Finite Group Databasebeta

← #13149520

Subgroups — #13149520

Order 1728.

1105 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 1105 conjugacy classes of subgroups — too many to draw. Only its 203 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×#88650C211yesmaximal, characteristic
2#11721760C211yesmaximal, characteristic, complemented
3#64101⋊C3C212yesmaximal, complemented
4#11721152C211yesmaximal, characteristic, complemented
5#11721138C212yesmaximal, complemented
6#11721129C212yesmaximal, complemented
7#11721123C214yesmaximal, complemented
8C4×#6196C212yesmaximal, complemented
9#10594511—32nomaximal
10C2×C4×#272C2211yescharacteristic
11C2×C4×(C3⋊D3)⋊C3C2212yes—
12C2×#6200C2212yes—
13C2×#6199C2212yes—
14C2×#6198C2211yescharacteristic
15C2×((C3⋊D3)⋊C3)⋊C4C2214yes—
16C2×#6196C2211yescharacteristic
17#88650C418yescomplemented
18C4⋊C4×H(3,𝔽3)C2211yescharacteristic
19#88542C2212yes—
20C4⋊#1084C2211yescharacteristic
21C4×#1084C2212yes—
22#3752⋊C3C2214yes—
23C4⋊((C3⋊Dic3)⋊C3)C2212yes—
24C4×(C3⋊Dic3)⋊C3C2214yes—
25#88449C2212yes—
26#88448C2214yes—
27C2×#3798—32no—
28#64101D312yescomplemented
29C3×#892—32no—
30C3×#892—32no—
31C3×#892—61no—
32C4×#627—34no—
33#63902—34no—
34#63900—32no—
35D3×C4⋊Dic3—32no—
36D3×C4⋊Dic3—32no—
37#63880—34no—
38#63880—34no—
39C22×#272C2311yescharacteristic
40C22×(C3⋊D3)⋊C3C2312yes—
41C2×C4×H(3,𝔽3)C2312yes—
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43C2×#1084C2312yes—
44C2×#1084C2311yescharacteristic
45C4×#272C2×C414yescomplemented
46C2×(C3⋊Dic3)⋊C3C2314yes—
47C2×(C3⋊Dic3)⋊C3C2312yes—
48C4×(C3⋊D3)⋊C3C2×C418yescomplemented
49#6200C2×C418yescomplemented
50#6199—42no—
51#6198C2×C414yescomplemented
52((C3⋊D3)⋊C3)⋊C4—44no—
53#6196C2×C414yescomplemented
54#5523—91no—
55#5523—92no—
56C2×C4×C3⋊D3C2×D312yes—
57C2×C3×C4×D3—32no—
58C2×C3×C4×D3—32no—
59C2×C3×C4×D3—61no—
60C2×#627—32no—
61C2×#626—32no—
62C2×#626—32no—
63C2×#625—32no—
64C2×#624—34no—
65C2×D3×Dic3—34no—
66C2×D3×Dic3—34no—
67#3798—316no—
68C32×C4⋊C4—21no—
69C32×C4⋊C4C2×D312yes—
70#3752C2×D314yes—
71C4⋊(C3⋊Dic3)C2×D312yes—
72C4×C3⋊Dic3C2×D314yes—
73C3×(C2×D3)⋊C4—34no—
74C3×(C2×D3)⋊C4—34no—
75C3×(C2×D3)⋊C4—62no—
76C3×C4⋊Dic3—32no—
77C3×C4⋊Dic3—32no—
78C3×C4⋊Dic3—61no—
79C3×C4×Dic3—34no—
80C3×C4×Dic3—34no—
81C3×C4×Dic3—62no—
82#3724—34no—
83Dic3⋊Dic3—34no—
84Dic3⋊Dic3—34no—
85C22×H(3,𝔽3)C2411yescharacteristic
86C2×#272—21no—
87C2×#272—21no—
88C2×#272C22×C412yes—
89C2×(C3⋊D3)⋊C3—22no—
90C2×(C3⋊D3)⋊C3—22no—
91C2×(C3⋊D3)⋊C3C22×C414yes—
92C4×H(3,𝔽3)—22no—
93C4×H(3,𝔽3)C22×C412yes—
94#1084—22no—
95#1084C22×C412yes—
96(C3⋊Dic3)⋊C3—24no—
97(C3⋊Dic3)⋊C3C22×C414yes—
98C2×#235—91no—
99C2×#235—92no—
100C3×#117—92no—
101C3×#117—91no—
102C2×C4⋊Dic3—91no—
103C2×C4⋊Dic3—92no—
104#892—32no—
105#892—32no—
106#892—61no—
107#892—92no—
108D3×C4⋊C4—92no—
109D3×C4⋊C4—92no—
110D3×C4⋊C4—92no—
111#67⋊C4—94no—
112#67⋊C4—94no—
113#67⋊C4—94no—
114C4×D12—92no—
115C4×D12—94no—
116C4×Dic6—92no—
117C4×Dic6—94no—
118C22×C3⋊D3C22×D312yes—
119C22×C3×D3—32no—
120C22×C3×D3—32no—
121C22×C3×D3—61no—
122C2×C32×C4—21no—
123C2×C32×C4—22no—
124C2×C32×C4C22×D314yes—
125C2×C32×C4C22×D312yes—
126C2×C3⋊Dic3C22×D314yes—
127C2×C3⋊Dic3C22×D312yes—
128C4×C3⋊D3C4×D318yescomplemented
129C2×C3×Dic3—32no—
130C2×C3×Dic3—32no—
131C2×C3×Dic3—34no—
132C2×C3×Dic3—34no—
133C2×C3×Dic3—61no—
134C2×C3×Dic3—62no—
135C3×C4×D3—38no—
136C3×C4×D3—38no—
137C3×C4×D3—64no—
138#627—38no—
139#626—38no—
140#626—38no—
141#625—38no—
142#624—124no—
143D3×Dic3—124no—
144D3×Dic3—124no—
145#520—271noSylow
146C2×H(3,𝔽3)C23×C411yescharacteristic
147C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
148C2×H(3,𝔽3)2−1+411yescharacteristic
149#272—22no—
150(C3⋊D3)⋊C3—24no—
151C22×C3×C4—92no—
152C22×C3×C4—91no—
153C2×#67—92no—
154C2×#67—92no—
155C2×#67—92no—
156C22×Dic3—92no—
157C22×Dic3—94no—
158#235—98no—
159#235—916no—
160C2×D12—91no—
161C2×D12—92no—
162C2×C4×D3—32no—
163C2×C4×D3—32no—
164C2×C4×D3—61no—
165C2×C4×D3—92no—
166C2×C4×D3—94no—
167C2×C4×D3—94no—
168C2×C4×D3—94no—
169C2×Dic6—91no—
170C2×Dic6—92no—
171C3×C4⋊C4—32no—
172C3×C4⋊C4—61no—
173C3×C4⋊C4—92no—
174C3×C4⋊C4—91no—
175C3×C4⋊C4D3211yescharacteristic
176C3×C22⋊C4—94no—
177C3×C22⋊C4—92no—
178C3×C42—94no—
179C3×C42—92no—
180(C2×D3)⋊C4—34no—
181(C2×D3)⋊C4—34no—
182(C2×D3)⋊C4—62no—
183(C2×D3)⋊C4—94no—
184C4⋊Dic3—32no—
185C4⋊Dic3—32no—
186C4⋊Dic3—61no—
187C4⋊Dic3—92no—
188C4⋊Dic3—92no—
189C4⋊Dic3—94no—
190Dic3⋊C4—94no—
191Dic3⋊C4—94no—
192Dic3⋊C4—94no—
193C4×Dic3—34no—
194C4×Dic3—34no—
195C4×Dic3—62no—
196C4×Dic3—94no—
197C22×C32—21no—
198C22×C32C23×D312yes—
199C2×C3⋊D3—22no—
200C2×C3⋊D3—22no—
201C2×C3⋊D3C2×C4×D314yes—
202C2×C3×D3—34no—
203C2×C3×D3—34no—
204C2×C3×D3—62no—
205C2×C3×D3—62no—
206C2×C3×D3—62no—
207C2×C3×D3—62no—
208C2×C3×D3—62no—
209C2×C3×D3—121no—
210C2×C3×D3—121no—
211C32×C4—24no—
212C32×C4—22no—
213C32×C4—42no—
214C32×C4C2×C4×D314yes—
215C3⋊Dic3—24no—
216C3⋊Dic3C2×C4×D314yes—
217C3×Dic3—34no—
218C3×Dic3—34no—
219C3×Dic3—62no—
220C3×Dic3—64no—
221C3×Dic3—64no—
222C3×Dic3—122no—
223C2×#41—271no—
224C4×Q8—272no—
225C4×D4—272no—
226C4×D4—274no—
227#117—91no—
228#117—92no—
229C2×C4⋊C4—272no—
230C2×C4⋊C4—271no—
231H(3,𝔽3)#52011yescharacteristic, complemented, Sylow
232C23×C3—92no—
233C23×C3—91no—
234C22×D3—32no—
235C22×D3—32no—
236C22×D3—61no—
237C22×D3—92no—
238C2×C3×C4—32no—
239C2×C3×C4—34no—
240C2×C3×C4—61no—
241C2×C3×C4—62no—
242C2×C3×C4—98no—
243C2×C3×C4—92no—
244C2×C3×C4—91no—
245C2×C3×C4—94no—
246C2×C3×C4—94no—
247C2×C3×C4—92no—
248C2×C3×C4C2×D3212yes—
249C2×C3×C4C2×D3211yescharacteristic
250#67—98no—
251#67—98no—
252#67—98no—
253C2×Dic3—32no—
254C2×Dic3—32no—
255C2×Dic3—34no—
256C2×Dic3—34no—
257C2×Dic3—61no—
258C2×Dic3—62no—
259C2×Dic3—92no—
260C2×Dic3—94no—
261C2×Dic3—362no—
262C2×Dic3—364no—
263D12—94no—
264D12—98no—
265C4×D3—38no—
266C4×D3—38no—
267C4×D3—64no—
268C4×D3—98no—
269C4×D3—364no—
270C4×D3—364no—
271C4×D3—364no—
272Dic6—94no—
273Dic6—98no—
274C2×C32—21no—
275C2×C32—21no—
276C2×C32—21no—
277C2×C32C22×C4×D312yes—
278C2×C32#100912yes—
279C2×C32#101012yes—
280C3⋊D3—24no—
281C3×D3—64no—
282C3×D3—64no—
283C3×D3—122no—
284#41—278no—
285C2×Q8—271no—
286C2×D4—271no—
287C2×D4—272no—
288C22×C4—91no—
289C22×C4—92no—
290C22×C4—274no—
291C22×C4—272no—
292C4⋊C4—91no—
293C4⋊C4—92no—
294C4⋊C4—272no—
295C4⋊C4—274no—
296C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
297C22⋊C4—92no—
298C22⋊C4—94no—
299C42—92no—
300C42—94no—
301C22×C3—32no—
302C22×C3—61no—
303C22×C3—94no—
304C22×C3—92no—
305C22×C3—182no—
306C22×C3—182no—
307C22×C3—181no—
308C22×C3—181no—
309C22×C3C22×D3211yescharacteristic
310C2×D3—34no—
311C2×D3—34no—
312C2×D3—62no—
313C2×D3—62no—
314C2×D3—62no—
315C2×D3—62no—
316C2×D3—62no—
317C2×D3—94no—
318C2×D3—121no—
319C2×D3—121no—
320C2×D3—182no—
321C2×D3—182no—
322C3×C4—22no—
323C3×C4—34no—
324C3×C4—64no—
325C3×C4—62no—
326C3×C4—94no—
327C3×C4—92no—
328C3×C4—122no—
329C3×C4—184no—
330C3×C4—182no—
331C3×C4C4×D3212yes—
332Dic3—34no—
333Dic3—34no—
334Dic3—62no—
335Dic3—64no—
336Dic3—64no—
337Dic3—94no—
338Dic3—122no—
339Dic3—184no—
340C32—21no—
341C32#552312yescomplemented
342C23—91no—
343C23—92no—
344Q8—274no—
345D4—274no—
346D4—278no—
347C2×C4—91no—
348C2×C4—92no—
349C2×C4—94no—
350C2×C4—98no—
351C2×C4—92no—
352C2×C4—94no—
353C2×C4—1084no—
354C2×C4—1082no—
355C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
356C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
357C2×C3—32no—
358C2×C3—32no—
359C2×C3—32no—
360C2×C3—61no—
361C2×C3—61no—
362C2×C3—61no—
363C2×C3—184no—
364C2×C3—182no—
365C2×C3C2×C4×D3211yescharacteristic
366C2×C3#6430911yescharacteristic
367C2×C3#6432011yescharacteristic
368D3—64no—
369D3—64no—
370D3—122no—
371D3—184no—
372C22—92no—
373C22—94no—
374C22—181no—
375C22—182no—
376C22—181no—
377C22—182no—
378C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
379C4—22no—
380C4—92no—
381C4—94no—
382C4—182no—
383C4—184no—
384C4C4×((C3⋊D3)⋊C3)⋊C212yes—
385C3—32no—
386C3—61no—
387C3#1059451111yescharacteristic
388C2—182no—
389C2—184no—
390C2C2×C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
391C2#1172250211yescharacteristic
392C2#1172252111yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C23×C4
Frattini, socleC22×C3C22×D32
FittingC4⋊C4×H(3,𝔽3)C22
prime radicalC2×#88650C2
cosocleC22×H(3,𝔽3)C24
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#520
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #11721760C2semidirect
3. #64101⋊C3C2semidirect
4. #11721152C2semidirect
5. #11721138C2semidirect
6. #11721129C2semidirect
7. #11721123C2semidirect
8. C4×#6196C2semidirect
17. #88650C4semidirect
28. #64101D3semidirect
45. C4×#272C2×C4semidirect
48. C4×(C3⋊D3)⋊C3C2×C4semidirect
49. #6200C2×C4semidirect
51. #6198C2×C4semidirect
53. #6196C2×C4semidirect
128. C4×C3⋊D3C4×D3semidirect
231. H(3,𝔽3)#520semidirect
341. C32#5523semidirect

Maximal subgroups

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

subgroupindex
C2×#886502
#117217602
#64101⋊C32
#117211522
#117211382
#117211292
#117211232
C4×#61962
#105945113