Finite Group Databasebeta

← #13160029

Subgroups — #13160029

Order 1728.

1412 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D3×C3⋊D3C211yesmaximal, characteristic
2C2×#88850C211yesmaximal, characteristic, complemented
3C2×#88788C211yesmaximal, characteristic, complemented
4C3×#63647C211yesmaximal, characteristic, complemented
5C32×(C2×D3)⋊C8C211yesmaximal, characteristic, complemented
6#11722023C211yesmaximal, characteristic, complemented
7(C2×D3)⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
8#10592425—31nomaximal
9#10590937—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
21#88850—22no—
22C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
23C2×C32×C3⋊C8C2211yescharacteristic, complemented
24#88788—22no—
25C22×C4×C3⋊D3—31no—
26C2×C4×D32—34no—
27C2×#3743—31no—
28C2×#3743—31no—
29C2×#3743—34no—
30C2×#3743—64no—
31C2×#3712—34no—
32C32×C22⋊C8—31no—
33#63657—31no—
34#63647—31no—
35#63647D311yescharacteristic, complemented
36C3×(C2×D3)⋊C8—34no—
37C3×(C2×D3)⋊C8D314yescomplemented
38(C2×D3)⋊(C3⋊C8)—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)D412yes—
49C4×C3⋊(C3⋊D3)D412yes—
50C2×C3×C3⋊Dic3C2×C411yescharacteristic
51C3×C4×C3⋊D3—21no—
52C3×C4×C3⋊D3—21no—
53C2×C32×Dic3C2×C411yescharacteristic
54C32×C4×D3—21no—
55C32×C4×D3—21no—
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#4726—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—34no—
71C2×C4×C3⋊D3—31no—
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—
84C2×C32×C8—24no—
85C2×C32×C8—31no—
86C2×C32×C8C2×D314yescomplemented
87C2×C32×C8C2×D311yescharacteristic, complemented
88C2×C3⋊(C3⋊C8)—31no—
89C2×C3⋊(C3⋊C8)—34no—
90C2×C3⋊(C3⋊C8)—64no—
91C2×C3⋊(C3⋊C8)C2×D311yescharacteristic, complemented
92#3743—62no—
93#3743—62no—
94#3743—68no—
95#3743—616no—
96C2×C3×C3⋊C8—34no—
97C2×C3×C3⋊C8C2×D314yescomplemented
98#3712—68no—
99C22×C33C22×C411yescharacteristic
100C2×C3⋊(C3⋊D3)—21no—
101C2×C3⋊(C3⋊D3)—21no—
102C2×C3⋊(C3⋊D3)C22⋊C412yes—
103C2×C3×C3⋊D3—21no—
104C2×C3×C3⋊D3—21no—
105C2×C3×C3⋊D3M1612yescomplemented
106C2×C32×D3—21no—
107C2×C32×D3—21no—
108C2×C32×D3M1612yescomplemented
109D3×C3⋊D3—28no—
110C33×C4C2×D411yescharacteristic
111C33×C4C2×D411yescharacteristic
112C3⋊(C3⋊Dic3)C22⋊C412yes—
113C3×C3⋊Dic3—21no—
114C32×Dic3—21no—
115C22×C4×D3—94no—
116C22×C4×D3—91no—
117C2×ΓL(1,𝔽25)—94no—
118C2×ΓL(1,𝔽25)—94no—
119C2×ΓL(1,𝔽25)—91no—
120C2×ΓL(1,𝔽25)—91no—
121C2×ΓL(1,𝔽25)—184no—
122C3×C22⋊C8—34no—
123C3×C22⋊C8—91no—
124#830—94no—
125#830—91no—
126(C2×D3)⋊C8—34no—
127(C2×D3)⋊C8—94no—
128(C2×D3)⋊C8—91no—
129(C2×D3)⋊C8C3⋊D311yescharacteristic, complemented
130C23×C32—31no—
131C22×C3⋊D3—31no—
132C22×C3⋊D3—34no—
133C22×C3⋊D3—64no—
134C22×C3⋊D3—62no—
135C22×C3⋊D3—62no—
136C22×C3⋊D3—62no—
137C22×C3⋊D3C4×D311yescharacteristic
138C22×C3×D3—34no—
139C22×C3×D3C4×D314yes—
140C2×D32—68no—
141C2×D32—68no—
142C2×D32—68no—
143C2×C32×C4—24no—
144C2×C32×C4—31no—
145C2×C32×C4—61no—
146C2×C32×C4—61no—
147C2×C32×C4C22×D314yes—
148C2×C32×C4C22×D311yescharacteristic
149C2×C3⋊Dic3—31no—
150C2×C3⋊Dic3—34no—
151C2×C3⋊Dic3—64no—
152C2×C3⋊Dic3—62no—
153C2×C3⋊Dic3C4×D311yescharacteristic
154C4×C3⋊D3—21no—
155C4×C3⋊D3—21no—
156C4×C3⋊D3—32no—
157C4×C3⋊D3—32no—
158C4×C3⋊D3—38no—
159C4×C3⋊D3—38no—
160C4×C3⋊D3—68no—
161C4×C3⋊D3—68no—
162C4×C3⋊D3—62no—
163C4×C3⋊D3—62no—
164C2×C3×Dic3—34no—
165C2×C3×Dic3C4×D314yes—
166C3×C4×D3—24no—
167C3×C4×D3—24no—
168C3×C4×D3—64no—
169C3×C4×D3—64no—
170#624—68no—
171D3×Dic3—68no—
172D3×Dic3—68no—
173C32×C8—28no—
174C32×C8—24no—
175C32×C8—21no—
176C32×C8—61no—
177C3⋊(C3⋊C8)—21no—
178C3⋊(C3⋊C8)—68no—
179C3⋊(C3⋊C8)—64no—
180C3⋊(C3⋊C8)—61no—
181C3×C3⋊C8—24no—
182C3×C3⋊C8—64no—
183#407—271noSylow
184C2×C33C2×C22⋊C411yescharacteristic
185C2×C33C2×M1611yescharacteristic
186C2×C33C2×M1611yescharacteristic
187C3⋊(C3⋊D3)—22no—
188C3×C3⋊D3—22no—
189C32×D3—22no—
190C23×D3—94no—
191C23×D3—91no—
192C22×C3×C4—34no—
193C22×C3×C4—91no—
194C22×Dic3—94no—
195C22×Dic3—91no—
196C2×C4×D3—34no—
197C2×C4×D3—94no—
198C2×C4×D3—91no—
199C2×C4×D3—94no—
200C2×C4×D3—94no—
201C2×C4×D3—91no—
202C2×C4×D3—91no—
203C2×C4×D3—184no—
204C2×C4×D3—182no—
205C2×C4×D3—182no—
206C2×C4×D3—188no—
207C2×C4×D3—188no—
208C2×C4×D3C2×C3⋊D311yescharacteristic
209C2×C3×C8—24no—
210C2×C3×C8—34no—
211C2×C3×C8—91no—
212C2×C3×C8D3214yescomplemented
213C2×C3×C8C2×C3⋊D311yescharacteristic, complemented
214C2×C3⋊C8—34no—
215C2×C3⋊C8—94no—
216C2×C3⋊C8—91no—
217C2×C3⋊C8—184no—
218C2×C3⋊C8C2×C3⋊D311yescharacteristic, complemented
219ΓL(1,𝔽25)—182no—
220ΓL(1,𝔽25)—188no—
221ΓL(1,𝔽25)—188no—
222ΓL(1,𝔽25)—182no—
223ΓL(1,𝔽25)—1816no—
224C22×C32—24no—
225C22×C32—32no—
226C22×C32—61no—
227C22×C32—61no—
228C22×C32C2×C4×D314yes—
229C22×C32C2×C4×D311yescharacteristic
230C2×C3⋊D3—21no—
231C2×C3⋊D3—21no—
232C2×C3⋊D3—32no—
233C2×C3⋊D3—38no—
234C2×C3⋊D3—68no—
235C2×C3⋊D3—68no—
236C2×C3⋊D3—61no—
237C2×C3⋊D3—61no—
238C2×C3⋊D3—64no—
239C2×C3⋊D3—64no—
240C2×C3⋊D3—124no—
241C2×C3⋊D3—124no—
242C2×C3⋊D3ΓL(1,𝔽25)12yescomplemented
243C2×C3×D3—24no—
244C2×C3×D3—24no—
245C2×C3×D3—38no—
246C2×C3×D3—64no—
247C2×C3×D3—64no—
248C2×C3×D3ΓL(1,𝔽25)18yescomplemented
249D32—632no—
250C32×C4—24no—
251C32×C4—24no—
252C32×C4—61no—
253C32×C4D3×D414yes—
254C32×C4D3×D414yes—
255C32×C4D3×D411yescharacteristic
256C32×C4D3×D411yescharacteristic
257C3⋊Dic3—21no—
258C3⋊Dic3—32no—
259C3⋊Dic3—38no—
260C3⋊Dic3—68no—
261C3×Dic3—24no—
262C3×Dic3—64no—
263C23×C4—271no—
264C2×M16—271no—
265C2×M16—271no—
266C22⋊C8—31no—
267C22⋊C8—91no—
268C22⋊C8—271no—
269C22⋊C8—271no—
270C33#40711yescharacteristic, complemented, Sylow
271C23×C3—34no—
272C23×C3—91no—
273C22×D3—34no—
274C22×D3—94no—
275C22×D3—91no—
276C22×D3—184no—
277C22×D3—182no—
278C22×D3—182no—
279C22×D3—188no—
280C22×D3—188no—
281C22×D3—188no—
282C22×D3—182no—
283C22×D3C4×C3⋊D311yescharacteristic
284C2×C3×C4—24no—
285C2×C3×C4—34no—
286C2×C3×C4—64no—
287C2×C3×C4—64no—
288C2×C3×C4—91no—
289C2×C3×C4—181no—
290C2×C3×C4—181no—
291C2×C3×C4C2×D3214yes—
292C2×C3×C4C22×C3⋊D311yescharacteristic
293C2×Dic3—34no—
294C2×Dic3—94no—
295C2×Dic3—91no—
296C2×Dic3—184no—
297C2×Dic3—182no—
298C2×Dic3—188no—
299C2×Dic3C4×C3⋊D311yescharacteristic
300C4×D3—21no—
301C4×D3—21no—
302C4×D3—64no—
303C4×D3—64no—
304C4×D3—98no—
305C4×D3—98no—
306C4×D3—92no—
307C4×D3—92no—
308C4×D3—188no—
309C4×D3—188no—
310C4×D3—182no—
311C4×D3—188no—
312C4×D3—188no—
313C4×D3—182no—
314C3×C8—28no—
315C3×C8—24no—
316C3×C8—21no—
317C3×C8—64no—
318C3×C8—181no—
319C3⋊C8—21no—
320C3⋊C8—64no—
321C3⋊C8—188no—
322C3⋊C8—184no—
323C3⋊C8—181no—
324C2×C32—24no—
325C2×C32—24no—
326C2×C32—24no—
327C2×C32—62no—
328C2×C32D3×C22⋊C414yes—
329C2×C32D3×C22⋊C411yescharacteristic
330C2×C32C2×ΓL(1,𝔽25)14yes—
331C2×C32C2×ΓL(1,𝔽25)11yescharacteristic
332C2×C32D3×M1614yes—
333C2×C32D3×M1611yescharacteristic
334C3⋊D3—22no—
335C3⋊D3—62no—
336C3⋊D3—68no—
337C3⋊D3—128no—
338C3×D3—28no—
339C3×D3—68no—
340C24—271no—
341C22×C4—31no—
342C22×C4—91no—
343C22×C4—271no—
344C22×C4—271no—
345C22×C4—271no—
346C22×C4—271no—
347C22×C4—542no—
348C22×C4—542no—
349M16—542no—
350M16—542no—
351C2×C8—31no—
352C2×C8—91no—
353C2×C8—271no—
354C2×C8D3×C3⋊D311yescharacteristic, complemented
355C22×C3—24no—
356C22×C3—38no—
357C22×C3—64no—
358C22×C3—64no—
359C22×C3—92no—
360C22×C3—181no—
361C22×C3—181no—
362C22×C3C4×D3214yes—
363C22×C3C2×C4×C3⋊D311yescharacteristic
364C2×D3—21no—
365C2×D3—21no—
366C2×D3—38no—
367C2×D3—64no—
368C2×D3—64no—
369C2×D3—92no—
370C2×D3—98no—
371C2×D3—188no—
372C2×D3—188no—
373C2×D3—1832no—
374C2×D3—184no—
375C2×D3—184no—
376C2×D3—181no—
377C2×D3—181no—
378C2×D3—364no—
379C2×D3—364no—
380C2×D3#374312yescomplemented
381C3×C4—24no—
382C3×C4—24no—
383C3×C4—64no—
384C3×C4—181no—
385C3×C4#380214yes—
386C3×C4#381114yes—
387C3×C4D4×C3⋊D311yescharacteristic
388C3×C4D4×C3⋊D311yescharacteristic
389Dic3—21no—
390Dic3—64no—
391Dic3—92no—
392Dic3—98no—
393Dic3—188no—
394C32—24no—
395C32#472614yescomplemented
396C32#472611yescharacteristic, complemented
397C23—31no—
398C23—91no—
399C23—271no—
400C23—542no—
401C23—542no—
402C23—542no—
403C2×C4—31no—
404C2×C4—61no—
405C2×C4—61no—
406C2×C4—91no—
407C2×C4—181no—
408C2×C4—181no—
409C2×C4—271no—
410C2×C4—272no—
411C2×C4—272no—
412C2×C4—542no—
413C2×C4—542no—
414C2×C4—542no—
415C2×C4C2×D3×C3⋊D311yescharacteristic
416C8—21no—
417C8—61no—
418C8—181no—
419C8—541no—
420C2×C3—24no—
421C2×C3—24no—
422C2×C3—24no—
423C2×C3—68no—
424C2×C3—182no—
425C2×C3D3×ΓL(1,𝔽25)14yes—
426C2×C3D3×ΓL(1,𝔽25)14yes—
427C2×C3#6392114yes—
428C2×C3C22⋊C4×C3⋊D311yescharacteristic
429C2×C3C2×#374311yescharacteristic
430C2×C3M16×C3⋊D311yescharacteristic
431D3—22no—
432D3—68no—
433D3—182no—
434D3—188no—
435D3—368no—
436C22—32no—
437C22—61no—
438C22—61no—
439C22—92no—
440C22—181no—
441C22—181no—
442C22—272no—
443C22—548no—
444C22—541no—
445C22—541no—
446C22C4×D3×C3⋊D311yescharacteristic
447C4—61no—
448C4—181no—
449C4—272no—
450C4#8902611yescharacteristic
451C4#8903911yescharacteristic
452C3—24no—
453C3#1059093714yescomplemented
454C3#1059242511yescharacteristic, complemented
455C2—62no—
456C2—182no—
457C2—542no—
458C2D3×#374311yescharacteristic
459C2C3⋊D3×ΓL(1,𝔽25)11yescharacteristic
460C2#1172353911yescharacteristic
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×#88850C2semidirect
3. C2×#88788C2semidirect
4. C3×#63647C2semidirect
5. C32×(C2×D3)⋊C8C2semidirect
6. #11722023C2semidirect
7. (C2×D3)⋊(C3⋊(C3⋊C8))C2semidirect
19. C2×C33×C8C22semidirect
20. C2×C3⋊(C3⋊(C3⋊C8))C22semidirect
22. C2×C3×C3⋊(C3⋊C8)C22semidirect
23. C2×C32×C3⋊C8C22semidirect
35. #63647D3semidirect
37. C3×(C2×D3)⋊C8D3semidirect
86. C2×C32×C8C2×D3semidirect
87. C2×C32×C8C2×D3semidirect
91. C2×C3⋊(C3⋊C8)C2×D3semidirect
97. C2×C3×C3⋊C8C2×D3semidirect
105. C2×C3×C3⋊D3M16semidirect
108. C2×C32×D3M16semidirect
129. (C2×D3)⋊C8C3⋊D3semidirect
212. C2×C3×C8D32semidirect
213. C2×C3×C8C2×C3⋊D3semidirect
218. C2×C3⋊C8C2×C3⋊D3semidirect
242. C2×C3⋊D3ΓL(1,𝔽25)semidirect
248. C2×C3×D3ΓL(1,𝔽25)semidirect
270. C33#407semidirect
354. C2×C8D3×C3⋊D3semidirect
380. C2×D3#3743semidirect
395. C32#4726semidirect
396. C32#4726semidirect
453. C3#10590937semidirect
454. C3#10592425semidirect

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×#888502
C2×#887882
C3×#636472
C32×(C2×D3)⋊C82
#117220232
(C2×D3)⋊(C3⋊(C3⋊C8))2
#105924253
#105909373