—94no—
171C2×C4×D3—94no—
172C2×C4×D3—91no—
173C2×C4×D3—91no—
174C2×C4×D3—184no—
175C2×C4×D3—184no—
176C2×C3×C8—34no—
177C2×C3×C8—34no—
178C2×C3×C8—91no—
179C2×C3×C8—91no—
180C3×C42—24no—
181C3×C42D3214yes—
182C3×C42C2×C3⋊D311yescharacteristic
183(C2×D3)⋊C4—94no—
184(C2×D3)⋊C4—94no—
185(C2×D3)⋊C4—91no—
186(C2×D3)⋊C4—91no—
187(C2×D3)⋊C4—184no—
188(C2×D3)⋊C4—184no—
189C4⋊Dic3—94no—
190C4⋊Dic3—91no—
191C4⋊Dic3—184no—
192C2×C3⋊C8—34no—
193C2×C3⋊C8—34no—
194C2×C3⋊C8C2×C3⋊D311yescharacteristic
195C2×C3⋊C8C2×C3⋊D311yescharacteristic
196ΓL(1,𝔽25)—182no—
197ΓL(1,𝔽25)—188no—
198D3×C8—182no—
199D3×C8—188no—
200C22×C32—24no—
201C22×C32C2×C4×D314yes—
202C22×C32C2×C4×D311yescharacteristic
203C2×C3⋊D3—32no—
204C2×C3⋊D3—38no—
205C2×C3⋊D3—68no—
206C2×C3⋊D3—61no—
207C2×C3⋊D3—61no—
208C2×C3⋊D3—68no—
209C2×C3⋊D3—61no—
210C2×C3⋊D3—64no—
211C2×C3⋊D3—61no—
212C2×C3⋊D3—61no—
213C2×C3⋊D3—64no—
214C2×C3⋊D3—64no—
215C2×C3⋊D3—64no—
216C2×C3⋊D3—64no—
217C2×C3⋊D3—124no—
218C2×C3⋊D3—124no—
219C2×C3⋊D3—124no—
220C2×C3⋊D3—124no—
221C32×C4—28no—
222C32×C4—24no—
223C32×C4—21no—
224C32×C4—21no—
225C32×C4—24no—
226C32×C4—24no—
227C32×C4—24no—
228C32×C4—44no—
229C32×C4C2×D1211yescharacteristic
230C32×C4#23511yescharacteristic
231C32×C4D3×D414yes—
232C32×C4#23914yes—
233C3⋊Dic3—61no—
234C3⋊Dic3—61no—
235C3⋊Dic3—64no—
236C3⋊Dic3—64no—
237C3⋊Dic3—124no—
238C3⋊Dic3—124no—
239C2×M16—271no—
240C22×C8—271no—
241C4×D4—271no—
242C4⋊C8—31no—
243C22⋊C8—271no—
244C22⋊C8—271no—
245#97—91no—
246C33#43511yescharacteristic, complemented, Sylow
247C22×D3—94no—
248C22×D3—91no—
249C22×D3—94no—
250C22×D3—91no—
251C22×D3—184no—
252C22×D3—184no—
253C2×C3×C4—24no—
254C2×C3×C4—24no—
255C2×C3×C4—24no—
256C2×C3×C4#62414yes—
257C2×C3×C4#62414yes—
258C2×C3×C4C4×C3⋊D311yescharacteristic
259C2×C3×C4C4×C3⋊D311yescharacteristic
260C2×C3×C4C2×D3214yes—
261C2×C3×C4C22×C3⋊D311yescharacteristic
262C2×Dic3—94no—
263C2×Dic3—94no—
264C2×Dic3—91no—
265C2×Dic3—91no—
266C2×Dic3—184no—
267C2×Dic3—184no—
268D12—188no—
269D12—182no—
270D12—1816no—
271C4×D3—184no—
272C4×D3—184no—
273C4×D3—184no—
274C4×D3—184no—
275C4×D3—181no—
276C4×D3—181no—
277C4×D3—181no—
278C4×D3—181no—
279C4×D3—364no—
280C4×D3—364no—
281C4×D3—364no—
282C4×D3—364no—
283C3×C8—64no—
284C3×C8—64no—
285C3×C8—92no—
286C3×C8—181no—
287C3⋊C8—21no—
288C3⋊C8—21no—
289C3⋊C8—38no—
290C3⋊C8—64no—
291C2×C32—24no—
292C2×C32—24no—
293C2×C32—24no—
294C2×C32C4×D1211yescharacteristic
295C2×C32#89314yes—
296C2×C32#90114yes—
297C2×C32D3×M1614yes—
298C2×C32D3×M1611yescharacteristic
299C2×C32#90711yescharacteristic
300C3⋊D3—62no—
301C3⋊D3—68no—
302C3⋊D3—128no—
303C3⋊D3—128no—
304C3⋊D3—121no—
305C3⋊D3—124no—
306C2×D4—271no—
307C22×C4—271no—
308C22×C4—271no—
309M16—542no—
310C2×C8—31no—
311C2×C8—31no—
312C2×C8—91no—
313C2×C8—91no—
314C2×C8—542no—
315C4⋊C4—271no—
316C22⋊C4—271no—
317C22⋊C4—271no—
318C42D3×C3⋊D311yescharacteristic
319C22×C3—24no—
320C22×C3C2×#62414yes—
321C22×C3C2×C4×C3⋊D311yescharacteristic
322C2×D3—98no—
323C2×D3—92no—
324C2×D3—188no—
325C2×D3—184no—
326C2×D3—181no—
327C2×D3—181no—
328C2×D3—184no—
329C2×D3—188no—
330C2×D3—184no—
331C2×D3—184no—
332C2×D3—181no—
333C2×D3—181no—
334C2×D3—181no—
335C2×D3—184no—
336C2×D3—364no—
337C2×D3—364no—
338C2×D3—364no—
339C2×D3—364no—
340C3×C4—28no—
341C3×C4—24no—
342C3×C4—24no—
343C3×C4—21no—
344C3×C4—21no—
345C3×C4—24no—
346C3×C4—24no—
347C3×C4—44no—
348C3×C4#379914yes—
349C3×C4D3×D1214yes—
350C3×C4D4×C3⋊D311yescharacteristic
351C3×C4#383211yescharacteristic
352Dic3—181no—
353Dic3—181no—
354Dic3—184no—
355Dic3—184no—
356Dic3—364no—
357Dic3—364no—
358C32—24no—
359C32#470611yescharacteristic, complemented
360C32#483614yescomplemented
361C23—271no—
362C23—271no—
363D4—542no—
364C2×C4—271no—
365C2×C4—271no—
366C2×C4—541no—
367C2×C4—541no—
368C2×C4—541no—
369C2×C4—541no—
370C2×C4#628911yescharacteristic
371C2×C4#628911yescharacteristic
372C2×C4C2×D3×C3⋊D311yescharacteristic
373C8—61no—
374C8—61no—
375C8—92no—
376C8—181no—
377C2×C3—24no—
378C2×C3—24no—
379C2×C3—24no—
380C2×C3#6381714yes—
381C2×C3#6381814yes—
382C2×C3#6389414yes—
383C2×C3#6410211yescharacteristic
384C2×C3#6411011yescharacteristic
385C2×C3M16×C3⋊D311yescharacteristic
386D3—182no—
387D3—188no—
388D3—368no—
389D3—368no—
390D3—361no—
391D3—364no—
392C22—272no—
393C22—541no—
394C22—541no—
395C22—541no—
396C22—541no—
397C22—541no—
398C22C2×#628911yescharacteristic
399C4—21no—
400C4—21no—
401C4—541no—
402C4—541no—
403C4#8901811yescharacteristic
404C4C3⋊D3×D1211yescharacteristic
405C3—24no—
406C3#1059032814yescomplemented
407C3#1059253511yescharacteristic, complemented
408C2—542no—
409C2—1081no—
410C2#1172335611yescharacteristic
411C2#1172335811yescharacteristic
412C2#1172349411yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×D3×C3⋊D3
commutator, socleC22×C33C22×C4
FittingC33×C42C22
prime radicalC2×C4⋊(C3⋊(C3⋊D3))C4
cosocleC2×C33×C4C23
hypercenterC42D3×C3⋊D3
hypocenterC33#435
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C2×#88788C2semidirect
3. C2×#88785C2semidirect
4. C3×#63630C2semidirect
5. C32×C4⋊(C3⋊C8)C2semidirect
6. #11722024C2semidirect
7. #11722024C2semidirect
33. #63630D3semidirect
34. C3×C4⋊(C3⋊C8)D3semidirect
51. C3×C3⋊(C3⋊C8)D4semidirect
95. C2×C3⋊(C3⋊D3)M16semidirect
115. C4⋊(C3⋊C8)C3⋊D3semidirect
156. C3⋊(C3⋊C8)D12semidirect
246. C33#435semidirect
359. C32#4706semidirect
360. C32#4836semidirect
406. C3#10590328semidirect
407. C3#10592535semidirect

Maximal subgroups

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

subgroupindex
C4×C4⋊(C3⋊(C3⋊D3))2
C2×#887882
C2×#887852
C3×#636302
C32×C4⋊(C3⋊C8)2
#117220242
#105925353
#105903283