Finite Group Databasebeta

← #13160382

Subgroups — #13160382

Order 1728.

1580 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89076C211yesmaximal, characteristic
2C3×#64135C211yesmaximal, characteristic, complemented
3C2×C32×#208C211yesmaximal, characteristic, complemented
4#11722047C214yesmaximal, complemented
5#10592838—31nomaximal
6#10591113—34nomaximal
7C22×C33×C4C2211yescharacteristic
8C2×#6312—21no—
9#89076—21no—
10#89076—21no—
11#89076C412yescomplemented
12#89076C2212yes—
13C2×C4⋊(C3⋊(C3⋊D3))C411yescharacteristic, complemented
14C2×C4×C3⋊(C3⋊D3)—21no—
15C2×#6308C411yescharacteristic, complemented
16C3×#3757—21no—
17C3×C4⋊(C3⋊Dic3)—21no—
18C3×C4×C3⋊Dic3C2212yescomplemented
19C32×#208—21no—
20C32×#208C2212yescomplemented
21C2×C32×C3⋊C8—21no—
22C2×#3828—31no—
23C2×#3828—34no—
24C2×#3828—64no—
25C2×C32×M16—31no—
26#64135D311yescharacteristic, complemented
27C3×#926—34no—
28C2×C3×#208D314yescomplemented
29#63652—34no—
30#63570—316no—
31C23×C33D411yescharacteristic
32C22×C3⋊(C3⋊D3)—21no—
33C2×C33×C4C2×C411yescharacteristic
34C2×C33×C4C2×C411yescharacteristic
35C2×C33×C4D411yescharacteristic
36C2×C33×C4D411yescharacteristic
37C2×C33×C4D411yescharacteristic
38C2×C33×C4C2311yescharacteristic
39#6312—22no—
40#6312—22no—
41C2×C3⋊(C3⋊Dic3)—21no—
42C4⋊(C3⋊(C3⋊D3))—21no—
43C4⋊(C3⋊(C3⋊D3))C2×C412yes—
44C4×C3⋊(C3⋊D3)—22no—
45C4×C3⋊(C3⋊D3)—22no—
46#6308—21no—
47#6308C2×C412yes—
48C2×C3×C3⋊Dic3—22no—
49C32×C3⋊C8—22no—
50#5139—94no—
51#5005—91no—
52C22×C32×C4—24no—
53C22×C32×C4C2×D311yescharacteristic
54C22×C32×C4C2×D314yes—
55C2×#638—68no—
56C2×#638—64no—
57C2×#638—61no—
58#3828—32no—
59#3828—38no—
60#3828—32no—
61#3828—38no—
62#3828—68no—
63#3828—68no—
64#3828—68no—
65#3828—68no—
66#3828—64no—
67#3828—61no—
68#3828—61no—
69#3828—64no—
70C2×C4⋊(C3⋊D3)—31no—
71C2×C4⋊(C3⋊D3)—34no—
72C2×C4⋊(C3⋊D3)—64no—
73C2×C4×C3⋊D3—68no—
74C2×C4×C3⋊D3—61no—
75C2×C4×C3⋊D3—64no—
76C2×#634—31no—
77C2×#634—34no—
78C2×#634—64no—
79C32×M16—32no—
80C32×M16—61no—
81C2×C32×C8—61no—
82#3757—21no—
83C4⋊(C3⋊Dic3)—21no—
84C4×C3⋊Dic3C2×D312yescomplemented
85C3×#217—64no—
86C3×C4⋊Dic3—64no—
87C3×C4×Dic3—38no—
88C3×#208—24no—
89C3×#208C2×D318yescomplemented
90C2×C3×C3⋊C8—24no—
91C22×C33—41no—
92C22×C33C22⋊C411yescharacteristic
93C22×C33C22⋊C411yescharacteristic
94C22×C33C2×D411yescharacteristic
95C2×C3⋊(C3⋊D3)—22no—
96C2×C3⋊(C3⋊D3)—42no—
97C33×C4C22⋊C411yescharacteristic
98C33×C4C22⋊C411yescharacteristic
99C33×C4C22×C411yescharacteristic
100C33×C4C2×D411yescharacteristic
101C3⋊(C3⋊Dic3)—22no—
102C3×C3⋊Dic3—42no—
103C2×#235—94no—
104C2×#235—91no—
105C2×#235—184no—
106C2×C3×M16—34no—
107C3×#117—91no—
108#926—34no—
109C2×#208C3⋊D311yescharacteristic, complemented
110#825—916no—
111#805—94no—
112C23×C32—24no—
113C23×C32D1214yes—
114C23×C32#6711yescharacteristic
115C22×C3⋊D3—68no—
116C22×C3⋊D3—64no—
117C22×C3⋊D3—61no—
118C2×C32×C4—24no—
119C2×C32×C4—24no—
120C2×C32×C4—24no—
121C2×C32×C4—24no—
122C2×C32×C4—24no—
123C2×C32×C4—24no—
124C2×C32×C4C4×D314yes—
125C2×C32×C4C4×D314yes—
126C2×C32×C4C4×D311yescharacteristic
127C2×C32×C4C4×D311yescharacteristic
128C2×C32×C4D1214yes—
129C2×C32×C4D1211yescharacteristic
130C2×C32×C4D1211yescharacteristic
131C2×C32×C4#6714yes—
132C2×C32×C4#6714yes—
133C2×C32×C4#6711yescharacteristic
134C2×C32×C4C22×D314yes—
135C2×C32×C4C22×D311yescharacteristic
136#638—616no—
137#638—616no—
138#638—68no—
139#638—62no—
140#638—68no—
141#638—62no—
142C2×C3⋊Dic3—22no—
143C2×C3⋊Dic3—68no—
144C2×C3⋊Dic3—64no—
145C2×C3⋊Dic3—61no—
146C4⋊(C3⋊D3)—32no—
147C4⋊(C3⋊D3)—38no—
148C4⋊(C3⋊D3)—68no—
149C4⋊(C3⋊D3)—68no—
150C4⋊(C3⋊D3)—64no—
151C4⋊(C3⋊D3)—61no—
152C4×C3⋊D3—616no—
153C4×C3⋊D3—616no—
154C4×C3⋊D3—68no—
155C4×C3⋊D3—62no—
156C4×C3⋊D3—62no—
157C4×C3⋊D3—68no—
158#634—32no—
159#634—38no—
160#634—68no—
161#634—68no—
162#634—64no—
163#634—61no—
164C2×C3×Dic3—68no—
165C32×C8—62no—
166C3×C3⋊C8—28no—
167#421—271noSylow
168C2×C33—21no—
169C2×C33—21no—
170C2×C33—21no—
171C2×C33C2×C22⋊C411yescharacteristic
172C3⋊(C3⋊D3)—42no—
173C22×C3×C4—24no—
174C22×C3×C4D3214yes—
175C22×C3×C4C2×C3⋊D311yescharacteristic
176C2×#67—188no—
177C2×#67—184no—
178C2×#67—181no—
179#235—98no—
180#235—92no—
181#235—98no—
182#235—92no—
183#235—188no—
184#235—188no—
185#235—188no—
186#235—188no—
187#235—184no—
188#235—184no—
189#235—181no—
190#235—181no—
191C2×D12—94no—
192C2×D12—91no—
193C2×D12—184no—
194C2×C4×D3—188no—
195C2×C4×D3—184no—
196C2×C4×D3—181no—
197C2×Dic6—94no—
198C2×Dic6—91no—
199C2×Dic6—184no—
200C3×M16—38no—
201C3×M16—64no—
202C2×C3×C8—64no—
203C3×C4⋊C4—181no—
204C3×C22⋊C4—181no—
205C3×C42—92no—
206#217—64no—
207C4⋊Dic3—64no—
208C4×Dic3—38no—
209#208—21no—
210#208C2×C3⋊D312yescomplemented
211C2×C3⋊C8—21no—
212C22×C32—24no—
213C22×C32—24no—
214C22×C32—24no—
215C22×C32—44no—
216C22×C32—41no—
217C22×C32—48no—
218C22×C32(C2×D3)⋊C414yes—
219C22×C32(C2×D3)⋊C414yes—
220C22×C32(C2×D3)⋊C411yescharacteristic
221C22×C32(C2×D3)⋊C411yescharacteristic
222C22×C32C2×D1214yes—
223C22×C32C2×#6711yescharacteristic
224C2×C3⋊D3—616no—
225C2×C3⋊D3—68no—
226C2×C3⋊D3—62no—
227C2×C3⋊D3—1216no—
228C2×C3⋊D3—128no—
229C2×C3⋊D3—122no—
230C32×C4—24no—
231C32×C4—24no—
232C32×C4—24no—
233C32×C4—24no—
234C32×C4(C2×D3)⋊C414yes—
235C32×C4(C2×D3)⋊C414yes—
236C32×C4(C2×D3)⋊C411yescharacteristic
237C32×C4(C2×D3)⋊C411yescharacteristic
238C32×C4C2×C4×D314yes—
239C32×C4C2×C4×D311yescharacteristic
240C32×C4C2×D1211yescharacteristic
241C32×C4C2×#6714yes—
242C3⋊Dic3—42no—
243C3⋊Dic3—616no—
244C3⋊Dic3—68no—
245C3⋊Dic3—62no—
246C3×Dic3—128no—
247C2×#41—271no—
248C2×M16—31no—
249#117—91no—
250#104—274no—
251C33#42111yescharacteristic, complemented, Sylow
252C23×C3—24no—
253C23×C3#62614yes—
254C23×C3C4⋊(C3⋊D3)11yescharacteristic
255C22×D3—188no—
256C22×D3—184no—
257C22×D3—181no—
258C2×C3×C4—24no—
259C2×C3×C4—24no—
260C2×C3×C4—24no—
261C2×C3×C4—24no—
262C2×C3×C4—24no—
263C2×C3×C4—24no—
264C2×C3×C4—182no—
265C2×C3×C4#62414yes—
266C2×C3×C4#62414yes—
267C2×C3×C4#62614yes—
268C2×C3×C4#62614yes—
269C2×C3×C4#62614yes—
270C2×C3×C4C4×C3⋊D311yescharacteristic
271C2×C3×C4C4×C3⋊D311yescharacteristic
272C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
273C2×C3×C4#63811yescharacteristic
274C2×C3×C4#63811yescharacteristic
275C2×C3×C4C2×D3214yes—
276C2×C3×C4C22×C3⋊D311yescharacteristic
277#67—1816no—
278#67—1816no—
279#67—188no—
280#67—182no—
281#67—188no—
282#67—182no—
283C2×Dic3—68no—
284C2×Dic3—188no—
285C2×Dic3—184no—
286C2×Dic3—181no—
287D12—98no—
288D12—92no—
289D12—188no—
290D12—188no—
291D12—184no—
292D12—181no—
293C4×D3—1816no—
294C4×D3—1816no—
295C4×D3—188no—
296C4×D3—182no—
297C4×D3—188no—
298C4×D3—182no—
299Dic6—98no—
300Dic6—92no—
301Dic6—188no—
302Dic6—188no—
303Dic6—184no—
304Dic6—181no—
305C3×C8—68no—
306C3⋊C8—22no—
307C2×C32—28no—
308C2×C32—24no—
309C2×C32—21no—
310C2×C32—21no—
311C2×C32—21no—
312C2×C32—24no—
313C2×C32—24no—
314C2×C32—24no—
315C2×C32—44no—
316C2×C32—44no—
317C2×C32C2×(C2×D3)⋊C414yes—
318C2×C32C2×(C2×D3)⋊C411yescharacteristic
319C3⋊D3—1216no—
320C3⋊D3—128no—
321C3⋊D3—122no—
322#41—272no—
323#41—272no—
324#41—541no—
325#41—541no—
326C2×Q8—271no—
327C2×D4—271no—
328C2×D4—541no—
329C22×C4—541no—
330C22×C4D3×C3⋊D311yescharacteristic
331M16—32no—
332M16—61no—
333C2×C8—61no—
334C4⋊C4—181no—
335C22⋊C4—181no—
336C42—92no—
337C22×C3—24no—
338C22×C3—24no—
339C22×C3—24no—
340C22×C3—44no—
341C22×C3—41no—
342C22×C3—48no—
343C22×C3#372214yes—
344C22×C3#372214yes—
345C22×C3#375211yescharacteristic
346C22×C3#375211yescharacteristic
347C22×C3C2×#62614yes—
348C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
349C2×D3—1816no—
350C2×D3—182no—
351C2×D3—188no—
352C2×D3—3616no—
353C2×D3—362no—
354C2×D3—368no—
355C3×C4—24no—
356C3×C4—24no—
357C3×C4—24no—
358C3×C4—24no—
359C3×C4—362no—
360C3×C4#372214yes—
361C3×C4#372214yes—
362C3×C4#375211yescharacteristic
363C3×C4#375211yescharacteristic
364C3×C4C2×#62414yes—
365C3×C4C2×#62614yes—
366C3×C4C2×C4×C3⋊D311yescharacteristic
367C3×C4C2×#63811yescharacteristic
368Dic3—128no—
369Dic3—1816no—
370Dic3—182no—
371Dic3—188no—
372C32—24no—
373C32#500511yescharacteristic, complemented
374C32#513914yescomplemented
375C23—541no—
376C23#629211yescharacteristic
377Q8—272no—
378Q8—541no—
379D4—272no—
380D4—542no—
381D4—542no—
382D4—541no—
383C2×C4—182no—
384C2×C4—541no—
385C2×C4—542no—
386C2×C4—542no—
387C2×C4#628911yescharacteristic
388C2×C4#628911yescharacteristic
389C2×C4#629111yescharacteristic
390C2×C4#629111yescharacteristic
391C2×C4#629211yescharacteristic
392C2×C4C2×D3×C3⋊D311yescharacteristic
393C8—62no—
394C2×C3—28no—
395C2×C3—21no—
396C2×C3—21no—
397C2×C3—21no—
398C2×C3—24no—
399C2×C3—24no—
400C2×C3—24no—
401C2×C3—24no—
402C2×C3—44no—
403C2×C3—44no—
404C2×C3C2×#372214yes—
405C2×C3C2×#375211yescharacteristic
406D3—3616no—
407D3—362no—
408D3—368no—
409C22—41no—
410C22—542no—
411C22—1082no—
412C22#8880411yescharacteristic
413C22#8880411yescharacteristic
414C22C2×#629211yescharacteristic
415C4—362no—
416C4—542no—
417C4#8880411yescharacteristic
418C4#8880411yescharacteristic
419C4C2×#628911yescharacteristic
420C4C2×#629111yescharacteristic
421C3—24no—
422C3#1059111314yescomplemented
423C3#1059283811yescharacteristic, complemented
424C2—21no—
425C2—21no—
426C2—21no—
427C2—1082no—
428C2C2×#8880411yescharacteristic
subgroupG/H
centerC4C2×#6291
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC22×C33×C4C22
socleC2×C33C2×C22⋊C4
prime radicalC2×#89076C2
cosocleC2×C33×C4C23
hypercenterC22×C4D3×C3⋊D3
hypocenterC33#421
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C3×#64135C2semidirect
3. C2×C32×#208C2semidirect
4. #11722047C2semidirect
11. #89076C4semidirect
13. C2×C4⋊(C3⋊(C3⋊D3))C4semidirect
15. C2×#6308C4semidirect
18. C3×C4×C3⋊Dic3C22semidirect
20. C32×#208C22semidirect
26. #64135D3semidirect
28. C2×C3×#208D3semidirect
84. C4×C3⋊Dic3C2×D3semidirect
89. C3×#208C2×D3semidirect
109. C2×#208C3⋊D3semidirect
210. #208C2×C3⋊D3semidirect
251. C33#421semidirect
373. C32#5005semidirect
374. C32#5139semidirect
422. C3#10591113semidirect
423. C3#10592838semidirect

Maximal subgroups

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

subgroupindex
C2×#890762
C3×#641352
C2×C32×#2082
#117220472
#105928383
#105911133