Finite Group Databasebeta

← #13161116

Subgroups — #13161116

Order 1728.

1348 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1348 conjugacy classes of subgroups — too many to draw. Only its 158 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×#89015C211yesmaximal, characteristic, complemented
2#11723979C211yesmaximal, characteristic, complemented
3C2×C3×#3754C211yesmaximal, characteristic, complemented
4C2×C32×#213C211yesmaximal, characteristic, complemented
5#11722040C211yesmaximal, characteristic, complemented
6#11722032C211yesmaximal, characteristic, complemented
7#11722024C211yesmaximal, characteristic, complemented
8#10592857—31nomaximal
9#10591520—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×#6312—21no—
12C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×#634C2211yescharacteristic
14C2×C32×D12C2211yescharacteristic
15C2×#6291—21no—
16C2×D3×C3⋊Dic3—21no—
17#89015—22no—
18#89015—22no—
19#88864—21no—
20C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
21C3×#3754—22no—
22C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
23C32×#213—22no—
24C2×C32×C3⋊C8C2211yescharacteristic
25C2×#3830—31no—
26C2×#3796—34no—
27C2×C32×D8—31no—
28#64147—31no—
29#64147—34no—
30#64147—64no—
31C2×#3754D311yescharacteristic, complemented
32C2×C3×#214—34no—
33C2×C3×#213D314yescomplemented
34#63659—31no—
35#63647—31no—
36#63643—31no—
37#63566—34no—
38#63563—34no—
39#63557—34no—
40C23×C33—21no—
41C22×C3⋊(C3⋊D3)D411yescharacteristic
42C22×C32×D3—21no—
43C33×D4—21no—
44C33×D4—21no—
45C2×C33×C4C2311yescharacteristic
46#6312—42no—
47C2×C3⋊(C3⋊Dic3)—21no—
48C2×C3⋊(C3⋊Dic3)D411yescharacteristic
49C4×C3⋊(C3⋊D3)—21no—
50C4×C3⋊(C3⋊D3)—21no—
51C2×C3×C3⋊Dic3—21no—
52C3×#634—21no—
53C3×#634D412yescomplemented
54C32×D12—21no—
55C32×D12D412yescomplemented
56#6291—24no—
57D3×C3⋊Dic3—24no—
58C3×C3⋊(C3⋊C8)—21no—
59C32×C3⋊C8—21no—
60#5172—91no—
61#5158—94no—
62C2×C32×D4—24no—
63C2×C32×D4—31no—
64C2×C32×D4C2×D314yes—
65C2×C32×D4C2×D311yescharacteristic
66C2×#638—61no—
67C2×#638—61no—
68C2×#638—64no—
69C2×#638—68no—
70C22×C3⋊Dic3—61no—
71#3830—62no—
72#3830—62no—
73C2×C4×C3⋊D3—31no—
74C2×C4×C3⋊D3—34no—
75C2×C4×C3⋊D3—64no—
76C2×#634C2×D311yescharacteristic
77C2×C3×D12C2×D314yes—
78C2×C3×Dic6—34no—
79C2×#626—64no—
80C2×D3×Dic3—64no—
81#3796—68no—
82#3796—68no—
83C32×D8—62no—
84C2×C32×C8—31no—
85#3757—61no—
86#3757—64no—
87#3757—68no—
88#3754—22no—
89C4⋊(C3⋊Dic3)—31no—
90C4⋊(C3⋊Dic3)—34no—
91C4⋊(C3⋊Dic3)—64no—
92C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
93C3×#214—68no—
94C3×#213—28no—
95C2×C3×C3⋊C8—34no—
96C2×C3×C3⋊C8C2×D314yes—
97C22×C33—41no—
98C22×C33—41no—
99C22×C33—41no—
100C22×C33C2×D411yescharacteristic
101C2×C3⋊(C3⋊D3)—21no—
102C2×C3⋊(C3⋊D3)—21no—
103C2×C3⋊(C3⋊D3)—21no—
104C2×C32×D3—22no—
105C2×C32×D3—41no—
106C2×C32×D3—41no—
107C33×C4C2×D411yescharacteristic
108C33×C4C2×D411yescharacteristic
109C3⋊(C3⋊Dic3)—21no—
110C3⋊(C3⋊Dic3)—41no—
111C3×C3⋊Dic3—22no—
112C2×#239—91no—
113C2×#237—94no—
114C2×C3×SD16—91no—
115C2×C3×D8—34no—
116#938—94no—
117#938—91no—
118#938—184no—
119C2×#214—34no—
120C2×#213C3⋊D311yescharacteristic, complemented
121#835—91no—
122#832—94no—
123#821—91no—
124(C2×D3)⋊C8—94no—
125(C2×D3)⋊C8—91no—
126#816—94no—
127C23×C32—28no—
128C23×C32—24no—
129C23×C32—21no—
130C23×C32—61no—
131C22×C3⋊D3—31no—
132C22×C3⋊D3—34no—
133C22×C3⋊D3—64no—
134C22×C3×D3—24no—
135C32×D4—21no—
136C32×D4—21no—
137C32×D4—24no—
138C32×D4—24no—
139C32×D4—32no—
140C32×D4—44no—
141C32×D4—44no—
142C32×D4—61no—
143C2×C32×C4—24no—
144C2×C32×C4C22×D314yes—
145C2×C32×C4C22×D311yescharacteristic
146#638—64no—
147#638—1216no—
148#638—128no—
149#638—122no—
150C2×C3⋊Dic3—21no—
151C2×C3⋊Dic3—31no—
152C2×C3⋊Dic3—34no—
153C2×C3⋊Dic3—64no—
154C2×C3⋊Dic3—64no—
155C2×C3⋊Dic3—68no—
156C2×C3⋊Dic3—64no—
157C2×C3⋊Dic3—61no—
158C4×C3⋊D3—61no—
159C4×C3⋊D3—61no—
160C4×C3⋊D3—64no—
161C4×C3⋊D3—64no—
162C4×C3⋊D3—124no—
163C4×C3⋊D3—124no—
164#634—21no—
165#634#6712yescomplemented
166C2×C3×Dic3—64no—
167C3×D12—24no—
168C3×D12#6718yescomplemented
169C3×Dic6—38no—
170C3×Dic6—64no—
171#626—616no—
172D3×Dic3—616no—
173C32×C8—61no—
174C3⋊(C3⋊C8)—21no—
175C3×C3⋊C8—24no—
176C3×C3⋊C8—64no—
177#449—271noSylow
178C2×C33—81no—
179C2×C33#12011yescharacteristic
180C2×C33#13511yescharacteristic
181C2×C33AGL(1,ℤ/8)11yescharacteristic
182C3⋊(C3⋊D3)—41no—
183C32×D3—42no—
184C2×C3×Q8—91no—
185C2×C3×D4—24no—
186C2×C3×D4—34no—
187C2×C3×D4D3214yes—
188C2×C3×D4C2×C3⋊D311yescharacteristic
189C2×#67—184no—
190C2×#67—184no—
191C2×#67—181no—
192C2×#67—188no—
193C22×Dic3—184no—
194#239—182no—
195#239—182no—
196#237—188no—
197#237—188no—
198C2×D12—181no—
199C2×D12C2×C3⋊D311yescharacteristic
200C2×C4×D3—94no—
201C2×C4×D3—91no—
202C2×C4×D3—184no—
203C2×C4×D3—181no—
204C2×Dic6—34no—
205C3×SD16—182no—
206C3×D8—68no—
207C2×C3×C8—34no—
208C2×C3×C8—91no—
209#217—184no—
210#217—181no—
211#217—188no—
212#214—68no—
213#213—22no—
214C4⋊Dic3—94no—
215C4⋊Dic3—91no—
216C4⋊Dic3—184no—
217C2×C3⋊C8—34no—
218C2×C3⋊C8C2×C3⋊D311yescharacteristic
219C22×C32—24no—
220C22×C32—48no—
221C22×C32—48no—
222C22×C32—48no—
223C22×C32—44no—
224C22×C32—44no—
225C22×C32—41no—
226C22×C32—41no—
227C22×C32—44no—
228C22×C32—41no—
229C22×C32—62no—
230C22×C32—121no—
231C22×C32—121no—
232C22×C32D3×D414yes—
233C22×C32D3×D411yescharacteristic
234C2×C3⋊D3—68no—
235C2×C3⋊D3—64no—
236C2×C3⋊D3—61no—
237C2×C3⋊D3—61no—
238C2×C3⋊D3—61no—
239C2×C3⋊D3—64no—
240C2×C3⋊D3—64no—
241C2×C3⋊D3—124no—
242C2×C3⋊D3—124no—
243C2×C3×D3—28no—
244C2×C3×D3—44no—
245C2×C3×D3—44no—
246C32×C4—24no—
247C32×C4—24no—
248C32×C4D3×D414yes—
249C32×C4D3×D411yescharacteristic
250C32×C4C2×#6714yes—
251C32×C4C2×#6711yescharacteristic
252C3⋊Dic3—22no—
253C3⋊Dic3—61no—
254C3⋊Dic3—64no—
255C3⋊Dic3—124no—
256C3⋊Dic3—128no—
257C3⋊Dic3—124no—
258C3⋊Dic3—121no—
259C3×Dic3—68no—
260C2×#41—271no—
261C2×SD16—91no—
262C2×D8—31no—
263#121—271no—
264#103—271no—
265#102—271no—
266C22⋊C8—271no—
267C33#44911yescharacteristic, complemented, Sylow
268C23×C3—28no—
269C23×C3—24no—
270C23×C3—21no—
271C23×C3—64no—
272C22×D3—21no—
273C22×D3—94no—
274C22×D3—91no—
275C22×D3—184no—
276C3×Q8—92no—
277C3×Q8—181no—
278C3×D4—24no—
279C3×D4—24no—
280C3×D4—21no—
281C3×D4—21no—
282C3×D4—38no—
283C3×D4—44no—
284C3×D4—44no—
285C3×D4—64no—
286C2×C3×C4—24no—
287C2×C3×C4—181no—
288C2×C3×C4C2×D3214yes—
289C2×C3×C4C22×C3⋊D311yescharacteristic
290#67—1816no—
291#67—3616no—
292#67—362no—
293#67—368no—
294C2×Dic3—64no—
295C2×Dic3—94no—
296C2×Dic3—91no—
297C2×Dic3—184no—
298C2×Dic3—1816no—
299C2×Dic3—188no—
300C2×Dic3—184no—
301C2×Dic3—181no—
302D12—21no—
303D12—184no—
304D12#63812yescomplemented
305C4×D3—184no—
306C4×D3—184no—
307C4×D3—184no—
308C4×D3—181no—
309C4×D3—181no—
310C4×D3—364no—
311C4×D3—364no—
312Dic6—38no—
313Dic6—64no—
314C3×C8—64no—
315C3×C8—181no—
316C3⋊C8—21no—
317C3⋊C8—64no—
318C2×C32—24no—
319C2×C32—24no—
320C2×C32—24no—
321C2×C32—88no—
322C2×C32—84no—
323C2×C32—81no—
324C2×C32—122no—
325C2×C32#91114yes—
326C2×C32#91214yes—
327C2×C32#91411yescharacteristic
328C2×C32#91611yescharacteristic
329C2×C32#93714yes—
330C2×C32#93711yescharacteristic
331C3⋊D3—128no—
332C3⋊D3—124no—
333C3⋊D3—121no—
334C3×D3—48no—
335#41—542no—
336#41—542no—
337C2×Q8—91no—
338C2×D4—31no—
339C2×D4—541no—
340C2×D4—541no—
341C2×D4D3×C3⋊D311yescharacteristic
342C22×C4—271no—
343C22×C4—541no—
344SD16—182no—
345D8—62no—
346C2×C8—31no—
347C2×C8—91no—
348C4⋊C4—271no—
349C22⋊C4—541no—
350C22×C3—24no—
351C22×C3—48no—
352C22×C3—48no—
353C22×C3—48no—
354C22×C3—41no—
355C22×C3—41no—
356C22×C3—41no—
357C22×C3—44no—
358C22×C3—44no—
359C22×C3—44no—
360C22×C3—68no—
361C22×C3—124no—
362C22×C3—124no—
363C22×C3#381114yes—
364C22×C3D4×C3⋊D311yescharacteristic
365C2×D3—22no—
366C2×D3—41no—
367C2×D3—41no—
368C2×D3—188no—
369C2×D3—181no—
370C2×D3—181no—
371C2×D3—181no—
372C2×D3—184no—
373C2×D3—184no—
374C2×D3—184no—
375C2×D3—364no—
376C2×D3—364no—
377C3×C4—24no—
378C3×C4—24no—
379C3×C4—182no—
380C3×C4D3×#6714yes—
381C3×C4D3×#6714yes—
382C3×C4D4×C3⋊D311yescharacteristic
383C3×C4C2×#63811yescharacteristic
384Dic3—68no—
385Dic3—181no—
386Dic3—184no—
387Dic3—364no—
388Dic3—368no—
389Dic3—361no—
390Dic3—364no—
391C32—24no—
392C32#515814yescomplemented
393C32#517211yescharacteristic, complemented
394C23—21no—
395C23—61no—
396C23—271no—
397Q8—92no—
398Q8—181no—
399D4—21no—
400D4—21no—
401D4—32no—
402D4—61no—
403D4—544no—
404D4—1082no—
405C2×C4—181no—
406C2×C4—271no—
407C2×C4—544no—
408C2×C4—541no—
409C2×C4—541no—
410C2×C4—541no—
411C2×C4C2×D3×C3⋊D311yescharacteristic
412C8—61no—
413C8—181no—
414C2×C3—24no—
415C2×C3—24no—
416C2×C3—24no—
417C2×C3—88no—
418C2×C3—84no—
419C2×C3—81no—
420C2×C3—128no—
421C2×C3#6393614yes—
422C2×C3#6393714yes—
423C2×C3#6398314yes—
424C2×C3#6412011yescharacteristic
425C2×C3#6412111yescharacteristic
426C2×C3#6414611yescharacteristic
427D3—42no—
428D3—368no—
429D3—364no—
430D3—361no—
431C22—41no—
432C22—41no—
433C22—41no—
434C22—62no—
435C22—121no—
436C22—121no—
437C22—541no—
438C22—541no—
439C22—541no—
440C22#8903911yescharacteristic
441C4—182no—
442C4—541no—
443C4—1081no—
444C4D3×#63811yescharacteristic
445C4C3⋊D3×#6711yescharacteristic
446C3—24no—
447C3#1059152014yescomplemented
448C3#1059285711yescharacteristic, complemented
449C2—81no—
450C2—122no—
451C2—1081no—
452C2#1172356311yescharacteristic
453C2#1172356411yescharacteristic
454C2#1172364311yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC22×C33C2×D4
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#449
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#89015C2semidirect
2. #11723979C2semidirect
3. C2×C3×#3754C2semidirect
4. C2×C32×#213C2semidirect
5. #11722040C2semidirect
6. #11722032C2semidirect
7. #11722024C2semidirect
31. C2×#3754D3semidirect
33. C2×C3×#213D3semidirect
53. C3×#634D4semidirect
55. C32×D12D4semidirect
120. C2×#213C3⋊D3semidirect
165. #634#67semidirect
168. C3×D12#67semidirect
267. C33#449semidirect
304. D12#638semidirect
392. C32#5158semidirect
393. C32#5172semidirect
447. C3#10591520semidirect
448. C3#10592857semidirect

Maximal subgroups

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

subgroupindex
C2×#890152
#117239792
C2×C3×#37542
C2×C32×#2132
#117220402
#117220322
#117220242
#105928573
#105915203