Finite Group Databasebeta

← #13150712

Subgroups — #13150712

Order 1728.

1121 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 1121 conjugacy classes of subgroups — too many to draw. Only its 147 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×#88672C212yesmaximal, complemented
2C2×#88540C211yesmaximal, characteristic, complemented
3#64139⋊C3C212yesmaximal, complemented
4#11721318C212yesmaximal, complemented
5#11721134C214yesmaximal, complemented
6#11721126C214yesmaximal, complemented
7#10594911—32nomaximal
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C22×#1084C2211yescharacteristic
10C22×#1084C2211yescharacteristic
11C2×#638⋊C3C2214yes—
12C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
13C2×#6198C2212yes—
14C2×#6198C2212yes—
15#88672—28no—
16C2×((C3⋊D3)⋊C3)⋊C4C2214yes—
17C2×#6196C2212yes—
18#88540C2214yes—
19#3752⋊C3C2214yes—
20C4⋊((C3⋊Dic3)⋊C3)C2212yes—
21#88450C2214yes—
22#88449C2212yes—
23C2×#3804—34no—
24#64139D312yescomplemented
25C3×#930—32no—
26C2×C3×Dic3⋊C4—32no—
27C2×C3×Dic3⋊C4—61no—
28#63978—34no—
29#63910—34no—
30#63895—34no—
31#63893—34no—
32#63886—34no—
33C23×H(3,𝔽3)C2311yescharacteristic
34C22×(C3⋊D3)⋊C3C2314yes—
35C2×C4×H(3,𝔽3)—41no—
36C2×C4×H(3,𝔽3)C2312yes—
37C2×#1084—41no—
38C2×#1084D414yescomplemented
39C2×#1084C2312yes—
40C2×#1084C2312yes—
41#638⋊C3—28no—
42C2×(C3⋊Dic3)⋊C3C2314yes—
43C4⋊((C3⋊D3)⋊C3)—42no—
44#6198—24no—
45#6198—42no—
46((C3⋊D3)⋊C3)⋊C4—28no—
47#6196—24no—
48#5597—92no—
49#5510—91no—
50C22×C32×C4—21no—
51C22×C32×C4C2×D312yes—
52C2×#638C2×D314yes—
53C2×C4⋊(C3⋊D3)C2×D312yes—
54C2×C3×#67—34no—
55C22×C3×Dic3—32no—
56C22×C3×Dic3—32no—
57C22×C3×Dic3—61no—
58C22×C3×Dic3—61no—
59C2×C3×D12—32no—
60C2×#627—34no—
61C2×#626—34no—
62C2×#626—34no—
63C2×#624—34no—
64#3804—616no—
65C2×D3×Dic3—34no—
66#3752C2×D314yes—
67C4⋊(C3⋊Dic3)C2×D312yes—
68C3×(C2×D3)⋊C4—34no—
69C3×C4⋊Dic3—32no—
70C3×Dic3⋊C4—34no—
71C3×Dic3⋊C4—34no—
72C3×Dic3⋊C4—64no—
73Dic3⋊Dic3—34no—
74#3722—34no—
75(C2×D3)⋊Dic3—34no—
76C22×H(3,𝔽3)—21no—
77C22×H(3,𝔽3)—21no—
78C22×H(3,𝔽3)C2×D412yes—
79C22×H(3,𝔽3)C2411yescharacteristic
80C2×(C3⋊D3)⋊C3—24no—
81C2×(C3⋊D3)⋊C3—24no—
82C2×(C3⋊D3)⋊C3—24no—
83C4×H(3,𝔽3)—22no—
84#1084—22no—
85#1084C2×D414yes—
86(C3⋊Dic3)⋊C3—24no—
87C2×#237—94no—
88C2×#235—92no—
89C3×#121—92no—
90C2×C3×C4⋊C4—91no—
91#939—94no—
92#939—94no—
93#930—32no—
94#930—92no—
95#930—92no—
96C2×Dic3⋊C4—32no—
97C2×Dic3⋊C4—61no—
98#897—94no—
99#896—94no—
100#895—94no—
101#884—94no—
102#878—94no—
103C23×C32—21no—
104C23×C32C22×D312yes—
105C22×C3⋊D3C22×D314yes—
106C22×C3×D3—34no—
107C2×C32×C4—22no—
108C2×C32×C4—42no—
109C2×C32×C4—42no—
110C2×C32×C4C22×D312yes—
111C2×C32×C4C22×D312yes—
112#638—28no—
113C2×C3⋊Dic3C22×D314yes—
114C4⋊(C3⋊D3)—42no—
115C3×#67—68no—
116C2×C3×Dic3—38no—
117C2×C3×Dic3—34no—
118C2×C3×Dic3—34no—
119C2×C3×Dic3—34no—
120C2×C3×Dic3—64no—
121C2×C3×Dic3—62no—
122C2×C3×Dic3—62no—
123C2×C3×Dic3—122no—
124C2×C3×Dic3—122no—
125C3×D12—122no—
126#627—68no—
127#626—68no—
128#626—124no—
129#624—68no—
130D3×Dic3—68no—
131#537—271noSylow
132C2×H(3,𝔽3)—22no—
133C2×H(3,𝔽3)C22×D411yescharacteristic
134C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
135C2×H(3,𝔽3)2−1+411yescharacteristic
136(C3⋊D3)⋊C3—44no—
137C2×C3×D4—92no—
138C2×C3×D4—94no—
139C22×C3×C4—32no—
140C22×C3×C4—61no—
141C22×C3×C4—91no—
142C22×C3×C4—91no—
143C22×C3×C4D3211yescharacteristic
144C2×#67—34no—
145C2×#67—94no—
146C2×#67—94no—
147C2×#67—94no—
148C22×Dic3—32no—
149C22×Dic3—32no—
150C22×Dic3—61no—
151C22×Dic3—61no—
152C22×Dic3—94no—
153#237—1816no—
154#235—188no—
155C2×D12—32no—
156C2×D12—92no—
157C2×D12—92no—
158C2×D12—92no—
159C2×C4×D3—94no—
160C2×C4×D3—94no—
161C2×Dic6—92no—
162C2×Dic6—94no—
163C3×C4⋊C4—92no—
164C3×C4⋊C4—94no—
165C3×C22⋊C4—94no—
166#217—94no—
167(C2×D3)⋊C4—34no—
168(C2×D3)⋊C4—94no—
169(C2×D3)⋊C4—94no—
170(C2×D3)⋊C4—94no—
171C4⋊Dic3—32no—
172C4⋊Dic3—92no—
173C4⋊Dic3—92no—
174Dic3⋊C4—34no—
175Dic3⋊C4—34no—
176Dic3⋊C4—64no—
177Dic3⋊C4—94no—
178C22×C32—22no—
179C22×C32—22no—
180C22×C32—22no—
181C22×C32—22no—
182C22×C32—21no—
183C22×C32—22no—
184C22×C32D3×D414yes—
185C22×C32C23×D312yes—
186C2×C3⋊D3—24no—
187C2×C3⋊D3—24no—
188C2×C3⋊D3—24no—
189C2×C3×D3—64no—
190C2×C3×D3—64no—
191C2×C3×D3—64no—
192C32×C4—22no—
193C32×C4—22no—
194C32×C4—42no—
195C3⋊Dic3—24no—
196C3×Dic3—38no—
197C3×Dic3—64no—
198C3×Dic3—64no—
199C3×Dic3—64no—
200C3×Dic3—122no—
201C2×#41—272no—
202C22⋊Q8—274no—
203#121—92no—
204#121—274no—
205#121—272no—
206C2×C4⋊C4—91no—
207H(3,𝔽3)#53711yescharacteristic, complemented, Sylow
208C23×C3—32no—
209C23×C3—61no—
210C23×C3—94no—
211C23×C3C2×D3211yescharacteristic
212C22×D3—34no—
213C22×D3—94no—
214C3×D4—188no—
215C3×D4—362no—
216C2×C3×C4—32no—
217C2×C3×C4—32no—
218C2×C3×C4—41no—
219C2×C3×C4—62no—
220C2×C3×C4—92no—
221C2×C3×C4—94no—
222C2×C3×C4—94no—
223C2×C3×C4—92no—
224C2×C3×C4—122no—
225C2×C3×C4—122no—
226C2×C3×C4—361no—
227C2×C3×C4C2×D3212yes—
228#67—68no—
229#67—188no—
230#67—188no—
231#67—364no—
232C2×Dic3—34no—
233C2×Dic3—38no—
234C2×Dic3—34no—
235C2×Dic3—34no—
236C2×Dic3—62no—
237C2×Dic3—64no—
238C2×Dic3—62no—
239C2×Dic3—94no—
240C2×Dic3—122no—
241C2×Dic3—122no—
242C2×Dic3—188no—
243D12—122no—
244D12—184no—
245D12—362no—
246D12—362no—
247C4×D3—188no—
248C4×D3—188no—
249Dic6—188no—
250Dic6—184no—
251C2×C32—24no—
252C2×C32—24no—
253C2×C32—21no—
254C2×C32—21no—
255C2×C32—21no—
256C2×C32C2×D3×D412yes—
257C2×C32#100412yes—
258C2×C32#101012yes—
259C3⋊D3—44no—
260C3×D3—124no—
261#41—548no—
262C2×Q8—272no—
263C2×D4—94no—
264C2×D4—92no—
265C2×D4—272no—
266C2×D4—272no—
267C22×C4—91no—
268C22×C4—91no—
269C22×C4—274no—
270C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
271C4⋊C4—94no—
272C4⋊C4—92no—
273C4⋊C4—272no—
274C22⋊C4—94no—
275C22⋊C4—274no—
276C22×C3—21no—
277C22×C3—21no—
278C22×C3—32no—
279C22×C3—34no—
280C22×C3—62no—
281C22×C3—62no—
282C22×C3—62no—
283C22×C3—62no—
284C22×C3—61no—
285C22×C3—62no—
286C22×C3—184no—
287C22×C3—184no—
288C22×C3—184no—
289C22×C3#380212yes—
290C22×C3C22×D3211yescharacteristic
291C2×D3—64no—
292C2×D3—64no—
293C2×D3—64no—
294C2×D3—184no—
295C2×D3—184no—
296C2×D3—184no—
297C3×C4—22no—
298C3×C4—62no—
299C3×C4—62no—
300C3×C4—94no—
301C3×C4—122no—
302C3×C4—184no—
303C3×C4—182no—
304Dic3—38no—
305Dic3—64no—
306Dic3—64no—
307Dic3—64no—
308Dic3—122no—
309Dic3—184no—
310C32—21no—
311C32#559712yescomplemented
312C23—94no—
313C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
314Q8—544no—
315D4—188no—
316D4—362no—
317D4—544no—
318D4—1082no—
319C2×C4—41no—
320C2×C4—92no—
321C2×C4—94no—
322C2×C4—94no—
323C2×C4—92no—
324C2×C4—361no—
325C2×C4—548no—
326C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
327C2×C3—22no—
328C2×C3—32no—
329C2×C3—32no—
330C2×C3—32no—
331C2×C3—64no—
332C2×C3—64no—
333C2×C3—61no—
334C2×C3—61no—
335C2×C3—61no—
336C2×C3—364no—
337C2×C3#6429811yescharacteristic
338C2×C3C2×#380211yescharacteristic
339C2×C3#6433011yescharacteristic
340D3—124no—
341D3—364no—
342C22—21no—
343C22—21no—
344C22—184no—
345C22—184no—
346C22—184no—
347C22#8865512yes—
348C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
349C4—22no—
350C4—94no—
351C4—182no—
352C4—184no—
353C3—32no—
354C3—61no—
355C3#1059493011yescharacteristic
356C2—22no—
357C2—364no—
358C2#1172248311yescharacteristic
359C2C2×#8865511yescharacteristic
360C2#1172254511yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC22×C4×H(3,𝔽3)C22
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#537
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88672C2semidirect
2. C2×#88540C2semidirect
3. #64139⋊C3C2semidirect
4. #11721318C2semidirect
5. #11721134C2semidirect
6. #11721126C2semidirect
24. #64139D3semidirect
38. C2×#1084D4semidirect
207. H(3,𝔽3)#537semidirect
311. C32#5597semidirect

Maximal subgroups

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

subgroupindex
C2×#886722
C2×#885402
#64139⋊C32
#117213182
#117211342
#117211262
#105949113