Finite Group Databasebeta

← #13159416

Subgroups — #13159416

Order 1728.

1444 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 1444 conjugacy classes of subgroups — too many to draw. Only its 170 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×#89026C211yesmaximal, characteristic
2#11723933C211yesmaximal, characteristic, complemented
3C2×#88788C211yesmaximal, characteristic, complemented
4C3×#63636C211yesmaximal, characteristic, complemented
5C32×#809C211yesmaximal, characteristic, complemented
6#11722034C211yesmaximal, characteristic, complemented
7#11722032C211yesmaximal, characteristic, complemented
8#10592469—31nomaximal
9#10590594—34nomaximal
10C22×D3×C3⋊D3—21no—
11C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×#6290—21no—
15#89026—22no—
16#89026C414yescomplemented
17C33×C4⋊C4C2211yescharacteristic, complemented
18#88859—21no—
19C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic, complemented
20C4×C3⋊(C3⋊Dic3)—21no—
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
22C2×C32×C3⋊C8C2211yescharacteristic, complemented
23#88788—22no—
24C2×D4×C3⋊D3—31no—
25C2×#3802—34no—
26C2×#3743—31no—
27#64101—31no—
28#64101—34no—
29#64101—64no—
30C2×#3712—34no—
31C32×#102—31no—
32#63659—31no—
33#63648—31no—
34#63636D311yescharacteristic, complemented
35C3×#809—34no—
36C3×#809D314yescomplemented
37#63563—34no—
38#63563—34no—
39C22×C3⋊(C3⋊D3)D411yescharacteristic
40C22×C3×C3⋊D3—21no—
41C22×C32×D3—21no—
42C2×D3×C3⋊D3—24no—
43C2×D3×C3⋊D3—42no—
44C2×D3×C3⋊D3—42no—
45C2×C33×C4—21no—
46C2×C33×C4C2311yescharacteristic
47C2×C3⋊(C3⋊Dic3)—21no—
48C2×C3⋊(C3⋊Dic3)D411yescharacteristic
49C4×C3⋊(C3⋊D3)C2×C412yes—
50C4×C3⋊(C3⋊D3)D412yes—
51C3×C4⋊(C3⋊D3)—21no—
52C3×C4⋊(C3⋊D3)C2×C412yes—
53C32×D12—21no—
54C32×D12C2×C412yes—
55#6290—24no—
56C3×C3⋊(C3⋊C8)—21no—
57C32×C3⋊C8—21no—
58#4770—94no—
59#4770—91no—
60C23×C3⋊D3—61no—
61C22×D32—64no—
62C2×C32×D4—31no—
63C2×#638—61no—
64D4×C3⋊D3—34no—
65D4×C3⋊D3—62no—
66C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
67C2×C4×C3⋊D3—31no—
68C2×C4×C3⋊D3—34no—
69C2×C4×C3⋊D3—64no—
70C2×C3×D12—34no—
71C2×C3×D12C2×D314yes—
72C2×#625—64no—
73#3802—316no—
74#3802—68no—
75C2×C32×C8—31no—
76C32×C4⋊C4—24no—
77C32×C4⋊C4C2×D314yescomplemented
78C32×C4⋊C4C2×D311yescharacteristic, complemented
79#3752—61no—
80#3752—64no—
81#3752—68no—
82C4⋊(C3⋊Dic3)—31no—
83C4⋊(C3⋊Dic3)—34no—
84C4⋊(C3⋊Dic3)—64no—
85C4×C3⋊Dic3—61no—
86C4×C3⋊Dic3—64no—
87C4×C3⋊Dic3—68no—
88C2×C3⋊(C3⋊C8)C2×D311yescharacteristic, complemented
89#3743—62no—
90C2×C3×C3⋊C8—34no—
91C2×C3×C3⋊C8C2×D314yescomplemented
92#3712—68no—
93C22×C33C2×D411yescharacteristic
94C2×C3⋊(C3⋊D3)—21no—
95C2×C3⋊(C3⋊D3)—21no—
96C2×C3⋊(C3⋊D3)C22⋊C412yes—
97C2×C3×C3⋊D3—22no—
98C2×C3×C3⋊D3—41no—
99C2×C3×C3⋊D3—41no—
100C2×C32×D3—22no—
101C2×C32×D3—41no—
102C2×C32×D3—41no—
103D3×C3⋊D3—48no—
104C33×C4—41no—
105C33×C4C22×C411yescharacteristic
106C33×C4C2×D411yescharacteristic
107C3⋊(C3⋊Dic3)—41no—
108C3⋊(C3⋊Dic3)C22⋊C412yes—
109C2×D3×D4—94no—
110C2×D3×D4—91no—
111C2×ΓL(1,𝔽25)—94no—
112C2×ΓL(1,𝔽25)—91no—
113#892—94no—
114#892—91no—
115#892—184no—
116C3×#102—34no—
117C3×#102—91no—
118#832—94no—
119#832—91no—
120#821—94no—
121#821—91no—
122#809—34no—
123#809C3⋊D311yescharacteristic, complemented
124C23×C32—61no—
125C22×C3⋊D3—21no—
126C22×C3⋊D3—31no—
127C22×C3⋊D3—34no—
128C22×C3⋊D3—64no—
129C22×C3⋊D3—64no—
130C22×C3⋊D3—122no—
131C22×C3⋊D3—122no—
132C22×C3×D3—24no—
133C22×C3×D3—64no—
134C2×D32—616no—
135C2×D32—128no—
136C2×D32—128no—
137C32×D4—32no—
138C32×D4—61no—
139C2×C32×C4—28no—
140C2×C32×C4—24no—
141C2×C32×C4—21no—
142C2×C32×C4—24no—
143C2×C32×C4C22×D314yes—
144C2×C32×C4C22×D311yescharacteristic
145#638—64no—
146C2×C3⋊Dic3—31no—
147C2×C3⋊Dic3—34no—
148C2×C3⋊Dic3—64no—
149C2×C3⋊Dic3—68no—
150C2×C3⋊Dic3—64no—
151C2×C3⋊Dic3—61no—
152C4⋊(C3⋊D3)—21no—
153C4⋊(C3⋊D3)C4×D312yes—
154C4×C3⋊D3—32no—
155C4×C3⋊D3—32no—
156C4×C3⋊D3—38no—
157C4×C3⋊D3—38no—
158C4×C3⋊D3—68no—
159C4×C3⋊D3—68no—
160C3×D12—24no—
161C3×D12—38no—
162C3×D12—64no—
163C3×D12C4×D318yes—
164#625—616no—
165C32×C8—61no—
166C3⋊(C3⋊C8)—21no—
167C3×C3⋊C8—24no—
168C3×C3⋊C8—64no—
169#418—271noSylow
170C2×C33C2×C22⋊C411yescharacteristic
171C2×C33AGL(1,ℤ/8)11yescharacteristic
172C2×C33AGL(1,ℤ/8)11yescharacteristic
173C3⋊(C3⋊D3)—22no—
174C3×C3⋊D3—42no—
175C32×D3—42no—
176C23×D3—181no—
177C23×D3—184no—
178C2×C3×D4—34no—
179C2×C3×D4—91no—
180C2×#67—181no—
181C2×#67—184no—
182D3×D4—916no—
183D3×D4—94no—
184D3×D4—182no—
185D3×D4—188no—
186C2×D12—34no—
187C2×D12C2×C3⋊D311yescharacteristic
188C2×C4×D3—94no—
189C2×C4×D3—91no—
190C2×C4×D3—184no—
191C2×C3×C8—34no—
192C2×C3×C8—91no—
193C3×C4⋊C4—24no—
194C3×C4⋊C4D3214yescomplemented
195C3×C4⋊C4C2×C3⋊D311yescharacteristic, complemented
196(C2×D3)⋊C4—184no—
197(C2×D3)⋊C4—181no—
198(C2×D3)⋊C4—188no—
199C4⋊Dic3—94no—
200C4⋊Dic3—91no—
201C4⋊Dic3—184no—
202C4×Dic3—184no—
203C4×Dic3—181no—
204C4×Dic3—188no—
205C2×C3⋊C8—34no—
206C2×C3⋊C8C2×C3⋊D311yescharacteristic, complemented
207ΓL(1,𝔽25)—182no—
208ΓL(1,𝔽25)—188no—
209C22×C32—24no—
210C22×C32—62no—
211C22×C32—121no—
212C22×C32—121no—
213C22×C32D3×D414yes—
214C22×C32D3×D411yescharacteristic
215C2×C3⋊D3—22no—
216C2×C3⋊D3—32no—
217C2×C3⋊D3—38no—
218C2×C3⋊D3—41no—
219C2×C3⋊D3—41no—
220C2×C3⋊D3—68no—
221C2×C3⋊D3—61no—
222C2×C3⋊D3—61no—
223C2×C3⋊D3—64no—
224C2×C3⋊D3—64no—
225C2×C3⋊D3—124no—
226C2×C3⋊D3—124no—
227C2×C3⋊D3—128no—
228C2×C3×D3—28no—
229C2×C3×D3—44no—
230C2×C3×D3—44no—
231C2×C3×D3—68no—
232C2×C3×D3—124no—
233C2×C3×D3—124no—
234D32—1232no—
235C32×C4—24no—
236C32×C4—24no—
237C32×C4—48no—
238C32×C4—44no—
239C32×C4—41no—
240C32×C4C2×C4×D314yes—
241C32×C4C2×C4×D311yescharacteristic
242C32×C4D3×D414yes—
243C32×C4D3×D411yescharacteristic
244C3⋊Dic3—32no—
245C3⋊Dic3—38no—
246C3⋊Dic3—68no—
247C3⋊Dic3—128no—
248C3⋊Dic3—124no—
249C3⋊Dic3—121no—
250C22×D4—271no—
251C2×M16—271no—
252#117—271no—
253#102—31no—
254#102—91no—
255#102—271no—
256#102—271no—
257C33#41811yescharacteristic, complemented, Sylow
258C23×C3—64no—
259C23×C3—181no—
260C22×D3—21no—
261C22×D3—64no—
262C22×D3—94no—
263C22×D3—91no—
264C22×D3—184no—
265C22×D3—184no—
266C22×D3—1816no—
267C22×D3—362no—
268C22×D3—362no—
269C22×D3—368no—
270C22×D3—368no—
271C3×D4—38no—
272C3×D4—64no—
273C3×D4—92no—
274C3×D4—181no—
275C2×C3×C4—28no—
276C2×C3×C4—24no—
277C2×C3×C4—21no—
278C2×C3×C4—24no—
279C2×C3×C4C2×D3214yes—
280C2×C3×C4C22×C3⋊D311yescharacteristic
281#67—184no—
282#67—1816no—
283C2×Dic3—94no—
284C2×Dic3—91no—
285C2×Dic3—184no—
286C2×Dic3—188no—
287C2×Dic3—184no—
288C2×Dic3—181no—
289D12—21no—
290D12—38no—
291D12—64no—
292D12C4×C3⋊D312yes—
293C4×D3—98no—
294C4×D3—98no—
295C4×D3—92no—
296C4×D3—92no—
297C4×D3—188no—
298C4×D3—188no—
299C3×C8—64no—
300C3×C8—181no—
301C3⋊C8—21no—
302C3⋊C8—64no—
303C2×C32—24no—
304C2×C32—24no—
305C2×C32—24no—
306C2×C32—122no—
307C2×C32D3×C22⋊C414yes—
308C2×C32D3×C22⋊C411yescharacteristic
309C2×C32#91114yes—
310C2×C32#91111yescharacteristic
311C2×C32#91414yes—
312C2×C32#91411yescharacteristic
313C3⋊D3—42no—
314C3⋊D3—62no—
315C3⋊D3—68no—
316C3⋊D3—128no—
317C3×D3—48no—
318C3×D3—128no—
319C24—541no—
320C2×D4—31no—
321C2×D4—91no—
322C2×D4—274no—
323C2×D4—542no—
324C2×D4—541no—
325C22×C4—271no—
326M16—542no—
327C2×C8—31no—
328C2×C8—91no—
329C4⋊C4—271no—
330C4⋊C4D3×C3⋊D311yescharacteristic, complemented
331C22⋊C4—541no—
332C42—541no—
333C22×C3—24no—
334C22×C3—68no—
335C22×C3—124no—
336C22×C3—124no—
337C22×C3—182no—
338C22×C3—361no—
339C22×C3—361no—
340C22×C3#381114yes—
341C22×C3D4×C3⋊D311yescharacteristic
342C2×D3—22no—
343C2×D3—41no—
344C2×D3—41no—
345C2×D3—68no—
346C2×D3—92no—
347C2×D3—98no—
348C2×D3—124no—
349C2×D3—124no—
350C2×D3—188no—
351C2×D3—181no—
352C2×D3—181no—
353C2×D3—184no—
354C2×D3—184no—
355C2×D3—364no—
356C2×D3—364no—
357C2×D3—368no—
358C2×D3—3632no—
359C3×C4—24no—
360C3×C4—24no—
361C3×C4—48no—
362C3×C4—41no—
363C3×C4—44no—
364C3×C4C4×D3214yes—
365C3×C4#380214yes—
366C3×C4C2×C4×C3⋊D311yescharacteristic
367C3×C4D4×C3⋊D311yescharacteristic
368Dic3—92no—
369Dic3—98no—
370Dic3—188no—
371Dic3—368no—
372Dic3—361no—
373Dic3—364no—
374C32—24no—
375C32#477014yescomplemented
376C32#477011yescharacteristic, complemented
377C23—61no—
378C23—181no—
379C23—271no—
380C23—544no—
381C23—1082no—
382C23—1082no—
383D4—32no—
384D4—61no—
385D4—92no—
386D4—181no—
387D4—544no—
388C2×C4—21no—
389C2×C4—271no—
390C2×C4—272no—
391C2×C4—272no—
392C2×C4—541no—
393C2×C4C2×D3×C3⋊D311yescharacteristic
394C8—61no—
395C8—181no—
396C2×C3—24no—
397C2×C3—24no—
398C2×C3—24no—
399C2×C3—128no—
400C2×C3—362no—
401C2×C3#6392114yes—
402C2×C3#6392714yes—
403C2×C3#6394114yes—
404C2×C3C22⋊C4×C3⋊D311yescharacteristic
405C2×C3#6412011yescharacteristic
406C2×C3#6412311yescharacteristic
407D3—42no—
408D3—128no—
409D3—188no—
410D3—182no—
411D3—368no—
412C22—62no—
413C22—121no—
414C22—121no—
415C22—182no—
416C22—272no—
417C22—361no—
418C22—361no—
419C22—541no—
420C22—541no—
421C22—1088no—
422C22#8903911yescharacteristic
423C4—41no—
424C4—272no—
425C4—1081no—
426C4C4×D3×C3⋊D311yescharacteristic
427C4#8902611yescharacteristic
428C3—24no—
429C3#1059059414yescomplemented
430C3#1059246911yescharacteristic, complemented
431C2—122no—
432C2—362no—
433C2—542no—
434C2#1172353911yescharacteristic
435C2#1172354811yescharacteristic
436C2#1172357211yescharacteristic
subgroupG/H
centerC22#89039
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×#89026C2
cosocleC2×C33×C4C23
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#418
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #11723933C2semidirect
3. C2×#88788C2semidirect
4. C3×#63636C2semidirect
5. C32×#809C2semidirect
6. #11722034C2semidirect
7. #11722032C2semidirect
16. #89026C4semidirect
17. C33×C4⋊C4C22semidirect
19. C4⋊(C3⋊(C3⋊Dic3))C22semidirect
21. C2×C3×C3⋊(C3⋊C8)C22semidirect
22. C2×C32×C3⋊C8C22semidirect
34. #63636D3semidirect
36. C3×#809D3semidirect
77. C32×C4⋊C4C2×D3semidirect
78. C32×C4⋊C4C2×D3semidirect
88. C2×C3⋊(C3⋊C8)C2×D3semidirect
91. C2×C3×C3⋊C8C2×D3semidirect
123. #809C3⋊D3semidirect
194. C3×C4⋊C4D32semidirect
195. C3×C4⋊C4C2×C3⋊D3semidirect
206. C2×C3⋊C8C2×C3⋊D3semidirect
257. C33#418semidirect
330. C4⋊C4D3×C3⋊D3semidirect
375. C32#4770semidirect
376. C32#4770semidirect
429. C3#10590594semidirect
430. C3#10592469semidirect

Maximal subgroups

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

subgroupindex
C2×#890262
#117239332
C2×#887882
C3×#636362
C32×#8092
#117220342
#117220322
#105924693
#105905943