Finite Group Databasebeta

← #13161134

Subgroups — #13161134

Order 1728.

1628 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 1628 conjugacy classes of subgroups — too many to draw. Only its 180 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
1C2×#89080C211yesmaximal, characteristic
2#11723503C211yesmaximal, characteristic, complemented
3C2×#88788C211yesmaximal, characteristic, complemented
4C3×#63662C211yesmaximal, characteristic, complemented
5C32×#835C211yesmaximal, characteristic, complemented
6#11722036C211yesmaximal, characteristic, complemented
7#11722035C211yesmaximal, characteristic, complemented
8#10592502—31nomaximal
9#10591554—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11#89080—22no—
12#89080C414yescomplemented
13C2×C4⋊(C3⋊(C3⋊D3))—21no—
14C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
15C2×C4×C3⋊(C3⋊D3)—21no—
16C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
17C2×#6289—21no—
18C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
19C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
20C32×C4⋊Dic3C2211yescharacteristic
21C2×C32×C3⋊C8C2211yescharacteristic
22#88806—21no—
23#88788—22no—
24C2×#3832—31no—
25C2×#3832—34no—
26C2×#3832—64no—
27C2×#3743—31no—
28C4⋊C4×C3⋊D3—31no—
29#63900—34no—
30C2×#3712—34no—
31C32×#103—31no—
32#63662D311yescharacteristic, complemented
33#63648—31no—
34#63636—31no—
35C3×#835—34no—
36C3×#835D314yescomplemented
37#63564—34no—
38#63564—34no—
39C22×C3⋊(C3⋊D3)—21no—
40C22×C3⋊(C3⋊D3)D411yescharacteristic
41C33×Q8—21no—
42C33×Q8C2×C412yescomplemented
43C2×C33×C4—21no—
44C2×C33×C4C2311yescharacteristic
45C2×C3⋊(C3⋊Dic3)D411yescharacteristic
46C4⋊(C3⋊(C3⋊D3))—24no—
47C4⋊(C3⋊(C3⋊D3))—21no—
48C4⋊(C3⋊(C3⋊D3))C2×C412yescomplemented
49C4×C3⋊(C3⋊D3)—24no—
50C4×C3⋊(C3⋊D3)C2×C412yes—
51C4×C3⋊(C3⋊D3)D412yes—
52C2×C3×C3⋊Dic3—21no—
53C2×C32×Dic3—21no—
54#6289—42no—
55C3×C3⋊(C3⋊C8)—21no—
56C32×C3⋊C8—21no—
57#4803—94no—
58#4803—91no—
59C2×C32×Q8—24no—
60C2×C32×Q8C2×D314yes—
61C2×C32×Q8C2×D311yescharacteristic
62#3832—34no—
63#3832—316no—
64#3832—62no—
65#3832—68no—
66#3832—616no—
67#3832—616no—
68C2×C4⋊(C3⋊D3)—31no—
69C2×C4⋊(C3⋊D3)—34no—
70C2×C4⋊(C3⋊D3)—61no—
71C2×C4⋊(C3⋊D3)—64no—
72C2×C4⋊(C3⋊D3)—64no—
73C2×C4⋊(C3⋊D3)—68no—
74C2×C4×C3⋊D3—31no—
75C2×C4×C3⋊D3—34no—
76C2×C4×C3⋊D3—61no—
77C2×C4×C3⋊D3—64no—
78C2×C4×C3⋊D3—64no—
79C2×C4×C3⋊D3—68no—
80C2×C4×C3⋊D3—61no—
81C2×#624—64no—
82C2×C32×C8—31no—
83C32×C4⋊C4—31no—
84C4⋊(C3⋊Dic3)C2×D311yescharacteristic
85#3750—61no—
86C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
87#3743—62no—
88C3×C4⋊Dic3—34no—
89C3×C4⋊Dic3C2×D314yes—
90C2×C3×C3⋊C8—34no—
91C2×C3×C3⋊C8C2×D314yes—
92#3724—64no—
93#3712—68no—
94C22×C33C2×D411yescharacteristic
95C2×C3⋊(C3⋊D3)—21no—
96C2×C3⋊(C3⋊D3)—21no—
97C2×C3⋊(C3⋊D3)—22no—
98C2×C3⋊(C3⋊D3)—41no—
99C2×C3⋊(C3⋊D3)—41no—
100C2×C3⋊(C3⋊D3)C22⋊C412yes—
101C33×C4—22no—
102C33×C4C22×C411yescharacteristic
103C33×C4C2×D411yescharacteristic
104C3⋊(C3⋊Dic3)C22⋊C412yes—
105C3×C3⋊Dic3—41no—
106C32×Dic3—41no—
107C2×#239—94no—
108C2×#239—91no—
109C2×#239—184no—
110C2×ΓL(1,𝔽25)—94no—
111C2×ΓL(1,𝔽25)—91no—
112D3×C4⋊C4—94no—
113D3×C4⋊C4—91no—
114C3×#103—34no—
115C3×#103—91no—
116#835—34no—
117#835C3⋊D311yescharacteristic, complemented
118#821—94no—
119#821—91no—
120#809—94no—
121#809—91no—
122C22×C3⋊D3—31no—
123C22×C3⋊D3—34no—
124C22×C3⋊D3—64no—
125C22×C3⋊D3—68no—
126C22×C3⋊D3—64no—
127C22×C3⋊D3—61no—
128C32×Q8—28no—
129C32×Q8—24no—
130C32×Q8—21no—
131C32×Q8—28no—
132C32×Q8C4×D318yescomplemented
133C32×Q8C4×D312yescomplemented
134C2×C32×C4—28no—
135C2×C32×C4—24no—
136C2×C32×C4—21no—
137C2×C32×C4—24no—
138C2×C32×C4—61no—
139C2×C32×C4C22×D314yes—
140C2×C32×C4C22×D311yescharacteristic
141C2×C3⋊Dic3—21no—
142C2×C3⋊Dic3—31no—
143C2×C3⋊Dic3—34no—
144C2×C3⋊Dic3—64no—
145C4⋊(C3⋊D3)—32no—
146C4⋊(C3⋊D3)—38no—
147C4⋊(C3⋊D3)—68no—
148C4⋊(C3⋊D3)—64no—
149C4⋊(C3⋊D3)—616no—
150C4⋊(C3⋊D3)—632no—
151C4⋊(C3⋊D3)—68no—
152C4⋊(C3⋊D3)—64no—
153C4⋊(C3⋊D3)—61no—
154C4×C3⋊D3—32no—
155C4×C3⋊D3—32no—
156C4×C3⋊D3—38no—
157C4×C3⋊D3—38no—
158C4×C3⋊D3—68no—
159C4×C3⋊D3—68no—
160C4×C3⋊D3—64no—
161C4×C3⋊D3—616no—
162C4×C3⋊D3—632no—
163C4×C3⋊D3—122no—
164C2×C3×Dic3—24no—
165C2×C3×Dic3—64no—
166#624—128no—
167C32×C8—61no—
168C3⋊(C3⋊C8)—21no—
169C3×C3⋊C8—24no—
170C3×C3⋊C8—64no—
171#417—271noSylow
172C2×C33C2×C22⋊C411yescharacteristic
173C2×C33AGL(1,ℤ/8)11yescharacteristic
174C2×C33#13711yescharacteristic
175C3⋊(C3⋊D3)—22no—
176C3⋊(C3⋊D3)—42no—
177C2×C3×Q8—24no—
178C2×C3×Q8D3214yes—
179C2×C3×Q8C2×C3⋊D311yescharacteristic
180#239—916no—
181#239—94no—
182#239—1816no—
183#239—188no—
184#239—182no—
185#239—1816no—
186C2×D12—94no—
187C2×D12—91no—
188C2×D12—184no—
189C2×D12—184no—
190C2×D12—181no—
191C2×D12—188no—
192C2×C4×D3—94no—
193C2×C4×D3—91no—
194C2×C4×D3—184no—
195C2×C4×D3—184no—
196C2×C4×D3—181no—
197C2×C4×D3—188no—
198C2×C4×D3—181no—
199C2×C4×D3—184no—
200C2×C3×C8—34no—
201C2×C3×C8—91no—
202C3×C4⋊C4—34no—
203C3×C4⋊C4—91no—
204C4⋊Dic3—34no—
205C4⋊Dic3C2×C3⋊D311yescharacteristic
206Dic3⋊C4—181no—
207Dic3⋊C4—184no—
208C2×C3⋊C8—34no—
209C2×C3⋊C8C2×C3⋊D311yescharacteristic
210ΓL(1,𝔽25)—182no—
211ΓL(1,𝔽25)—188no—
212C22×C32—24no—
213C22×C32D3×D414yes—
214C22×C32D3×D411yescharacteristic
215C2×C3⋊D3—32no—
216C2×C3⋊D3—38no—
217C2×C3⋊D3—68no—
218C2×C3⋊D3—61no—
219C2×C3⋊D3—61no—
220C2×C3⋊D3—616no—
221C2×C3⋊D3—68no—
222C2×C3⋊D3—62no—
223C2×C3⋊D3—64no—
224C2×C3⋊D3—64no—
225C2×C3⋊D3—124no—
226C2×C3⋊D3—124no—
227C2×C3⋊D3—128no—
228C2×C3⋊D3—128no—
229C2×C3⋊D3—124no—
230C2×C3⋊D3—121no—
231C2×C3⋊D3—124no—
232C2×C3⋊D3—121no—
233C32×C4—216no—
234C32×C4—28no—
235C32×C4—22no—
236C32×C4—24no—
237C32×C4—24no—
238C32×C4—121no—
239C32×C4C2×C4×D314yes—
240C32×C4C2×C4×D311yescharacteristic
241C32×C4D3×D414yes—
242C32×C4D3×D411yescharacteristic
243C3⋊Dic3—32no—
244C3⋊Dic3—38no—
245C3⋊Dic3—41no—
246C3⋊Dic3—68no—
247C3×Dic3—44no—
248C3×Dic3—124no—
249C2×#41—271no—
250C2×M16—271no—
251C2×C4⋊C4—271no—
252#103—31no—
253#103—91no—
254#102—271no—
255#102—271no—
256C33#41711yescharacteristic, complemented, Sylow
257C22×D3—94no—
258C22×D3—91no—
259C22×D3—184no—
260C22×D3—188no—
261C22×D3—184no—
262C22×D3—181no—
263C3×Q8—28no—
264C3×Q8—24no—
265C3×Q8—21no—
266C3×Q8—28no—
267C3×Q8#62418yescomplemented
268C3×Q8C4×C3⋊D312yescomplemented
269C2×C3×C4—28no—
270C2×C3×C4—24no—
271C2×C3×C4—21no—
272C2×C3×C4—24no—
273C2×C3×C4—64no—
274C2×C3×C4—181no—
275C2×C3×C4C2×D3214yes—
276C2×C3×C4C22×C3⋊D311yescharacteristic
277C2×Dic3—21no—
278C2×Dic3—64no—
279C2×Dic3—94no—
280C2×Dic3—91no—
281C2×Dic3—184no—
282D12—98no—
283D12—92no—
284D12—188no—
285D12—184no—
286D12—1816no—
287D12—1832no—
288D12—188no—
289D12—184no—
290D12—181no—
291C4×D3—98no—
292C4×D3—98no—
293C4×D3—92no—
294C4×D3—92no—
295C4×D3—188no—
296C4×D3—188no—
297C4×D3—184no—
298C4×D3—1816no—
299C4×D3—1832no—
300C4×D3—362no—
301C4×D3—368no—
302C3×C8—64no—
303C3×C8—181no—
304C3⋊C8—21no—
305C3⋊C8—64no—
306C2×C32—24no—
307C2×C32—24no—
308C2×C32—24no—
309C2×C32D3×C22⋊C414yes—
310C2×C32D3×C22⋊C411yescharacteristic
311C2×C32#91414yes—
312C2×C32#91411yescharacteristic
313C2×C32#91814yes—
314C2×C32#91811yescharacteristic
315C3⋊D3—62no—
316C3⋊D3—68no—
317C3⋊D3—128no—
318C3⋊D3—1216no—
319C3⋊D3—128no—
320C3⋊D3—122no—
321#41—274no—
322#41—542no—
323C2×Q8D3×C3⋊D311yescharacteristic
324C2×D4—271no—
325C2×D4—541no—
326C22×C4—271no—
327C22×C4—541no—
328C22×C4—541no—
329M16—542no—
330C2×C8—31no—
331C2×C8—91no—
332C4⋊C4—31no—
333C4⋊C4—91no—
334C4⋊C4—541no—
335C22×C3—24no—
336C22×C3#381114yes—
337C22×C3D4×C3⋊D311yescharacteristic
338C2×D3—92no—
339C2×D3—98no—
340C2×D3—188no—
341C2×D3—181no—
342C2×D3—181no—
343C2×D3—184no—
344C2×D3—184no—
345C2×D3—1816no—
346C2×D3—182no—
347C2×D3—188no—
348C2×D3—364no—
349C2×D3—364no—
350C2×D3—368no—
351C2×D3—368no—
352C2×D3—361no—
353C2×D3—361no—
354C2×D3—364no—
355C2×D3—364no—
356C3×C4—216no—
357C3×C4—22no—
358C3×C4—28no—
359C3×C4—24no—
360C3×C4—24no—
361C3×C4—124no—
362C3×C4—361no—
363C3×C4C2×#62414yes—
364C3×C4#381114yes—
365C3×C4C2×C4×C3⋊D311yescharacteristic
366C3×C4D4×C3⋊D311yescharacteristic
367Dic3—41no—
368Dic3—92no—
369Dic3—98no—
370Dic3—124no—
371Dic3—188no—
372C32—24no—
373C32#480314yescomplemented
374C32#480311yescharacteristic, complemented
375C23—271no—
376C23—541no—
377Q8—21no—
378Q8#628912yescomplemented
379D4—272no—
380D4—544no—
381D4—541no—
382C2×C4—21no—
383C2×C4—61no—
384C2×C4—181no—
385C2×C4—271no—
386C2×C4—272no—
387C2×C4—272no—
388C2×C4—544no—
389C2×C4—1082no—
390C2×C4C2×D3×C3⋊D311yescharacteristic
391C8—61no—
392C8—181no—
393C2×C3—24no—
394C2×C3—24no—
395C2×C3—24no—
396C2×C3#6394114yes—
397C2×C3#6394514yes—
398C2×C3#6397414yes—
399C2×C3C22⋊C4×C3⋊D311yescharacteristic
400C2×C3#6412311yescharacteristic
401C2×C3#6412711yescharacteristic
402D3—188no—
403D3—182no—
404D3—368no—
405D3—3616no—
406D3—368no—
407D3—362no—
408C22—272no—
409C22—541no—
410C22—541no—
411C22—542no—
412C22—1081no—
413C22—1081no—
414C22#8903911yescharacteristic
415C4—22no—
416C4—121no—
417C4—272no—
418C4—361no—
419C4C2×#628911yescharacteristic
420C4#8903911yescharacteristic
421C3—24no—
422C3#1059155414yescomplemented
423C3#1059250211yescharacteristic, complemented
424C2—542no—
425C2—1082no—
426C2#1172357211yescharacteristic
427C2#1172357811yescharacteristic
428C2#1172362711yescharacteristic
subgroupG/H
centerC22#89039
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×#89080C2
cosocleC2×C33×C4C23
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#417
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. #11723503C2semidirect
3. C2×#88788C2semidirect
4. C3×#63662C2semidirect
5. C32×#835C2semidirect
6. #11722036C2semidirect
7. #11722035C2semidirect
12. #89080C4semidirect
32. #63662D3semidirect
36. C3×#835D3semidirect
42. C33×Q8C2×C4semidirect
48. C4⋊(C3⋊(C3⋊D3))C2×C4semidirect
117. #835C3⋊D3semidirect
132. C32×Q8C4×D3semidirect
133. C32×Q8C4×D3semidirect
256. C33#417semidirect
267. C3×Q8#624semidirect
268. C3×Q8C4×C3⋊D3semidirect
373. C32#4803semidirect
374. C32#4803semidirect
378. Q8#6289semidirect
422. C3#10591554semidirect
423. C3#10592502semidirect

Maximal subgroups

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

subgroupindex
C2×#890802
#117235032
C2×#887882
C3×#636622
C32×#8352
#117220362
#117220352
#105925023
#105915543