Finite Group Databasebeta

← #13160028

Subgroups — #13160028

Order 1728.

1340 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D3×C3⋊D3C211yesmaximal, characteristic
2C2×C32×ΓL(1,𝔽25)C211yesmaximal, characteristic, complemented
3C2×#88786C211yesmaximal, characteristic, complemented
4#11722190C211yesmaximal, characteristic, complemented
5C3×#63647C211yesmaximal, characteristic, complemented
6#11722024C211yesmaximal, characteristic, complemented
7#11722023C211yesmaximal, characteristic, complemented
8#10592825—31nomaximal
9#10590936—34nomaximal
10C22×D3×C3⋊D3C411yescharacteristic
11C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C32×C4×D3C2211yescharacteristic
14C2×#6289C411yescharacteristic
15C2×Dic3×C3⋊D3C411yescharacteristic
16C2×D3×C3⋊Dic3C411yescharacteristic
17C4×D3×C3⋊D3—22no—
18C4×D3×C3⋊D3—22no—
19C2×C33×C8C2211yescharacteristic, complemented
20C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
22C2×C32×C3⋊C8C2211yescharacteristic, complemented
23C32×ΓL(1,𝔽25)—22no—
24#88786—22no—
25C22×C4×C3⋊D3—31no—
26C2×C4×D32—34no—
27C2×C32×M16—31no—
28C2×#3748—31no—
29C2×C3×ΓL(1,𝔽25)D314yescomplemented
30C2×#3711—34no—
31#63647—31no—
32#63647—31no—
33#63647—31no—
34#63647—34no—
35#63647—64no—
36#63647D311yescharacteristic, complemented
37C3×(C2×D3)⋊C8—34no—
38#63557—34no—
39(C2×D3)⋊(C3⋊C8)—34no—
40C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
41C22×C3×C3⋊D3C2×C411yescharacteristic
42C22×C32×D3C2×C411yescharacteristic
43C2×D3×C3⋊D3—22no—
44C2×D3×C3⋊D3—22no—
45C2×D3×C3⋊D3—22no—
46C2×C33×C4C2311yescharacteristic
47C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
48C4×C3⋊(C3⋊D3)—21no—
49C4×C3⋊(C3⋊D3)—21no—
50C2×C3×C3⋊Dic3C2×C411yescharacteristic
51C3×C4×C3⋊D3—21no—
52C3×C4×C3⋊D3—21no—
53C2×C32×Dic3C2×C411yescharacteristic
54C32×C4×D3D412yes—
55C32×C4×D3D412yes—
56#6289—22no—
57Dic3×C3⋊D3—22no—
58D3×C3⋊Dic3—22no—
59C33×C8—21no—
60C3⋊(C3⋊(C3⋊C8))—21no—
61C3×C3⋊(C3⋊C8)—21no—
62C32×C3⋊C8—21no—
63#5126—94no—
64#4726—91no—
65C23×C3⋊D3—31no—
66C22×D32—34no—
67C22×C32×C4—31no—
68C22×C3⋊Dic3—31no—
69C2×C4×C3⋊D3—31no—
70C2×C4×C3⋊D3—31no—
71C2×C4×C3⋊D3—34no—
72C2×C4×C3⋊D3—31no—
73C2×C4×C3⋊D3—62no—
74C2×C4×C3⋊D3—62no—
75C2×C4×C3⋊D3—64no—
76C2×C4×C3⋊D3C2×D311yescharacteristic
77C2×C3×C4×D3—34no—
78C2×C3×C4×D3C2×D314yes—
79C2×#624—34no—
80C2×D3×Dic3—34no—
81C2×D3×Dic3—34no—
82C4×D32—68no—
83C4×D32—68no—
84C32×M16—62no—
85C2×C32×C8—24no—
86C2×C32×C8—31no—
87C2×C32×C8C2×D314yescomplemented
88C2×C32×C8C2×D311yescharacteristic, complemented
89#3748—62no—
90C2×C3⋊(C3⋊C8)—31no—
91C2×C3⋊(C3⋊C8)—34no—
92C2×C3⋊(C3⋊C8)—64no—
93C2×C3⋊(C3⋊C8)C2×D311yescharacteristic, complemented
94C2×C3×C3⋊C8—34no—
95C2×C3×C3⋊C8C2×D314yescomplemented
96C3×ΓL(1,𝔽25)—28no—
97#3711—68no—
98C22×C33C22×C411yescharacteristic
99C2×C3⋊(C3⋊D3)—21no—
100C2×C3⋊(C3⋊D3)—21no—
101C2×C3⋊(C3⋊D3)M1612yescomplemented
102C2×C3×C3⋊D3—21no—
103C2×C3×C3⋊D3—21no—
104C2×C3×C3⋊D3M1612yescomplemented
105C2×C32×D3—21no—
106C2×C32×D3—21no—
107C2×C32×D3C22⋊C412yes—
108D3×C3⋊D3—28no—
109C33×C4C2×D411yescharacteristic
110C33×C4C2×D411yescharacteristic
111C3⋊(C3⋊Dic3)—21no—
112C3×C3⋊Dic3—21no—
113C32×Dic3C22⋊C412yes—
114C22×C4×D3—94no—
115C22×C4×D3—91no—
116C2×C3×M16—34no—
117C2×#208—94no—
118C2×ΓL(1,𝔽25)—91no—
119C2×ΓL(1,𝔽25)C3⋊D311yescharacteristic, complemented
120C3×C22⋊C8—91no—
121#830—91no—
122(C2×D3)⋊C8—34no—
123(C2×D3)⋊C8—94no—
124(C2×D3)⋊C8—94no—
125(C2×D3)⋊C8—94no—
126(C2×D3)⋊C8—91no—
127(C2×D3)⋊C8—91no—
128(C2×D3)⋊C8—184no—
129C23×C32—31no—
130C22×C3⋊D3—31no—
131C22×C3⋊D3—34no—
132C22×C3⋊D3—64no—
133C22×C3⋊D3—62no—
134C22×C3⋊D3—62no—
135C22×C3⋊D3—62no—
136C22×C3⋊D3C4×D311yescharacteristic
137C22×C3×D3—34no—
138C22×C3×D3C4×D314yes—
139C2×D32—68no—
140C2×D32—68no—
141C2×D32—68no—
142C2×C32×C4—24no—
143C2×C32×C4—31no—
144C2×C32×C4—32no—
145C2×C32×C4—32no—
146C2×C32×C4C22×D314yes—
147C2×C32×C4C22×D311yescharacteristic
148C2×C3⋊Dic3—31no—
149C2×C3⋊Dic3—34no—
150C2×C3⋊Dic3—64no—
151C2×C3⋊Dic3—62no—
152C2×C3⋊Dic3C4×D311yescharacteristic
153C4×C3⋊D3—21no—
154C4×C3⋊D3—21no—
155C4×C3⋊D3—62no—
156C4×C3⋊D3—62no—
157C4×C3⋊D3—61no—
158C4×C3⋊D3—61no—
159C4×C3⋊D3—64no—
160C4×C3⋊D3—64no—
161C4×C3⋊D3—124no—
162C4×C3⋊D3—124no—
163C2×C3×Dic3—34no—
164C2×C3×Dic3C4×D314yes—
165C3×C4×D3—64no—
166C3×C4×D3—64no—
167C3×C4×D3D1218yes—
168C3×C4×D3#6718yes—
169#624—68no—
170D3×Dic3—68no—
171D3×Dic3—68no—
172C32×C8—21no—
173C32×C8—24no—
174C32×C8—44no—
175C32×C8—61no—
176C3⋊(C3⋊C8)—21no—
177C3⋊(C3⋊C8)—61no—
178C3⋊(C3⋊C8)—64no—
179C3⋊(C3⋊C8)—124no—
180C3×C3⋊C8—24no—
181C3×C3⋊C8—64no—
182#407—271noSylow
183C2×C33C2×C22⋊C411yescharacteristic
184C2×C33C2×M1611yescharacteristic
185C2×C33C2×M1611yescharacteristic
186C3⋊(C3⋊D3)—22no—
187C3×C3⋊D3—22no—
188C32×D3—22no—
189C23×D3—94no—
190C23×D3—91no—
191C22×C3×C4—34no—
192C22×C3×C4—91no—
193C22×Dic3—94no—
194C22×Dic3—91no—
195C2×C4×D3—34no—
196C2×C4×D3—94no—
197C2×C4×D3—94no—
198C2×C4×D3—91no—
199C2×C4×D3—94no—
200C2×C4×D3—91no—
201C2×C4×D3—91no—
202C2×C4×D3—184no—
203C2×C4×D3—188no—
204C2×C4×D3—188no—
205C2×C4×D3—182no—
206C2×C4×D3—182no—
207C2×C4×D3C2×C3⋊D311yescharacteristic
208C3×M16—68no—
209C2×C3×C8—24no—
210C2×C3×C8—34no—
211C2×C3×C8—91no—
212C2×C3×C8D3214yescomplemented
213C2×C3×C8C2×C3⋊D311yescharacteristic, complemented
214#208—188no—
215C2×C3⋊C8—34no—
216C2×C3⋊C8—94no—
217C2×C3⋊C8—91no—
218C2×C3⋊C8—184no—
219C2×C3⋊C8C2×C3⋊D311yescharacteristic, complemented
220ΓL(1,𝔽25)—22no—
221ΓL(1,𝔽25)—182no—
222C22×C32—24no—
223C22×C32—32no—
224C22×C32—61no—
225C22×C32—61no—
226C22×C32C2×C4×D314yes—
227C22×C32C2×C4×D311yescharacteristic
228C2×C3⋊D3—21no—
229C2×C3⋊D3—21no—
230C2×C3⋊D3—32no—
231C2×C3⋊D3—38no—
232C2×C3⋊D3—68no—
233C2×C3⋊D3—68no—
234C2×C3⋊D3—61no—
235C2×C3⋊D3—61no—
236C2×C3⋊D3—64no—
237C2×C3⋊D3—64no—
238C2×C3⋊D3—124no—
239C2×C3⋊D3—124no—
240C2×C3⋊D3ΓL(1,𝔽25)12yescomplemented
241C2×C3×D3—24no—
242C2×C3×D3—24no—
243C2×C3×D3—38no—
244C2×C3×D3—64no—
245C2×C3×D3—64no—
246C2×C3×D3(C2×D3)⋊C418yes—
247D32—632no—
248C32×C4—24no—
249C32×C4—24no—
250C32×C4—32no—
251C32×C4C2×D1214yes—
252C32×C4D3×D411yescharacteristic
253C32×C4D3×D411yescharacteristic
254C32×C4C2×#6714yes—
255C3⋊Dic3—21no—
256C3⋊Dic3—61no—
257C3⋊Dic3—64no—
258C3⋊Dic3—124no—
259C3×Dic3—64no—
260C3×Dic3(C2×D3)⋊C418yes—
261C23×C4—271no—
262C2×M16—31no—
263C2×M16—271no—
264C22⋊C8—91no—
265C22⋊C8—271no—
266C22⋊C8—271no—
267C22⋊C8—271no—
268C33#40711yescharacteristic, complemented, Sylow
269C23×C3—34no—
270C23×C3—91no—
271C22×D3—34no—
272C22×D3—94no—
273C22×D3—91no—
274C22×D3—184no—
275C22×D3—182no—
276C22×D3—182no—
277C22×D3—188no—
278C22×D3—188no—
279C22×D3—188no—
280C22×D3—182no—
281C22×D3C4×C3⋊D311yescharacteristic
282C2×C3×C4—24no—
283C2×C3×C4—34no—
284C2×C3×C4—38no—
285C2×C3×C4—38no—
286C2×C3×C4—91no—
287C2×C3×C4—181no—
288C2×C3×C4—181no—
289C2×C3×C4C2×D3214yes—
290C2×C3×C4C22×C3⋊D311yescharacteristic
291C2×Dic3—34no—
292C2×Dic3—94no—
293C2×Dic3—91no—
294C2×Dic3—184no—
295C2×Dic3—188no—
296C2×Dic3—182no—
297C2×Dic3C4×C3⋊D311yescharacteristic
298C4×D3—64no—
299C4×D3—64no—
300C4×D3—188no—
301C4×D3—188no—
302C4×D3—182no—
303C4×D3—184no—
304C4×D3—184no—
305C4×D3—181no—
306C4×D3—181no—
307C4×D3—182no—
308C4×D3—364no—
309C4×D3—364no—
310C4×D3C4⋊(C3⋊D3)12yes—
311C4×D3#63812yes—
312C3×C8—24no—
313C3×C8—21no—
314C3×C8—44no—
315C3×C8—64no—
316C3×C8—181no—
317C3⋊C8—21no—
318C3⋊C8—64no—
319C3⋊C8—184no—
320C3⋊C8—181no—
321C3⋊C8—364no—
322C2×C32—24no—
323C2×C32—24no—
324C2×C32—24no—
325C2×C32—62no—
326C2×C32D3×C22⋊C411yescharacteristic
327C2×C32C2×ΓL(1,𝔽25)11yescharacteristic
328C2×C32D3×M1614yes—
329C2×C32D3×M1614yes—
330C2×C32D3×M1611yescharacteristic
331C2×C32C2×(C2×D3)⋊C414yes—
332C3⋊D3—22no—
333C3⋊D3—62no—
334C3⋊D3—68no—
335C3⋊D3—128no—
336C3×D3—28no—
337C3×D3—68no—
338C24—271no—
339C22×C4—31no—
340C22×C4—91no—
341C22×C4—271no—
342C22×C4—271no—
343C22×C4—271no—
344C22×C4—271no—
345C22×C4—542no—
346C22×C4—542no—
347M16—62no—
348M16—542no—
349C2×C8—31no—
350C2×C8—91no—
351C2×C8—271no—
352C2×C8D3×C3⋊D311yescharacteristic, complemented
353C22×C3—24no—
354C22×C3—38no—
355C22×C3—64no—
356C22×C3—64no—
357C22×C3—92no—
358C22×C3—181no—
359C22×C3—181no—
360C22×C3C4×D3214yes—
361C22×C3C2×C4×C3⋊D311yescharacteristic
362C2×D3—21no—
363C2×D3—21no—
364C2×D3—38no—
365C2×D3—64no—
366C2×D3—64no—
367C2×D3—98no—
368C2×D3—92no—
369C2×D3—188no—
370C2×D3—188no—
371C2×D3—1832no—
372C2×D3—184no—
373C2×D3—181no—
374C2×D3—181no—
375C2×D3—184no—
376C2×D3—364no—
377C2×D3—364no—
378C2×D3#375212yes—
379C3×C4—24no—
380C3×C4—24no—
381C3×C4—38no—
382C3×C4—181no—
383C3×C4D3×D1214yes—
384C3×C4D3×#6714yes—
385C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
386C3×C4C2×#63811yescharacteristic
387Dic3—64no—
388Dic3—181no—
389Dic3—184no—
390Dic3—364no—
391Dic3#375212yes—
392C32—24no—
393C32#472611yescharacteristic, complemented
394C32#512614yescomplemented
395C23—31no—
396C23—91no—
397C23—271no—
398C23—542no—
399C23—542no—
400C23—542no—
401C2×C4—31no—
402C2×C4—32no—
403C2×C4—32no—
404C2×C4—91no—
405C2×C4—181no—
406C2×C4—181no—
407C2×C4—271no—
408C2×C4—542no—
409C2×C4—542no—
410C2×C4—542no—
411C2×C4—541no—
412C2×C4—541no—
413C2×C4C2×D3×C3⋊D311yescharacteristic
414C8—21no—
415C8—61no—
416C8—181no—
417C8—541no—
418C2×C3—24no—
419C2×C3—24no—
420C2×C3—24no—
421C2×C3—68no—
422C2×C3—182no—
423C2×C3D3×ΓL(1,𝔽25)14yes—
424C2×C3#6379114yes—
425C2×C3D3×(C2×D3)⋊C414yes—
426C2×C3M16×C3⋊D311yescharacteristic
427C2×C3M16×C3⋊D311yescharacteristic
428C2×C3C2×#375211yescharacteristic
429D3—22no—
430D3—68no—
431D3—182no—
432D3—188no—
433D3—368no—
434C22—32no—
435C22—61no—
436C22—61no—
437C22—92no—
438C22—181no—
439C22—181no—
440C22—272no—
441C22—548no—
442C22—541no—
443C22—541no—
444C22C4×D3×C3⋊D311yescharacteristic
445C4—32no—
446C4—181no—
447C4—541no—
448C4D3×C4⋊(C3⋊D3)11yescharacteristic
449C4D3×#63811yescharacteristic
450C3—24no—
451C3#1059093614yescomplemented
452C3#1059282511yescharacteristic, complemented
453C2—62no—
454C2—182no—
455C2—542no—
456C2C3⋊D3×ΓL(1,𝔽25)11yescharacteristic
457C2#1172331211yescharacteristic
458C2D3×#375211yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×D3×C3⋊D3
commutator, socleC22×C33C22×C4
FittingC2×C33×C8C22
prime radicalC22×D3×C3⋊D3C4
cosocleC2×C33×C4C23
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#407
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C2×C32×ΓL(1,𝔽25)C2semidirect
3. C2×#88786C2semidirect
4. #11722190C2semidirect
5. C3×#63647C2semidirect
6. #11722024C2semidirect
7. #11722023C2semidirect
19. C2×C33×C8C22semidirect
20. C2×C3⋊(C3⋊(C3⋊C8))C22semidirect
21. C2×C3×C3⋊(C3⋊C8)C22semidirect
22. C2×C32×C3⋊C8C22semidirect
29. C2×C3×ΓL(1,𝔽25)D3semidirect
36. #63647D3semidirect
87. C2×C32×C8C2×D3semidirect
88. C2×C32×C8C2×D3semidirect
93. C2×C3⋊(C3⋊C8)C2×D3semidirect
95. C2×C3×C3⋊C8C2×D3semidirect
101. C2×C3⋊(C3⋊D3)M16semidirect
104. C2×C3×C3⋊D3M16semidirect
119. C2×ΓL(1,𝔽25)C3⋊D3semidirect
212. C2×C3×C8D32semidirect
213. C2×C3×C8C2×C3⋊D3semidirect
219. C2×C3⋊C8C2×C3⋊D3semidirect
240. C2×C3⋊D3ΓL(1,𝔽25)semidirect
268. C33#407semidirect
352. C2×C8D3×C3⋊D3semidirect
393. C32#4726semidirect
394. C32#5126semidirect
451. C3#10590936semidirect
452. C3#10592825semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D3×C3⋊D32
C2×C32×ΓL(1,𝔽25)2
C2×#887862
#117221902
C3×#636472
#117220242
#117220232
#105928253
#105909363