Finite Group Databasebeta

← #13153857

Subgroups — #13153857

Order 1728.

1606 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88740C212yesmaximal, complemented
2C2×C4⋊C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
3#11721830C211yesmaximal, characteristic, complemented
4#11721829C212yesmaximal, complemented
5C4×#6235C211yesmaximal, characteristic, complemented
6#11721764C212yesmaximal, complemented
7#11721763C211yesmaximal, characteristic, complemented
8#11721762C212yesmaximal, complemented
9#11721761C211yesmaximal, characteristic, complemented
10C4⋊C4×#272C211yesmaximal, characteristic, complemented
11#11721756C211yesmaximal, characteristic, complemented
12C3×#5507—34nomaximal
13C22×C4×H(3,𝔽3)C2211yescharacteristic
14C22×C4×H(3,𝔽3)C2212yes—
15C2×#6235C2212yes—
16C2×#6235C2211yescharacteristic
17#88740—28no—
18C2×C4⋊#272C2211yescharacteristic
19C2×C4⋊#272C2212yes—
20C2×C4×#272C2212yes—
21C2×C4×#272C2211yescharacteristic
22C2×C4×#272C2212yes—
23C2×#6229C2212yes—
24C4⋊C4×H(3,𝔽3)C2211yescharacteristic
25C4⋊C4×H(3,𝔽3)C2211yescharacteristic
26C4⋊C4×H(3,𝔽3)C2212yes—
27#88552C2211yescharacteristic
28#88542C2212yes—
29#88542C2211yescharacteristic
30#88542C2212yes—
31#88542C2212yes—
32C4⋊#1084C2211yescharacteristic
33#88540C2212yes—
34#88540C2212yes—
35#88540C2211yescharacteristic
36C4×#1084C2211yescharacteristic
37C2×C3×#235—38no—
38C2×C32×C4⋊C4D314yescomplemented
39C3×#930—34no—
40C3×#929—38no—
41C3×C4×#67—34no—
42C3×#896—38no—
43C3×#895—34no—
44C3×#894—38no—
45C3×#893—34no—
46C3×D3×C4⋊C4—34no—
47C3×#888—34no—
48C23×H(3,𝔽3)C2311yescharacteristic
49C22×#272C2312yes—
50C22×#272C2311yescharacteristic
51C2×C4×H(3,𝔽3)—24no—
52C2×C4×H(3,𝔽3)—22no—
53C2×C4×H(3,𝔽3)C2312yes—
54C2×C4×H(3,𝔽3)C2312yes—
55C2×C4×H(3,𝔽3)C2311yescharacteristic
56C2×C4×H(3,𝔽3)C2311yescharacteristic
57#6235—24no—
58#6235D414yescomplemented
59C2×#1084C2312yes—
60C2×#1084C2311yescharacteristic
61C2×#1084C2311yescharacteristic
62C2×#1084C2311yescharacteristic
63C4⋊#272—24no—
64C4⋊#272—22no—
65C4×#272—24no—
66C4×#272—22no—
67C4×#272—24no—
68#6229—24no—
69C3×#547—91no—
70#5507—34no—
71C22×C32×C4C2×D314yes—
72C22×C32×C4C2×D318yes—
73C2×C3×#67—34no—
74C2×C3×#67—38no—
75C3×#235—632no—
76C2×C3×D12—34no—
77C2×C3×D12—38no—
78C2×C3×C4×D3—38no—
79C2×C3×C4×D3—34no—
80C2×C3×C4×D3—38no—
81C2×C3×Dic6—38no—
82C32×C4⋊C4C2×D314yes—
83C32×C4⋊C4C2×D314yes—
84C32×C4⋊C4C2×D318yes—
85C3×#217—34no—
86C3×(C2×D3)⋊C4—38no—
87C3×(C2×D3)⋊C4—34no—
88C3×(C2×D3)⋊C4—38no—
89C3×(C2×D3)⋊C4—38no—
90C3×C4⋊Dic3—34no—
91C3×Dic3⋊C4—38no—
92C3×Dic3⋊C4—38no—
93C3×Dic3⋊C4—34no—
94C3×C4×Dic3—34no—
95C22×H(3,𝔽3)—21no—
96C22×H(3,𝔽3)—21no—
97C22×H(3,𝔽3)C2×D412yes—
98C22×H(3,𝔽3)C2411yescharacteristic
99C2×#272—21no—
100C2×#272—21no—
101C2×#272—22no—
102C2×#272—22no—
103C2×#272—22no—
104C2×#272C2×D412yes—
105C4×H(3,𝔽3)—22no—
106C4×H(3,𝔽3)—22no—
107C4×H(3,𝔽3)—21no—
108C4×H(3,𝔽3)#4112yescomplemented
109#1084—22no—
110#1084—21no—
111#1084—21no—
112#1084C2×D412yes—
113C2×C3×#41—92no—
114C2×#235—38no—
115C3×C4⋊Q8—91no—
116C3×#123—92no—
117C3×#123—92no—
118C3×C22⋊Q8—92no—
119C3×#121—91no—
120C3×#121—91no—
121C3×C4×D4—91no—
122C3×C4×D4—91no—
123C2×C3×C4⋊C4—34no—
124C2×C3×C4⋊C4—91no—
125C2×C3×C4⋊C4C3⋊D311yescharacteristic
126#930—34no—
127#929—38no—
128C4×#67—34no—
129#896—38no—
130#895—34no—
131#894—38no—
132#893—34no—
133D3×C4⋊C4—34no—
134#888—34no—
135C23×C32C22×D314yes—
136C22×C3×D3—34no—
137C22×C3×D3—38no—
138C2×C32×C4—216no—
139C2×C32×C4—28no—
140C2×C32×C4C22×D314yes—
141C2×C32×C4C22×D318yes—
142C2×C32×C4C22×D318yes—
143C2×C32×C4C22×D314yes—
144C3×#67—316no—
145C3×#67—616no—
146C2×C3×Dic3—34no—
147C2×C3×Dic3—34no—
148C2×C3×Dic3—34no—
149C2×C3×Dic3—38no—
150C3×D12—616no—
151C3×D12—68no—
152C3×C4×D3—616no—
153C3×C4×D3—616no—
154C3×C4×D3—68no—
155C3×Dic6—616no—
156#547—91noSylow
157C2×H(3,𝔽3)—22no—
158C2×H(3,𝔽3)C22×D411yescharacteristic
159C2×H(3,𝔽3)C2×#4111yescharacteristic
160C2×H(3,𝔽3)2−1+411yescharacteristic
161#272—22no—
162#272—42no—
163C3×#41—188no—
164C2×C3×Q8—92no—
165C2×C3×D4—91no—
166C2×C3×D4—92no—
167C2×C3×D4—91no—
168C2×C3×D4—92no—
169C22×C3×C4—34no—
170C22×C3×C4—38no—
171C22×C3×C4—91no—
172C22×C3×C4—92no—
173C22×C3×C4—92no—
174C22×C3×C4C2×C3⋊D311yescharacteristic
175C22×C3×C4C2×C3⋊D312yes—
176C2×#67—34no—
177C2×#67—38no—
178#235—632no—
179C2×D12—34no—
180C2×D12—38no—
181C2×C4×D3—38no—
182C2×C4×D3—34no—
183C2×C4×D3—38no—
184C2×Dic6—38no—
185C3×C4⋊C4—34no—
186C3×C4⋊C4—34no—
187C3×C4⋊C4—38no—
188C3×C4⋊C4—92no—
189C3×C4⋊C4—91no—
190C3×C4⋊C4—92no—
191C3×C4⋊C4—91no—
192C3×C4⋊C4C2×C3⋊D311yescharacteristic
193C3×C4⋊C4C2×C3⋊D311yescharacteristic
194C3×C4⋊C4C2×C3⋊D312yes—
195C3×C22⋊C4—91no—
196C3×C22⋊C4—91no—
197C3×C22⋊C4—92no—
198C3×C22⋊C4—92no—
199C3×C22⋊C4—92no—
200C3×C42—91no—
201#217—34no—
202(C2×D3)⋊C4—38no—
203(C2×D3)⋊C4—34no—
204(C2×D3)⋊C4—38no—
205(C2×D3)⋊C4—38no—
206C4⋊Dic3—34no—
207Dic3⋊C4—38no—
208Dic3⋊C4—38no—
209Dic3⋊C4—34no—
210C4×Dic3—34no—
211C22×C32—24no—
212C22×C32—24no—
213C22×C32D3×D418yes—
214C22×C32C23×D314yes—
215C2×C3×D3—38no—
216C2×C3×D3—64no—
217C2×C3×D3—64no—
218C2×C3×D3—68no—
219C2×C3×D3—68no—
220C2×C3×D3—68no—
221C32×C4—24no—
222C32×C4—28no—
223C32×C4—28no—
224C32×C4#23518yescomplemented
225C3×Dic3—38no—
226C3×Dic3—64no—
227C3×Dic3—68no—
228C3×Dic3—64no—
229C2×#41—92no—
230C4⋊Q8—91no—
231#123—92no—
232#123—92no—
233C22⋊Q8—92no—
234#121—91no—
235#121—91no—
236C4×D4—91no—
237C4×D4—91no—
238C2×C4⋊C4—91no—
239C2×C4⋊C4#27211yescharacteristic, complemented
240H(3,𝔽3)#54711yescharacteristic, complemented, Sylow
241C23×C3—34no—
242C23×C3—91no—
243C23×C3—92no—
244C23×C3C22×C3⋊D311yescharacteristic
245C22×D3—34no—
246C22×D3—38no—
247C3×Q8—184no—
248C3×D4—94no—
249C3×D4—184no—
250C3×D4—184no—
251C3×D4—182no—
252C2×C3×C4—24no—
253C2×C3×C4—22no—
254C2×C3×C4—34no—
255C2×C3×C4—34no—
256C2×C3×C4—38no—
257C2×C3×C4—38no—
258C2×C3×C4—68no—
259C2×C3×C4—616no—
260C2×C3×C4—91no—
261C2×C3×C4—91no—
262C2×C3×C4—91no—
263C2×C3×C4—92no—
264C2×C3×C4—184no—
265C2×C3×C4—184no—
266C2×C3×C4—182no—
267C2×C3×C4C22×C3⋊D311yescharacteristic
268C2×C3×C4C22×C3⋊D312yes—
269C2×C3×C4C22×C3⋊D312yes—
270C2×C3×C4C22×C3⋊D311yescharacteristic
271#67—316no—
272#67—616no—
273C2×Dic3—34no—
274C2×Dic3—34no—
275C2×Dic3—34no—
276C2×Dic3—38no—
277D12—68no—
278D12—616no—
279C4×D3—616no—
280C4×D3—68no—
281C4×D3—616no—
282Dic6—616no—
283C2×C32—28no—
284C2×C32C2×#23514yes—
285C2×C32C2×D3×D414yes—
286C2×C32#100714yes—
287C3×D3—68no—
288C3×D3—128no—
289#41—188no—
290C2×Q8—92no—
291C2×D4—91no—
292C2×D4—92no—
293C2×D4—91no—
294C2×D4—92no—
295C22×C4—91no—
296C22×C4—92no—
297C22×C4—92no—
298C22×C4C2×#27211yescharacteristic
299C22×C4C2×#27212yes—
300C4⋊C4—91no—
301C4⋊C4—92no—
302C4⋊C4—91no—
303C4⋊C4—92no—
304C4⋊C4C2×#27211yescharacteristic
305C4⋊C4C2×#27211yescharacteristic
306C4⋊C4C2×#27212yes—
307C22⋊C4—91no—
308C22⋊C4—91no—
309C22⋊C4—92no—
310C22⋊C4—92no—
311C22⋊C4—92no—
312C42—91no—
313C22×C3—21no—
314C22×C3—21no—
315C22×C3—34no—
316C22×C3—38no—
317C22×C3—64no—
318C22×C3—64no—
319C22×C3—92no—
320C22×C3—181no—
321C22×C3—181no—
322C22×C3—182no—
323C22×C3—182no—
324C22×C3—182no—
325C22×C3D4×C3⋊D312yes—
326C22×C3C23×C3⋊D311yescharacteristic
327C2×D3—38no—
328C2×D3—64no—
329C2×D3—64no—
330C2×D3—68no—
331C2×D3—68no—
332C2×D3—68no—
333C3×C4—21no—
334C3×C4—22no—
335C3×C4—22no—
336C3×C4—38no—
337C3×C4—68no—
338C3×C4—68no—
339C3×C4—64no—
340C3×C4—92no—
341C3×C4—181no—
342C3×C4—182no—
343C3×C4—181no—
344C3×C4#382812yes—
345Dic3—38no—
346Dic3—68no—
347Dic3—64no—
348Dic3—64no—
349C32#550714yescomplemented
350C23—91no—
351C23—92no—
352C23C22×#27211yescharacteristic
353Q8—184no—
354D4—94no—
355D4—184no—
356D4—184no—
357D4—182no—
358C2×C4—24no—
359C2×C4—22no—
360C2×C4—91no—
361C2×C4—91no—
362C2×C4—91no—
363C2×C4—92no—
364C2×C4—184no—
365C2×C4—184no—
366C2×C4—182no—
367C2×C4C22×#27211yescharacteristic
368C2×C4C22×#27212yes—
369C2×C4C22×#27212yes—
370C2×C4C22×#27211yescharacteristic
371C2×C3—22no—
372C2×C3—34no—
373C2×C3—34no—
374C2×C3—34no—
375C2×C3—68no—
376C2×C3—182no—
377C2×C3—362no—
378C2×C3C2×#382811yescharacteristic
379C2×C3C2×D4×C3⋊D311yescharacteristic
380C2×C3#6436411yescharacteristic
381D3—68no—
382D3—128no—
383C22—21no—
384C22—21no—
385C22—92no—
386C22—181no—
387C22—181no—
388C22—182no—
389C22—182no—
390C22—182no—
391C22D4×#27212yes—
392C22C23×#27211yescharacteristic
393C4—21no—
394C4—22no—
395C4—22no—
396C4—92no—
397C4—181no—
398C4—182no—
399C4—181no—
400C4#8874012yescomplemented
401C3—34no—
402C3#1059597811yescharacteristic
403C2—22no—
404C2—182no—
405C2—362no—
406C2C2×#8874011yescharacteristic
407C2C2×D4×#27211yescharacteristic
408C2#1172267011yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C23×C3⋊D3
commutator, cosocleC22×H(3,𝔽3)C24
FittingC2×C4⋊C4×H(3,𝔽3)C2
hypercenterC2×C3×C4⋊C4C3⋊D3
hypocenterH(3,𝔽3)#547
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#88740C2semidirect
2. C2×C4⋊C4×H(3,𝔽3)C2semidirect
3. #11721830C2semidirect
4. #11721829C2semidirect
5. C4×#6235C2semidirect
6. #11721764C2semidirect
7. #11721763C2semidirect
8. #11721762C2semidirect
9. #11721761C2semidirect
10. C4⋊C4×#272C2semidirect
11. #11721756C2semidirect
38. C2×C32×C4⋊C4D3semidirect
58. #6235D4semidirect
108. C4×H(3,𝔽3)#41semidirect
224. C32×C4#235semidirect
239. C2×C4⋊C4#272semidirect
240. H(3,𝔽3)#547semidirect
349. C32#5507semidirect
400. C4#88740semidirect

Maximal subgroups

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

subgroupindex
C2×#887402
C2×C4⋊C4×H(3,𝔽3)2
#117218302
#117218292
C4×#62352
#117217642
#117217632
#117217622
#117217612
C4⋊C4×#2722
#117217562
C3×#55073