Finite Group Databasebeta

← #13147412

Subgroups — #13147412

Order 1728.

901 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 901 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C23×#1088C211yesmaximal, characteristic, complemented
2C2×D4×#272C211yesmaximal, characteristic
3C2×C22⋊#1088C211yesmaximal, characteristic, complemented
4C2×C22⋊#1088C212yesmaximal, complemented
5#11720787C212yesmaximal, complemented
6C23×#272C2211yescharacteristic
7C23×#272C2211yescharacteristic
8C22×#1088—22no—
9C22×#1088—24no—
10C22×#1088C2212yes—
11C22×#1088C2212yes—
12C2×D4×H(3,𝔽3)C411yescharacteristic, complemented
13C2×#6235C411yescharacteristic, complemented
14C2×#6235C411yescharacteristic, complemented
15D4×#272—22no—
16D4×#272—22no—
17C2×C4⋊#272C411yescharacteristic, complemented
18C2×C4×#272C2211yescharacteristic
19C22⋊#1088—22no—
20C22⋊#1088—24no—
21C2×C3×D3×D4—62no—
22C23×H(3,𝔽3)C2×C411yescharacteristic
23C23×H(3,𝔽3)C2×C411yescharacteristic, complemented
24C22×#272—22no—
25C22×#272—22no—
26C22×#272—21no—
27C22×#272—21no—
28C22×#272—22no—
29C22×#272—22no—
30C22×#272C2×C411yescharacteristic
31C22×#272C2×C411yescharacteristic, complemented
32C22×#272D412yes—
33C22×#272D412yes—
34C22×#272C2311yescharacteristic
35C2×#1088—28no—
36C2×#1088—22no—
37C2×#1088—22no—
38C2×#1088—22no—
39C2×#1088—22no—
40C2×#1088—22no—
41C2×#1088D414yescomplemented
42D4×H(3,𝔽3)—22no—
43C2×C4×H(3,𝔽3)C2×C411yescharacteristic, complemented
44#6235—22no—
45#6235—22no—
46C2×#1084C2×C411yescharacteristic, complemented
47C4⋊#272—22no—
48C4×#272—21no—
49C4×#272—21no—
50C3×#386—91nomaximal
51C23×C3×D3—62no—
52C23×C3×D3—62no—
53C2×C32×D4—22no—
54C2×C3×#67—62no—
55C2×C3×#67—62no—
56C3×D3×D4—68no—
57C3×D3×D4—68no—
58C2×C3×D12—62no—
59C2×C3×C4×D3—62no—
60C22×H(3,𝔽3)—22no—
61C22×H(3,𝔽3)—21no—
62C22×H(3,𝔽3)—22no—
63C22×H(3,𝔽3)C22⋊C412yescomplemented
64C22×H(3,𝔽3)C22×C411yescharacteristic
65C2×#272—22no—
66C2×#272—22no—
67C2×#272—24no—
68C2×#272—22no—
69C2×#272—21no—
70C2×#272—22no—
71C2×#272—41no—
72C2×#272—42no—
73C2×#272—41no—
74C2×#272C22⋊C412yescomplemented
75C2×#272C2×D412yes—
76C2×#272C2×D411yescharacteristic
77C2×#272C2×D411yescharacteristic
78C2×#272#4112yes—
79#1088—24no—
80#1088—42no—
81C4×H(3,𝔽3)—21no—
82#1084—21no—
83C22×C3×D4—91no—
84C23×C3×C4—91no—
85C2×D3×D4—62no—
86C2×C3×C22⋊C4—92no—
87C2×C3×C22⋊C4—91no—
88C3×#95—92no—
89C23×C32—22no—
90C23×C32—22no—
91C22×C3×D3—62no—
92C22×C3×D3—62no—
93C22×C3×D3—62no—
94C22×C3×D3—64no—
95C22×C3×D3—64no—
96C22×C3×D3—64no—
97C22×C3×D3—64no—
98C22×C3×D3—124no—
99C22×C3×D3—124no—
100C22×C3×D3—124no—
101C22×C3×D3—124no—
102C32×D4—28no—
103C2×C32×C4—22no—
104C3×#67—68no—
105C3×#67—68no—
106C2×C3×Dic3—62no—
107C3×D12—68no—
108C3×C4×D3—64no—
109C3×C4×D3—64no—
110#386—91noSylow
111C2×H(3,𝔽3)—22no—
112C2×H(3,𝔽3)—41no—
113C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
114C2×H(3,𝔽3)C4×D412yes—
115#272—22no—
116#272—41no—
117#272#12011yescharacteristic
118#272#12112yes—
119#272#12311yescharacteristic
120C24×C3—91no—
121C24×C3—91no—
122C23×D3—62no—
123C23×D3—62no—
124C2×C3×D4—62no—
125C2×C3×D4—91no—
126C2×C3×D4—91no—
127C2×C3×D4—91no—
128C2×C3×D4—182no—
129C2×C3×D4—182no—
130C2×C3×D4#15711yescharacteristic
131C22×C3×C4—92no—
132C22×C3×C4—91no—
133C22×C3×C4—92no—
134C22×C3×C4—184no—
135C22×C3×C4—182no—
136C2×#67—62no—
137C2×#67—62no—
138D3×D4—68no—
139D3×D4—68no—
140C2×D12—62no—
141C2×C4×D3—62no—
142C3×C22⋊C4—184no—
143C3×C22⋊C4—182no—
144C22×C32—24no—
145C22×C32—22no—
146C22×C32—24no—
147C22×C32—44no—
148C22×C32—44no—
149C2×C3×D3—62no—
150C2×C3×D3—62no—
151C2×C3×D3—64no—
152C2×C3×D3—64no—
153C2×C3×D3—64no—
154C2×C3×D3—64no—
155C2×C3×D3—124no—
156C2×C3×D3—128no—
157C2×C3×D3—124no—
158C2×C3×D3—124no—
159C2×C3×D3—124no—
160C2×C3×D3—124no—
161C2×C3×D3—128no—
162C2×C3×D3—124no—
163C32×C4—24no—
164C3×Dic3—64no—
165C22×D4—91no—
166C23×C4—91no—
167C2×C22⋊C4—92no—
168C2×C22⋊C4—91no—
169#95—92no—
170H(3,𝔽3)#38611yescharacteristic, complemented, Sylow
171C23×C3—62no—
172C23×C3—62no—
173C23×C3—91no—
174C23×C3—91no—
175C23×C3—92no—
176C23×C3—92no—
177C23×C3—91no—
178C23×C3—182no—
179C23×C3—182no—
180C23×C3—181no—
181C23×C3—181no—
182C23×C3—182no—
183C23×C3—182no—
184C23×C3C2×#15711yescharacteristic
185C23×C3C2×#15711yescharacteristic
186C22×D3—62no—
187C22×D3—62no—
188C22×D3—62no—
189C22×D3—64no—
190C22×D3—64no—
191C22×D3—64no—
192C22×D3—64no—
193C22×D3—124no—
194C22×D3—124no—
195C22×D3—124no—
196C22×D3—124no—
197C3×D4—22no—
198C3×D4—68no—
199C3×D4—182no—
200C3×D4—182no—
201C3×D4—182no—
202C2×C3×C4—62no—
203C2×C3×C4—94no—
204C2×C3×C4—91no—
205C2×C3×C4—181no—
206C2×C3×C4—181no—
207C2×C3×C4—182no—
208C2×C3×C4—182no—
209C2×C3×C4—188no—
210C2×C3×C4—182no—
211C2×C3×C4—182no—
212C2×C3×C4—182no—
213C2×C3×C4C2×#15711yescharacteristic
214#67—68no—
215#67—68no—
216C2×Dic3—62no—
217D12—68no—
218C4×D3—64no—
219C4×D3—64no—
220C2×C32—24no—
221C2×C32—22no—
222C2×C32—44no—
223C2×C32—44no—
224C3×D3—62no—
225C3×D3—64no—
226C3×D3—62no—
227C3×D3—124no—
228C3×D3—124no—
229C24—91no—
230C24—91no—
231C2×D4—91no—
232C2×D4—91no—
233C2×D4—91no—
234C2×D4—182no—
235C2×D4—182no—
236C2×D4#108811yescharacteristic, complemented
237C22×C4—92no—
238C22×C4—91no—
239C22×C4—92no—
240C22×C4—184no—
241C22×C4—182no—
242C22⋊C4—184no—
243C22⋊C4—182no—
244C22×C3—22no—
245C22×C3—22no—
246C22×C3—21no—
247C22×C3—64no—
248C22×C3—64no—
249C22×C3—62no—
250C22×C3—92no—
251C22×C3—92no—
252C22×C3—91no—
253C22×C3—91no—
254C22×C3—92no—
255C22×C3—124no—
256C22×C3—124no—
257C22×C3—182no—
258C22×C3—184no—
259C22×C3—182no—
260C22×C3—182no—
261C22×C3—182no—
262C22×C3—181no—
263C22×C3—361no—
264C22×C3—362no—
265C22×C3—361no—
266C22×C3C22⋊#15712yes—
267C22×C3C22×#15711yescharacteristic
268C2×D3—62no—
269C2×D3—62no—
270C2×D3—64no—
271C2×D3—64no—
272C2×D3—64no—
273C2×D3—64no—
274C2×D3—124no—
275C2×D3—128no—
276C2×D3—124no—
277C2×D3—124no—
278C2×D3—124no—
279C2×D3—124no—
280C2×D3—128no—
281C2×D3—124no—
282C3×C4—21no—
283C3×C4—64no—
284C3×C4—184no—
285C3×C4—181no—
286C3×C4—362no—
287Dic3—64no—
288C32—22no—
289C23—91no—
290C23—92no—
291C23—92no—
292C23—91no—
293C23—91no—
294C23—182no—
295C23—182no—
296C23—182no—
297C23—181no—
298C23—181no—
299C23—182no—
300C23C2×#108811yescharacteristic
301C23C2×#108811yescharacteristic, complemented
302D4—22no—
303D4—182no—
304D4—182no—
305D4—182no—
306C2×C4—94no—
307C2×C4—91no—
308C2×C4—181no—
309C2×C4—181no—
310C2×C4—182no—
311C2×C4—182no—
312C2×C4—188no—
313C2×C4—182no—
314C2×C4—182no—
315C2×C4—182no—
316C2×C4C2×#108811yescharacteristic, complemented
317C2×C3—22no—
318C2×C3—41no—
319C2×C3—64no—
320C2×C3—62no—
321C2×C3—91no—
322C2×C3—92no—
323C2×C3—91no—
324C2×C3—124no—
325C2×C3—124no—
326C2×C3—182no—
327C2×C3—361no—
328C2×C3D4×#15712yes—
329C2×C3C2×C22⋊#15711yescharacteristic
330D3—62no—
331D3—64no—
332D3—62no—
333D3—124no—
334D3—124no—
335C22—22no—
336C22—22no—
337C22—21no—
338C22—92no—
339C22—92no—
340C22—91no—
341C22—91no—
342C22—92no—
343C22—182no—
344C22—184no—
345C22—182no—
346C22—182no—
347C22—182no—
348C22—181no—
349C22—361no—
350C22—362no—
351C22—361no—
352C22C22⋊#108812yescomplemented
353C22C22×#108811yescharacteristic
354C4—21no—
355C4—184no—
356C4—181no—
357C4—362no—
358C3—62no—
359C3#1059382611yescharacteristic
360C2—22no—
361C2—41no—
362C2—91no—
363C2—92no—
364C2—91no—
365C2—182no—
366C2—361no—
367C2D4×#108812yes—
368C2C2×C22⋊#108811yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C22×#157
commutatorC22×H(3,𝔽3)C22×C4
FittingC2×D4×H(3,𝔽3)C4
prime radicalC2×D4×#272C2
cosocleC22×#272C23
hypercenterC2×C3×D4#157
hypocenterH(3,𝔽3)#386
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
1. C23×#1088C2semidirect
3. C2×C22⋊#1088C2semidirect
4. C2×C22⋊#1088C2semidirect
5. #11720787C2semidirect
12. C2×D4×H(3,𝔽3)C4semidirect
13. C2×#6235C4semidirect
14. C2×#6235C4semidirect
17. C2×C4⋊#272C4semidirect
23. C23×H(3,𝔽3)C2×C4semidirect
31. C22×#272C2×C4semidirect
41. C2×#1088D4semidirect
43. C2×C4×H(3,𝔽3)C2×C4semidirect
46. C2×#1084C2×C4semidirect
63. C22×H(3,𝔽3)C22⋊C4semidirect
74. C2×#272C22⋊C4semidirect
170. H(3,𝔽3)#386semidirect
236. C2×D4#1088semidirect
301. C23C2×#1088semidirect
316. C2×C4C2×#1088semidirect
352. C22C22⋊#1088semidirect

Maximal subgroups

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

subgroupindex
C23×#10882
C2×D4×#2722
C2×C22⋊#10882
#117207872
C3×#3869