Finite Group Databasebeta

← #13153863

Subgroups — #13153863

Order 1728.

1309 conjugacy classes of subgroups.

The subgroup lattice has 395 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2#11721830C211yesmaximal, characteristic, complemented
3#11721830C212yesmaximal, complemented
4C4×#6235C211yesmaximal, characteristic, complemented
5C4×#6235C212yesmaximal, complemented
6#11721766C212yesmaximal, complemented
7#11721763C211yesmaximal, characteristic, complemented
8#11721762C212yesmaximal, complemented
9#11721761C211yesmaximal, characteristic, complemented
10#11721760C211yesmaximal, characteristic, complemented
11#11721758C211yesmaximal, characteristic, complemented
12C3×#5514—34nomaximal
13C22×C4×H(3,𝔽3)C2211yescharacteristic
14C22×C4×H(3,𝔽3)C2212yes—
15C2×#6235C2211yescharacteristic
16C2×#6235C2212yes—
17C2×C4⋊#272C2211yescharacteristic
18C2×C4⋊#272C2212yes—
19C2×C4×#272C2211yescharacteristic
20C2×C4×#272C2212yes—
21C4⋊C4×H(3,𝔽3)C2211yescharacteristic
22C4⋊C4×H(3,𝔽3)C2211yescharacteristic
23C4⋊C4×H(3,𝔽3)C2212yes—
24#88552C2211yescharacteristic
25#88542C2212yes—
26#88542C2212yes—
27#88542C2211yescharacteristic
28#88542C2212yes—
29#88542C2212yes—
30C4⋊#1084C2211yescharacteristic
31C4⋊#1084C2212yes—
32#88540C2212yes—
33#88540C2211yescharacteristic
34C4×#1084C2211yescharacteristic
35C4×#1084C2212yes—
36C2×C32×C4⋊C4D314yescomplemented
37C3×#930—34no—
38C3×#930—38no—
39C3×C4×#67—34no—
40C3×C4×#67—38no—
41C3×#898—38no—
42C3×#895—34no—
43C3×#894—38no—
44C3×#893—34no—
45C3×#892—34no—
46C3×#890—34no—
47C23×H(3,𝔽3)C2311yescharacteristic
48C22×#272C2311yescharacteristic
49C22×#272C2312yes—
50C2×C4×H(3,𝔽3)—22no—
51C2×C4×H(3,𝔽3)—42no—
52C2×C4×H(3,𝔽3)C2312yes—
53C2×C4×H(3,𝔽3)C2312yes—
54C2×C4×H(3,𝔽3)C2311yescharacteristic
55C2×C4×H(3,𝔽3)C2311yescharacteristic
56#6235—22no—
57#6235—24no—
58C2×#1084C2311yescharacteristic
59C2×#1084C2311yescharacteristic
60C2×#1084C2311yescharacteristic
61C2×#1084C2312yes—
62C4⋊#272—22no—
63C4⋊#272—42no—
64C4×#272—22no—
65C4×#272—42no—
66C3×#553—91no—
67#5514—34no—
68C22×C32×C4C2×D314yes—
69C22×C32×C4C2×D318yes—
70C2×C3×#67—34no—
71C2×C3×#67—38no—
72C2×C3×D12—34no—
73C2×C3×D12—38no—
74C2×C3×C4×D3—34no—
75C2×C3×C4×D3—38no—
76C32×C4⋊C4C2×D314yes—
77C32×C4⋊C4C2×D314yes—
78C32×C4⋊C4C2×D318yes—
79C3×#217—34no—
80C3×(C2×D3)⋊C4—38no—
81C3×(C2×D3)⋊C4—38no—
82C3×(C2×D3)⋊C4—34no—
83C3×(C2×D3)⋊C4—38no—
84C3×(C2×D3)⋊C4—38no—
85C3×C4⋊Dic3—34no—
86C3×C4⋊Dic3—38no—
87C3×Dic3⋊C4—38no—
88C3×Dic3⋊C4—34no—
89C3×C4×Dic3—34no—
90C3×C4×Dic3—38no—
91C22×H(3,𝔽3)—21no—
92C22×H(3,𝔽3)—21no—
93C22×H(3,𝔽3)#4112yes—
94C22×H(3,𝔽3)C2411yescharacteristic
95C2×#272—21no—
96C2×#272—22no—
97C2×#272—21no—
98C2×#272—22no—
99C2×#272—22no—
100C2×#272—21no—
101C4×H(3,𝔽3)—21no—
102C4×H(3,𝔽3)—22no—
103C4×H(3,𝔽3)—22no—
104C4×H(3,𝔽3)#4112yes—
105#1084—21no—
106#1084—21no—
107#1084—22no—
108#1084—21no—
109C3×#126—92no—
110C3×#125—91no—
111C3×#123—92no—
112C3×#121—92no—
113C3×#121—91no—
114C3×#121—91no—
115C3×C4×D4—92no—
116C3×C4×D4—91no—
117C3×C4×D4—91no—
118C3×#117—91no—
119C2×C3×C4⋊C4—34no—
120C2×C3×C4⋊C4C3⋊D311yescharacteristic
121#930—34no—
122#930—38no—
123C4×#67—34no—
124C4×#67—38no—
125#898—38no—
126#895—34no—
127#894—38no—
128#893—34no—
129#892—34no—
130#890—34no—
131C23×C32C22×D314yes—
132C22×C3×D3—34no—
133C22×C3×D3—38no—
134C2×C32×C4—28no—
135C2×C32×C4—48no—
136C2×C32×C4C22×D318yes—
137C2×C32×C4C22×D318yes—
138C2×C32×C4C22×D314yes—
139C2×C32×C4C22×D314yes—
140C3×#67—68no—
141C3×#67—616no—
142C2×C3×Dic3—34no—
143C2×C3×Dic3—34no—
144C2×C3×Dic3—34no—
145C2×C3×Dic3—38no—
146C3×D12—68no—
147C3×D12—128no—
148C3×C4×D3—68no—
149C3×C4×D3—128no—
150#553—91noSylow
151C2×H(3,𝔽3)—22no—
152C2×H(3,𝔽3)C2×#4111yescharacteristic
153C2×H(3,𝔽3)C2×#4111yescharacteristic
154C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
155#272—41no—
156#272—42no—
157C2×C3×D4—92no—
158C2×C3×D4—91no—
159C2×C3×D4—91no—
160C2×C3×D4—92no—
161C22×C3×C4—34no—
162C22×C3×C4—38no—
163C22×C3×C4—91no—
164C22×C3×C4—92no—
165C22×C3×C4C2×C3⋊D311yescharacteristic
166C22×C3×C4C2×C3⋊D312yes—
167C2×#67—34no—
168C2×#67—38no—
169C2×D12—34no—
170C2×D12—38no—
171C2×C4×D3—34no—
172C2×C4×D3—38no—
173C3×C4⋊C4—34no—
174C3×C4⋊C4—34no—
175C3×C4⋊C4—38no—
176C3×C4⋊C4—92no—
177C3×C4⋊C4—91no—
178C3×C4⋊C4—91no—
179C3×C4⋊C4—92no—
180C3×C4⋊C4C2×C3⋊D311yescharacteristic
181C3×C4⋊C4C2×C3⋊D311yescharacteristic
182C3×C4⋊C4C2×C3⋊D312yes—
183C3×C22⋊C4—91no—
184C3×C22⋊C4—92no—
185C3×C22⋊C4—92no—
186C3×C22⋊C4—92no—
187C3×C22⋊C4—91no—
188C3×C22⋊C4—92no—
189C3×C42—92no—
190C3×C42—91no—
191#217—34no—
192(C2×D3)⋊C4—38no—
193(C2×D3)⋊C4—38no—
194(C2×D3)⋊C4—34no—
195(C2×D3)⋊C4—38no—
196(C2×D3)⋊C4—38no—
197C4⋊Dic3—34no—
198C4⋊Dic3—38no—
199Dic3⋊C4—38no—
200Dic3⋊C4—34no—
201C4×Dic3—34no—
202C4×Dic3—38no—
203C22×C32—24no—
204C22×C32—24no—
205C22×C32#23918yes—
206C22×C32C23×D314yes—
207C2×C3×D3—64no—
208C2×C3×D3—68no—
209C2×C3×D3—68no—
210C2×C3×D3—64no—
211C2×C3×D3—64no—
212C2×C3×D3—68no—
213C32×C4—24no—
214C32×C4—28no—
215C32×C4—28no—
216C32×C4#23518yes—
217C3×Dic3—64no—
218C3×Dic3—64no—
219C3×Dic3—64no—
220C3×Dic3—68no—
221#126—92no—
222#125—91no—
223#123—92no—
224#121—92no—
225#121—91no—
226#121—91no—
227C4×D4—92no—
228C4×D4—91no—
229C4×D4—91no—
230#117—91no—
231C2×C4⋊C4#27211yescharacteristic, complemented
232H(3,𝔽3)#55311yescharacteristic, complemented, Sylow
233C23×C3—34no—
234C23×C3—91no—
235C23×C3—92no—
236C23×C3C22×C3⋊D311yescharacteristic
237C22×D3—34no—
238C22×D3—38no—
239C3×D4—184no—
240C3×D4—182no—
241C3×D4—182no—
242C3×D4—362no—
243C2×C3×C4—22no—
244C2×C3×C4—34no—
245C2×C3×C4—38no—
246C2×C3×C4—38no—
247C2×C3×C4—34no—
248C2×C3×C4—42no—
249C2×C3×C4—68no—
250C2×C3×C4—91no—
251C2×C3×C4—92no—
252C2×C3×C4—91no—
253C2×C3×C4—91no—
254C2×C3×C4—128no—
255C2×C3×C4—182no—
256C2×C3×C4—362no—
257C2×C3×C4C22×C3⋊D312yes—
258C2×C3×C4C22×C3⋊D311yescharacteristic
259C2×C3×C4C22×C3⋊D312yes—
260C2×C3×C4C22×C3⋊D311yescharacteristic
261#67—68no—
262#67—616no—
263C2×Dic3—34no—
264C2×Dic3—34no—
265C2×Dic3—34no—
266C2×Dic3—38no—
267D12—68no—
268D12—128no—
269C4×D3—68no—
270C4×D3—128no—
271C2×C32—28no—
272C2×C32C2×#23514yes—
273C2×C32#100414yes—
274C2×C32C2×#23914yes—
275C3×D3—124no—
276C3×D3—128no—
277C2×D4—92no—
278C2×D4—91no—
279C2×D4—91no—
280C2×D4—92no—
281C22×C4—91no—
282C22×C4—92no—
283C22×C4C2×#27211yescharacteristic
284C22×C4C2×#27212yes—
285C4⋊C4—91no—
286C4⋊C4—92no—
287C4⋊C4—91no—
288C4⋊C4—92no—
289C4⋊C4C2×#27211yescharacteristic
290C4⋊C4C2×#27211yescharacteristic
291C4⋊C4C2×#27212yes—
292C22⋊C4—91no—
293C22⋊C4—92no—
294C22⋊C4—92no—
295C22⋊C4—92no—
296C22⋊C4—91no—
297C22⋊C4—92no—
298C42—92no—
299C42—91no—
300C22×C3—21no—
301C22×C3—21no—
302C22×C3—34no—
303C22×C3—38no—
304C22×C3—64no—
305C22×C3—64no—
306C22×C3—181no—
307C22×C3—182no—
308C22×C3—182no—
309C22×C3—181no—
310C22×C3—181no—
311C22×C3—182no—
312C22×C3#383212yes—
313C22×C3C23×C3⋊D311yescharacteristic
314C2×D3—64no—
315C2×D3—68no—
316C2×D3—68no—
317C2×D3—64no—
318C2×D3—64no—
319C2×D3—68no—
320C3×C4—21no—
321C3×C4—22no—
322C3×C4—22no—
323C3×C4—38no—
324C3×C4—68no—
325C3×C4—68no—
326C3×C4—64no—
327C3×C4—181no—
328C3×C4—181no—
329C3×C4—182no—
330C3×C4—181no—
331C3×C4#382812yes—
332Dic3—64no—
333Dic3—64no—
334Dic3—64no—
335Dic3—68no—
336C32#551414yescomplemented
337C23—91no—
338C23—92no—
339C23C22×#27211yescharacteristic
340D4—184no—
341D4—182no—
342D4—182no—
343D4—362no—
344C2×C4—22no—
345C2×C4—42no—
346C2×C4—91no—
347C2×C4—92no—
348C2×C4—91no—
349C2×C4—91no—
350C2×C4—182no—
351C2×C4—362no—
352C2×C4C22×#27212yes—
353C2×C4C22×#27211yescharacteristic
354C2×C4C22×#27212yes—
355C2×C4C22×#27211yescharacteristic
356C2×C3—22no—
357C2×C3—34no—
358C2×C3—34no—
359C2×C3—34no—
360C2×C3—68no—
361C2×C3—361no—
362C2×C3—362no—
363C2×C3C2×#382811yescharacteristic
364C2×C3#6436111yescharacteristic
365C2×C3C2×#383211yescharacteristic
366D3—124no—
367D3—128no—
368C22—21no—
369C22—21no—
370C22—181no—
371C22—182no—
372C22—182no—
373C22—181no—
374C22—181no—
375C22—182no—
376C22#8874812yes—
377C22C23×#27211yescharacteristic
378C4—21no—
379C4—22no—
380C4—22no—
381C4—181no—
382C4—181no—
383C4—182no—
384C4—181no—
385C4#8874012yes—
386C3—34no—
387C3#1059598511yescharacteristic
388C2—22no—
389C2—361no—
390C2—362no—
391C2C2×#8874011yescharacteristic
392C2#1172266411yescharacteristic
393C2C2×#8874811yescharacteristic
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)#553
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×C4⋊C4×H(3,𝔽3)C2semidirect
2. #11721830C2semidirect
3. #11721830C2semidirect
4. C4×#6235C2semidirect
5. C4×#6235C2semidirect
6. #11721766C2semidirect
7. #11721763C2semidirect
8. #11721762C2semidirect
9. #11721761C2semidirect
10. #11721760C2semidirect
11. #11721758C2semidirect
36. C2×C32×C4⋊C4D3semidirect
231. C2×C4⋊C4#272semidirect
232. H(3,𝔽3)#553semidirect
336. C32#5514semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×H(3,𝔽3)2
#117218302
C4×#62352
#117217662
#117217632
#117217622
#117217612
#117217602
#117217582
C3×#55143