Finite Group Databasebeta

← #13160086

Subgroups — #13160086

Order 1728.

1172 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1172 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
2C2×C3×#3744C211yesmaximal, characteristic, complemented
3#11723532C211yesmaximal, characteristic, complemented
4C2×#88789C211yesmaximal, characteristic, complemented
5#11722190C211yesmaximal, characteristic, complemented
6C32×#821C211yesmaximal, characteristic, complemented
7#11722039C211yesmaximal, characteristic, complemented
8#10592480—31nomaximal
9#10590964—34nomaximal
10C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×#6292—21no—
15C2×#6291—21no—
16C2×D3×C3⋊Dic3—21no—
17#89015—22no—
18#89015—22no—
19C2×C33×C8C2211yescharacteristic
20C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
21C3×#3744—22no—
22C32×C4⋊Dic3C2211yescharacteristic
23#88803—21no—
24#88789—22no—
25C2×#3830—31no—
26C2×#3796—34no—
27C2×#3753—31no—
28C2×#3744D311yescharacteristic, complemented
29#64104—31no—
30C2×C3×#204—34no—
31#63916—34no—
32C2×#3713—34no—
33C32×#102—31no—
34#63647—31no—
35#63647—34no—
36#63647—64no—
37#63637—31no—
38C3×#821D314yescomplemented
39#63565—34no—
40C22×C3⋊(C3⋊D3)D411yescharacteristic
41C22×C3×C3⋊D3—21no—
42C22×C32×D3—21no—
43C2×C33×C4C2311yescharacteristic
44C2×C3⋊(C3⋊Dic3)D411yescharacteristic
45C4×C3⋊(C3⋊D3)—21no—
46C4×C3⋊(C3⋊D3)—21no—
47C2×C3×C3⋊Dic3—21no—
48C3×C4⋊(C3⋊D3)—21no—
49C3×C4⋊(C3⋊D3)—21no—
50C3×#634—21no—
51C3×#634D412yescomplemented
52C2×C32×Dic3—21no—
53C32×D12—21no—
54C32×D12D412yescomplemented
55#6292—42no—
56#6291—24no—
57D3×C3⋊Dic3—24no—
58C33×C8—21no—
59C3⋊(C3⋊(C3⋊C8))—21no—
60#5172—91no—
61#4781—94no—
62C2×C32×D4—31no—
63C2×#638—61no—
64C22×C3⋊Dic3—61no—
65#3830—62no—
66#3830—62no—
67C2×C4⋊(C3⋊D3)—61no—
68C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
69C2×C4×C3⋊D3—31no—
70C2×C4×C3⋊D3—34no—
71C2×C4×C3⋊D3—64no—
72C2×#634C2×D311yescharacteristic
73C2×C3×D12—34no—
74C2×C3×D12C2×D314yes—
75C2×C3×Dic6—34no—
76C2×#626—64no—
77C2×#626—64no—
78C2×D3×Dic3—64no—
79#3796—68no—
80#3796—68no—
81C2×C32×C8—24no—
82C2×C32×C8C2×D314yes—
83C2×C32×C8C2×D311yescharacteristic
84C32×C4⋊C4—31no—
85#3753—62no—
86#3752—61no—
87C2×C3⋊(C3⋊C8)—31no—
88C2×C3⋊(C3⋊C8)—34no—
89C2×C3⋊(C3⋊C8)—64no—
90#3744—22no—
91C3×C4⋊Dic3C2×D314yes—
92C3×#204—68no—
93(C2×D3)⋊Dic3—64no—
94#3713—68no—
95C22×C33C2×D411yescharacteristic
96C2×C3⋊(C3⋊D3)—21no—
97C2×C3⋊(C3⋊D3)—21no—
98C2×C3⋊(C3⋊D3)—21no—
99C2×C3×C3⋊D3—41no—
100C2×C3×C3⋊D3—41no—
101C2×C3×C3⋊D3—41no—
102C2×C32×D3—22no—
103C2×C32×D3—41no—
104C2×C32×D3—41no—
105C33×C4C2×D411yescharacteristic
106C33×C4C2×D411yescharacteristic
107C3⋊(C3⋊Dic3)—21no—
108C3×C3⋊Dic3—22no—
109C32×Dic3—41no—
110C2×#239—91no—
111C2×#237—94no—
112C2×C3×SD16—91no—
113#938—91no—
114C2×#213—94no—
115C2×#213—91no—
116C2×#204—34no—
117#895—94no—
118C3×#102—34no—
119#835—91no—
120#821C3⋊D311yescharacteristic, complemented
121(C2×D3)⋊C8—94no—
122(C2×D3)⋊C8—91no—
123(C2×D3)⋊C8—184no—
124#810—94no—
125C23×C32—61no—
126C22×C3⋊D3—21no—
127C22×C3⋊D3—31no—
128C22×C3⋊D3—34no—
129C22×C3⋊D3—64no—
130C22×C3×D3—24no—
131C22×C3×D3—64no—
132C32×D4—32no—
133C32×D4—61no—
134C2×C32×C4—24no—
135C2×C32×C4—61no—
136C2×C32×C4C22×D314yes—
137C2×C32×C4C22×D311yescharacteristic
138#638—64no—
139C2×C3⋊Dic3—21no—
140C2×C3⋊Dic3—31no—
141C2×C3⋊Dic3—34no—
142C2×C3⋊Dic3—64no—
143C2×C3⋊Dic3—64no—
144C4⋊(C3⋊D3)—21no—
145C4⋊(C3⋊D3)—21no—
146C4⋊(C3⋊D3)—122no—
147C4×C3⋊D3—61no—
148C4×C3⋊D3—61no—
149C4×C3⋊D3—64no—
150C4×C3⋊D3—64no—
151C4×C3⋊D3—124no—
152C4×C3⋊D3—124no—
153#634—21no—
154#634#6712yescomplemented
155C2×C3×Dic3—24no—
156C2×C3×Dic3—64no—
157C3×D12—24no—
158C3×D12—64no—
159C3×D12—64no—
160C3×D12D1218yescomplemented
161C3×Dic6—38no—
162C3×Dic6—64no—
163#626—616no—
164#626—128no—
165D3×Dic3—616no—
166C32×C8—21no—
167C32×C8—24no—
168C32×C8—44no—
169C3⋊(C3⋊C8)—61no—
170C3⋊(C3⋊C8)—64no—
171C3⋊(C3⋊C8)—124no—
172#449—271noSylow
173C2×C33#12011yescharacteristic
174C2×C33#13511yescharacteristic
175C2×C33AGL(1,ℤ/8)11yescharacteristic
176C3⋊(C3⋊D3)—41no—
177C3×C3⋊D3—81no—
178C32×D3—42no—
179C2×C3×Q8—91no—
180C2×C3×D4—34no—
181C2×C3×D4—91no—
182C2×#67—181no—
183C2×#67—184no—
184C22×Dic3—184no—
185#239—182no—
186#239—182no—
187#237—188no—
188#237—188no—
189C2×D12—34no—
190C2×D12—181no—
191C2×D12—184no—
192C2×D12C2×C3⋊D311yescharacteristic
193C2×C4×D3—94no—
194C2×C4×D3—91no—
195C2×C4×D3—184no—
196C2×C4×D3—181no—
197C2×Dic6—34no—
198C3×SD16—182no—
199C2×C3×C8—24no—
200C2×C3×C8D3214yes—
201C2×C3×C8C2×C3⋊D311yescharacteristic
202C3×C4⋊C4—34no—
203#217—181no—
204#213—188no—
205#213—182no—
206(C2×D3)⋊C4—184no—
207C4⋊Dic3C2×C3⋊D311yescharacteristic
208C2×C3⋊C8—94no—
209C2×C3⋊C8—91no—
210C2×C3⋊C8—184no—
211#204—68no—
212C22×C32—24no—
213C22×C32—62no—
214C22×C32—121no—
215C22×C32—121no—
216C22×C32D3×D414yes—
217C22×C32D3×D411yescharacteristic
218C2×C3⋊D3—41no—
219C2×C3⋊D3—41no—
220C2×C3⋊D3—41no—
221C2×C3⋊D3—68no—
222C2×C3⋊D3—61no—
223C2×C3⋊D3—64no—
224C2×C3⋊D3—61no—
225C2×C3⋊D3—61no—
226C2×C3⋊D3—64no—
227C2×C3⋊D3—64no—
228C2×C3⋊D3—124no—
229C2×C3⋊D3—124no—
230C2×C3×D3—28no—
231C2×C3×D3—44no—
232C2×C3×D3—44no—
233C2×C3×D3—124no—
234C2×C3×D3—124no—
235C2×C3×D3—124no—
236C32×C4—24no—
237C32×C4—24no—
238C32×C4—121no—
239C32×C4C2×D1214yes—
240C32×C4D3×D414yes—
241C32×C4D3×D411yescharacteristic
242C32×C4C2×#6711yescharacteristic
243C3⋊Dic3—22no—
244C3⋊Dic3—61no—
245C3⋊Dic3—64no—
246C3⋊Dic3—124no—
247C3×Dic3—44no—
248C3×Dic3—68no—
249C2×#41—271no—
250C2×SD16—91no—
251C2×D8—271no—
252#121—271no—
253#103—271no—
254#102—31no—
255C22⋊C8—271no—
256C33#44911yescharacteristic, complemented, Sylow
257C23×C3—64no—
258C23×C3—181no—
259C22×D3—21no—
260C22×D3—64no—
261C22×D3—94no—
262C22×D3—91no—
263C22×D3—184no—
264C3×Q8—92no—
265C3×Q8—181no—
266C3×D4—38no—
267C3×D4—64no—
268C3×D4—181no—
269C3×D4—181no—
270C2×C3×C4—24no—
271C2×C3×C4—64no—
272C2×C3×C4—181no—
273C2×C3×C4C2×D3214yes—
274C2×C3×C4C22×C3⋊D311yescharacteristic
275#67—1816no—
276#67—362no—
277C2×Dic3—21no—
278C2×Dic3—64no—
279C2×Dic3—94no—
280C2×Dic3—91no—
281C2×Dic3—184no—
282C2×Dic3—1816no—
283D12—21no—
284D12—64no—
285D12—64no—
286D12—184no—
287D12—368no—
288D12C4⋊(C3⋊D3)12yescomplemented
289C4×D3—184no—
290C4×D3—184no—
291C4×D3—184no—
292C4×D3—181no—
293C4×D3—181no—
294C4×D3—364no—
295C4×D3—364no—
296Dic6—38no—
297Dic6—64no—
298C3×C8—24no—
299C3×C8—21no—
300C3×C8—44no—
301C3⋊C8—184no—
302C3⋊C8—181no—
303C3⋊C8—364no—
304C2×C32—24no—
305C2×C32—24no—
306C2×C32—24no—
307C2×C32—122no—
308C2×C32#88214yes—
309C2×C32#91114yes—
310C2×C32#91411yescharacteristic
311C2×C32#91614yes—
312C2×C32#91611yescharacteristic
313C2×C32#93711yescharacteristic
314C3⋊D3—81no—
315C3⋊D3—128no—
316C3⋊D3—121no—
317C3⋊D3—124no—
318C3×D3—48no—
319C3×D3—244no—
320#41—542no—
321#41—542no—
322C2×Q8—91no—
323C2×D4—31no—
324C2×D4—91no—
325C2×D4—541no—
326C2×D4—541no—
327C22×C4—271no—
328C22×C4—541no—
329SD16—182no—
330D8—542no—
331C2×C8—271no—
332C2×C8D3×C3⋊D311yescharacteristic
333C4⋊C4—31no—
334C22⋊C4—541no—
335C22×C3—24no—
336C22×C3—68no—
337C22×C3—124no—
338C22×C3—124no—
339C22×C3—361no—
340C22×C3—361no—
341C22×C3—361no—
342C22×C3#380214yes—
343C22×C3D4×C3⋊D311yescharacteristic
344C2×D3—22no—
345C2×D3—41no—
346C2×D3—41no—
347C2×D3—124no—
348C2×D3—124no—
349C2×D3—124no—
350C2×D3—188no—
351C2×D3—184no—
352C2×D3—184no—
353C2×D3—181no—
354C2×D3—181no—
355C2×D3—181no—
356C2×D3—184no—
357C2×D3—364no—
358C2×D3—364no—
359C3×C4—24no—
360C3×C4—24no—
361C3×C4—124no—
362C3×C4—182no—
363C3×C4D3×D1214yes—
364C3×C4D3×#6714yes—
365C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
366C3×C4D4×C3⋊D311yescharacteristic
367Dic3—41no—
368Dic3—68no—
369Dic3—181no—
370Dic3—184no—
371Dic3—364no—
372C32—24no—
373C32#478114yescomplemented
374C32#517211yescharacteristic, complemented
375C23—61no—
376C23—181no—
377C23—271no—
378Q8—92no—
379Q8—181no—
380D4—32no—
381D4—61no—
382D4—181no—
383D4—181no—
384D4—544no—
385D4—1082no—
386C2×C4—61no—
387C2×C4—181no—
388C2×C4—271no—
389C2×C4—544no—
390C2×C4—541no—
391C2×C4—541no—
392C2×C4C2×D3×C3⋊D311yescharacteristic
393C8—21no—
394C8—541no—
395C2×C3—24no—
396C2×C3—24no—
397C2×C3—24no—
398C2×C3—128no—
399C2×C3—721no—
400C2×C3#6379814yes—
401C2×C3#6380914yes—
402C2×C3#6392214yes—
403C2×C3#6409111yescharacteristic
404C2×C3#6412011yescharacteristic
405C2×C3#6412511yescharacteristic
406D3—42no—
407D3—244no—
408D3—368no—
409D3—361no—
410D3—364no—
411C22—62no—
412C22—121no—
413C22—121no—
414C22—361no—
415C22—361no—
416C22—361no—
417C22—541no—
418C22—541no—
419C22—541no—
420C22#8902611yescharacteristic
421C4—121no—
422C4—182no—
423C4—541no—
424C4D3×C4⋊(C3⋊D3)11yescharacteristic
425C4C3⋊D3×#6711yescharacteristic
426C3—24no—
427C3#1059096414yescomplemented
428C3#1059248011yescharacteristic, complemented
429C2—122no—
430C2—721no—
431C2—1081no—
432C2#1172332411yescharacteristic
433C2#1172334211yescharacteristic
434C2#1172354111yescharacteristic
subgroupG/H
centerC22#89026
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#449
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. C2×#89015C2semidirect
2. C2×C3×#3744C2semidirect
3. #11723532C2semidirect
4. C2×#88789C2semidirect
5. #11722190C2semidirect
6. C32×#821C2semidirect
7. #11722039C2semidirect
28. C2×#3744D3semidirect
38. C3×#821D3semidirect
51. C3×#634D4semidirect
54. C32×D12D4semidirect
120. #821C3⋊D3semidirect
154. #634#67semidirect
160. C3×D12D12semidirect
256. C33#449semidirect
288. D12C4⋊(C3⋊D3)semidirect
373. C32#4781semidirect
374. C32#5172semidirect
427. C3#10590964semidirect
428. C3#10592480semidirect

Maximal subgroups

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

subgroupindex
C2×#890152
C2×C3×#37442
#117235322
C2×#887892
#117221902
C32×#8212
#117220392
#105924803
#105909643