Finite Group Databasebeta

← #13160316

Subgroups — #13160316

Order 1728.

1556 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
2C2×C3×#3748C211yesmaximal, characteristic, complemented
3C2×C32×#208C211yesmaximal, characteristic, complemented
4#11722030C212yesmaximal, complemented
5#11722027C212yesmaximal, complemented
6#10592831—31nomaximal
7#10591078—34nomaximal
8C22×C33×C4C2211yescharacteristic
9C2×#6312C412yes—
10#89076—21no—
11#89076—21no—
12#89076—21no—
13#89076—21no—
14C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
15C2×C4×C3⋊(C3⋊D3)C412yes—
16C2×#6308C2211yescharacteristic
17C3×#3748—21no—
18C3×#3748C2212yes—
19C2×C3×C3⋊(C3⋊C8)—21no—
20C32×#208—21no—
21C32×#208C2212yes—
22C2×C32×C3⋊C8—21no—
23C2×#3828—31no—
24C2×#3828—34no—
25C2×#3828—64no—
26C2×C32×M16—31no—
27C2×#3748D311yescharacteristic, complemented
28C2×C3×#208—34no—
29C2×C3×#208D314yescomplemented
30#63651—32no—
31#63650—32no—
32#63561—38no—
33#63559—38no—
34C23×C33C2×C411yescharacteristic
35C22×C3⋊(C3⋊D3)C2×C412yes—
36C2×C33×C4C2×C411yescharacteristic
37C2×C33×C4D411yescharacteristic
38C2×C33×C4D411yescharacteristic
39C2×C33×C4D411yescharacteristic
40C2×C33×C4D411yescharacteristic
41C2×C33×C4C2311yescharacteristic
42#6312—22no—
43#6312—22no—
44C2×C3⋊(C3⋊Dic3)C2×C412yes—
45C4⋊(C3⋊(C3⋊D3))—21no—
46C4⋊(C3⋊(C3⋊D3))—21no—
47C4×C3⋊(C3⋊D3)—22no—
48C4×C3⋊(C3⋊D3)—22no—
49#6308—21no—
50#6308—21no—
51C3×C3⋊(C3⋊C8)—22no—
52C32×C3⋊C8—22no—
53#5132—94no—
54#5132—91no—
55C22×C32×C4—24no—
56C22×C32×C4C2×D314yes—
57C22×C32×C4C2×D311yescharacteristic
58C2×#638—32no—
59C2×#638—38no—
60C2×#638—68no—
61#3828—68no—
62#3828—68no—
63#3828—68no—
64#3828—68no—
65#3828—64no—
66#3828—64no—
67#3828—61no—
68#3828—61no—
69#3828—64no—
70#3828—64no—
71#3828—61no—
72#3828—61no—
73C2×C4⋊(C3⋊D3)—31no—
74C2×C4⋊(C3⋊D3)—34no—
75C2×C4⋊(C3⋊D3)—64no—
76C2×C4×C3⋊D3—32no—
77C2×C4×C3⋊D3—38no—
78C2×C4×C3⋊D3—68no—
79C2×#634—31no—
80C2×#634—34no—
81C2×#634—64no—
82C32×M16—32no—
83C32×M16—61no—
84C2×C32×C8—61no—
85#3748—21no—
86#3748C2×D312yes—
87C2×C3⋊(C3⋊C8)—21no—
88C3×#208—24no—
89C3×#208—38no—
90C3×#208—64no—
91C3×#208C2×D318yes—
92C2×C3×C3⋊C8—24no—
93C2×C3×C3⋊C8—64no—
94C22×C33—41no—
95C22×C33C22⋊C411yescharacteristic
96C22×C33C22⋊C411yescharacteristic
97C22×C33C22×C411yescharacteristic
98C2×C3⋊(C3⋊D3)—22no—
99C2×C3⋊(C3⋊D3)—42no—
100C33×C4C22⋊C411yescharacteristic
101C33×C4C22⋊C411yescharacteristic
102C33×C4C2×D411yescharacteristic
103C33×C4C2×D411yescharacteristic
104C3⋊(C3⋊Dic3)—22no—
105C2×#235—94no—
106C2×#235—91no—
107C2×#235—184no—
108C2×C3×M16—34no—
109C2×C3×M16—91no—
110C2×#208—34no—
111C2×#208C3⋊D311yescharacteristic, complemented
112#824—98no—
113#824—92no—
114#823—98no—
115#823—92no—
116C23×C32—24no—
117C23×C32C4×D314yes—
118C23×C32C4×D311yescharacteristic
119C22×C3⋊D3—32no—
120C22×C3⋊D3—38no—
121C22×C3⋊D3—68no—
122C2×C32×C4—24no—
123C2×C32×C4—24no—
124C2×C32×C4—24no—
125C2×C32×C4—24no—
126C2×C32×C4—24no—
127C2×C32×C4—24no—
128C2×C32×C4C4×D314yes—
129C2×C32×C4C4×D311yescharacteristic
130C2×C32×C4D1214yes—
131C2×C32×C4D1214yes—
132C2×C32×C4D1211yescharacteristic
133C2×C32×C4D1211yescharacteristic
134C2×C32×C4#6714yes—
135C2×C32×C4#6714yes—
136C2×C32×C4#6711yescharacteristic
137C2×C32×C4#6711yescharacteristic
138C2×C32×C4C22×D314yes—
139C2×C32×C4C22×D311yescharacteristic
140#638—616no—
141#638—616no—
142#638—68no—
143#638—68no—
144#638—62no—
145#638—62no—
146C2×C3⋊Dic3—32no—
147C2×C3⋊Dic3—38no—
148C2×C3⋊Dic3—68no—
149C4⋊(C3⋊D3)—68no—
150C4⋊(C3⋊D3)—68no—
151C4⋊(C3⋊D3)—64no—
152C4⋊(C3⋊D3)—61no—
153C4⋊(C3⋊D3)—64no—
154C4⋊(C3⋊D3)—61no—
155C4×C3⋊D3—616no—
156C4×C3⋊D3—616no—
157C4×C3⋊D3—68no—
158C4×C3⋊D3—68no—
159C4×C3⋊D3—62no—
160C4×C3⋊D3—62no—
161#634—68no—
162#634—68no—
163#634—64no—
164#634—61no—
165#634—64no—
166#634—61no—
167C32×C8—62no—
168C3⋊(C3⋊C8)—22no—
169C3×C3⋊C8—28no—
170C3×C3⋊C8—68no—
171#413—271noSylow
172C2×C33—21no—
173C2×C33—21no—
174C2×C33—21no—
175C2×C33C2×C22⋊C411yescharacteristic
176C3⋊(C3⋊D3)—42no—
177C22×C3×C4—24no—
178C22×C3×C4D3214yes—
179C22×C3×C4C2×C3⋊D311yescharacteristic
180C2×#67—98no—
181C2×#67—92no—
182C2×#67—188no—
183#235—188no—
184#235—188no—
185#235—188no—
186#235—188no—
187#235—184no—
188#235—184no—
189#235—184no—
190#235—184no—
191#235—181no—
192#235—181no—
193#235—181no—
194#235—181no—
195C2×D12—94no—
196C2×D12—91no—
197C2×D12—184no—
198C2×C4×D3—98no—
199C2×C4×D3—92no—
200C2×C4×D3—188no—
201C2×Dic6—94no—
202C2×Dic6—91no—
203C2×Dic6—184no—
204C3×M16—38no—
205C3×M16—64no—
206C3×M16—92no—
207C3×M16—181no—
208C2×C3×C8—64no—
209C2×C3×C8—181no—
210#208—21no—
211#208—38no—
212#208—64no—
213#208C2×C3⋊D312yes—
214C2×C3⋊C8—21no—
215C2×C3⋊C8—64no—
216C22×C32—24no—
217C22×C32—24no—
218C22×C32—24no—
219C22×C32—41no—
220C22×C32—44no—
221C22×C32—48no—
222C22×C32(C2×D3)⋊C414yes—
223C22×C32(C2×D3)⋊C414yes—
224C22×C32(C2×D3)⋊C411yescharacteristic
225C22×C32(C2×D3)⋊C411yescharacteristic
226C22×C32C2×C4×D314yes—
227C22×C32C2×C4×D311yescharacteristic
228C2×C3⋊D3—616no—
229C2×C3⋊D3—68no—
230C2×C3⋊D3—62no—
231C2×C3⋊D3—1216no—
232C2×C3⋊D3—128no—
233C2×C3⋊D3—122no—
234C32×C4—24no—
235C32×C4—24no—
236C32×C4—24no—
237C32×C4—24no—
238C32×C4(C2×D3)⋊C414yes—
239C32×C4(C2×D3)⋊C414yes—
240C32×C4(C2×D3)⋊C411yescharacteristic
241C32×C4(C2×D3)⋊C411yescharacteristic
242C32×C4C2×D1214yes—
243C32×C4C2×D1211yescharacteristic
244C32×C4C2×#6714yes—
245C32×C4C2×#6711yescharacteristic
246C3⋊Dic3—616no—
247C3⋊Dic3—68no—
248C3⋊Dic3—62no—
249C2×#41—271no—
250C2×M16—31no—
251C2×M16—91no—
252#101—272no—
253#100—272no—
254C33#41311yescharacteristic, complemented, Sylow
255C23×C3—24no—
256C23×C3#62414yes—
257C23×C3C4×C3⋊D311yescharacteristic
258C22×D3—98no—
259C22×D3—92no—
260C22×D3—188no—
261C2×C3×C4—24no—
262C2×C3×C4—24no—
263C2×C3×C4—24no—
264C2×C3×C4—24no—
265C2×C3×C4—24no—
266C2×C3×C4—24no—
267C2×C3×C4#62414yes—
268C2×C3×C4#62614yes—
269C2×C3×C4#62614yes—
270C2×C3×C4#62614yes—
271C2×C3×C4#62614yes—
272C2×C3×C4C4×C3⋊D311yescharacteristic
273C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
274C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
275C2×C3×C4#63811yescharacteristic
276C2×C3×C4#63811yescharacteristic
277C2×C3×C4C2×D3214yes—
278C2×C3×C4C22×C3⋊D311yescharacteristic
279#67—1816no—
280#67—1816no—
281#67—188no—
282#67—188no—
283#67—182no—
284#67—182no—
285C2×Dic3—98no—
286C2×Dic3—92no—
287C2×Dic3—188no—
288D12—188no—
289D12—188no—
290D12—184no—
291D12—184no—
292D12—181no—
293D12—181no—
294C4×D3—1816no—
295C4×D3—1816no—
296C4×D3—188no—
297C4×D3—188no—
298C4×D3—182no—
299C4×D3—182no—
300Dic6—188no—
301Dic6—188no—
302Dic6—184no—
303Dic6—184no—
304Dic6—181no—
305Dic6—181no—
306C3×C8—68no—
307C3×C8—182no—
308C3⋊C8—22no—
309C3⋊C8—68no—
310C2×C32—28no—
311C2×C32—24no—
312C2×C32—21no—
313C2×C32—21no—
314C2×C32—21no—
315C2×C32—24no—
316C2×C32—24no—
317C2×C32—24no—
318C2×C32—44no—
319C2×C32—44no—
320C2×C32C2×(C2×D3)⋊C414yes—
321C2×C32C2×(C2×D3)⋊C411yescharacteristic
322C3⋊D3—1216no—
323C3⋊D3—128no—
324C3⋊D3—122no—
325#41—541no—
326#41—541no—
327#41—541no—
328#41—541no—
329C2×Q8—271no—
330C2×D4—272no—
331C2×D4—271no—
332C22×C4—272no—
333C22×C4D3×C3⋊D311yescharacteristic
334M16—32no—
335M16—61no—
336M16—92no—
337M16—181no—
338C2×C8—61no—
339C2×C8—181no—
340C22×C3—24no—
341C22×C3—24no—
342C22×C3—24no—
343C22×C3—44no—
344C22×C3—41no—
345C22×C3—48no—
346C22×C3#372214yes—
347C22×C3#372214yes—
348C22×C3#375211yescharacteristic
349C22×C3#375211yescharacteristic
350C22×C3C2×#62414yes—
351C22×C3C2×C4×C3⋊D311yescharacteristic
352C2×D3—1816no—
353C2×D3—182no—
354C2×D3—188no—
355C2×D3—3616no—
356C2×D3—362no—
357C2×D3—368no—
358C3×C4—24no—
359C3×C4—24no—
360C3×C4—24no—
361C3×C4—24no—
362C3×C4#372214yes—
363C3×C4#372214yes—
364C3×C4#375211yescharacteristic
365C3×C4#375211yescharacteristic
366C3×C4C2×#62614yes—
367C3×C4C2×#62614yes—
368C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
369C3×C4C2×#63811yescharacteristic
370Dic3—1816no—
371Dic3—182no—
372Dic3—188no—
373C32—24no—
374C32#513214yescomplemented
375C32#513211yescharacteristic, complemented
376C23—272no—
377C23#628911yescharacteristic
378Q8—541no—
379Q8—541no—
380D4—542no—
381D4—542no—
382D4—541no—
383D4—541no—
384C2×C4—272no—
385C2×C4—542no—
386C2×C4—542no—
387C2×C4#628911yescharacteristic
388C2×C4#629111yescharacteristic
389C2×C4#629111yescharacteristic
390C2×C4#629211yescharacteristic
391C2×C4#629211yescharacteristic
392C2×C4C2×D3×C3⋊D311yescharacteristic
393C8—62no—
394C8—182no—
395C2×C3—28no—
396C2×C3—21no—
397C2×C3—21no—
398C2×C3—21no—
399C2×C3—24no—
400C2×C3—24no—
401C2×C3—24no—
402C2×C3—24no—
403C2×C3—44no—
404C2×C3—44no—
405C2×C3C2×#372214yes—
406C2×C3C2×#375211yescharacteristic
407D3—3616no—
408D3—362no—
409D3—368no—
410C22—41no—
411C22—542no—
412C22—1082no—
413C22#8880411yescharacteristic
414C22#8880411yescharacteristic
415C22C2×#628911yescharacteristic
416C4—542no—
417C4#8880411yescharacteristic
418C4#8880411yescharacteristic
419C4C2×#629111yescharacteristic
420C4C2×#629211yescharacteristic
421C3—24no—
422C3#1059107814yescomplemented
423C3#1059283111yescharacteristic, complemented
424C2—21no—
425C2—21no—
426C2—21no—
427C2—1082no—
428C2C2×#8880411yescharacteristic
subgroupG/H
centerC4#88804
commutatorC22×C33C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC22×C33×C4C22
socleC2×C33C2×C22⋊C4
prime radicalC2×#89076C2
cosocleC2×C33×C4C23
hypercenterC22×C4D3×C3⋊D3
hypocenterC33#413
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3748C2semidirect
3. C2×C32×#208C2semidirect
4. #11722030C2semidirect
5. #11722027C2semidirect
27. C2×#3748D3semidirect
29. C2×C3×#208D3semidirect
111. C2×#208C3⋊D3semidirect
254. C33#413semidirect
374. C32#5132semidirect
375. C32#5132semidirect
422. C3#10591078semidirect
423. C3#10592831semidirect

Maximal subgroups

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

subgroupindex
C2×#890762
C2×C3×#37482
C2×C32×#2082
#117220302
#117220272
#105928313
#105910783