Finite Group Databasebeta

← #13147427

Subgroups — #13147427

Order 1728.

571 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 571 conjugacy classes of subgroups — too many to draw. Only its 59 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
1#41×#272C211yesmaximal, characteristic
2C2×C4⋊#1088C211yesmaximal, characteristic, complemented
3#11722445C211yesmaximal, characteristic, complemented
4#11720790C212yesmaximal, complemented
5#11720788C212yesmaximal, complemented
6C22×#1088—21no—
7#41×H(3,𝔽3)C411yescharacteristic, complemented
8#88748C411yescharacteristic, complemented
9Q8×#272C2211yescharacteristic
10#88744—21no—
11#88744C411yescharacteristic, complemented
12D4×#272—21no—
13D4×#272C2211yescharacteristic
14#88740—21no—
15#88740C411yescharacteristic, complemented
16C2×C4×#272—21no—
17C2×C4×#272C2211yescharacteristic
18#88631—21no—
19C4⋊#1088—21no—
20C4⋊#1088C2212yes—
21#88627—21no—
22#88626C2212yes—
23C3×D3×#41—62no—
24C22×#272—21no—
25C22×#272D411yescharacteristic
26C2×#1088—22no—
27C2×#1088—42no—
28Q8×H(3,𝔽3)C2×C411yescharacteristic, complemented
29D4×H(3,𝔽3)—21no—
30D4×H(3,𝔽3)C2×C411yescharacteristic, complemented
31C2×C4×H(3,𝔽3)—21no—
32C2×C4×H(3,𝔽3)C2×C411yescharacteristic
33#6235—21no—
34#6235—21no—
35#6235—21no—
36C2×#1084—21no—
37C2×#1084D411yescharacteristic
38C4⋊#272—21no—
39C4⋊#272C2×C411yescharacteristic, complemented
40C4×#272—21no—
41C4×#272—21no—
42C4×#272—21no—
43C4×#272C2×C411yescharacteristic
44C4×#272D411yescharacteristic
45C4×#272D411yescharacteristic
46C4×#272C2311yescharacteristic
47#6229—21no—
48#6229C2×C411yescharacteristic, complemented
49#6188—22no—
50C3×#417—91nomaximal
51C32×#41—22no—
52C3×#239—62no—
53C3×D3×Q8—62no—
54C3×#237—62no—
55C3×#237—64no—
56C3×D3×D4—62no—
57C3×D3×D4—64no—
58C3×#235—62no—
59C3×#235—64no—
60C2×C3×C4×D3—62no—
61C2×C3×C4×D3—64no—
62C22×H(3,𝔽3)—21no—
63C22×H(3,𝔽3)C22⋊C411yescharacteristic
64C2×#272—21no—
65C2×#272—21no—
66C2×#272—21no—
67C2×#272—41no—
68C2×#272—41no—
69C2×#272C22⋊C411yescharacteristic
70C2×#272C2×D411yescharacteristic
71#1088—42no—
72C4×H(3,𝔽3)—21no—
73C4×H(3,𝔽3)C22⋊C411yescharacteristic
74C4×H(3,𝔽3)C22×C411yescharacteristic
75#1084—21no—
76#1084C22⋊C411yescharacteristic
77#1084C2×D411yescharacteristic
78C2×C3×#41—91no—
79D3×#41—62no—
80C2×C3×M16—91no—
81C2×C3×C4⋊C4—91no—
82C3×#103—92no—
83C3×#102—92no—
84C22×C3×D3—62no—
85C22×C3×D3—64no—
86C32×Q8—22no—
87C32×D4—24no—
88C32×D4—22no—
89C2×C32×C4—24no—
90C2×C32×C4—22no—
91C3×#67—64no—
92C3×#67—64no—
93C3×#67—64no—
94C2×C3×Dic3—62no—
95C2×C3×Dic3—64no—
96C3×D12—62no—
97C3×D12—64no—
98C3×C4×D3—62no—
99C3×C4×D3—62no—
100C3×C4×D3—62no—
101C3×C4×D3—62no—
102C3×C4×D3—64no—
103C3×C4×D3—64no—
104C3×C4×D3—64no—
105C3×Dic6—62no—
106C3×Dic6—64no—
107#417—91noSylow
108C2×H(3,𝔽3)—21no—
109C2×H(3,𝔽3)—41no—
110C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
111#272—21no—
112#272—41no—
113#272AGL(1,ℤ/8)11yescharacteristic
114#272#13711yescharacteristic
115C3×#41—62no—
116C3×#41—91no—
117C3×#41—91no—
118C3×#41—91no—
119C3×#41—181no—
120C3×#41—181no—
121C3×#41#15711yescharacteristic
122C2×C3×Q8—91no—
123C2×C3×D4—91no—
124C2×C3×D4—181no—
125C22×C3×C4—91no—
126C22×C3×C4—181no—
127C22×C3×C4—181no—
128#239—62no—
129D3×Q8—62no—
130#237—62no—
131#237—64no—
132D3×D4—62no—
133D3×D4—64no—
134#235—62no—
135#235—64no—
136C2×C4×D3—62no—
137C2×C4×D3—64no—
138C3×M16—181no—
139C3×M16—181no—
140C2×C3×C8—92no—
141C3×C4⋊C4—92no—
142C3×C4⋊C4—181no—
143C22×C32—24no—
144C22×C32—22no—
145C2×C3×D3—62no—
146C2×C3×D3—62no—
147C2×C3×D3—64no—
148C2×C3×D3—122no—
149C2×C3×D3—122no—
150C2×C3×D3—124no—
151C2×C3×D3—124no—
152C32×C4—24no—
153C32×C4—22no—
154C32×C4—22no—
155C3×Dic3—62no—
156C3×Dic3—62no—
157C3×Dic3—64no—
158C2×#41—91no—
159C2×M16—91no—
160C2×C4⋊C4—91no—
161#103—92no—
162#102—92no—
163H(3,𝔽3)#41711yescharacteristic, complemented, Sylow
164C23×C3—91no—
165C23×C3—181no—
166C22×D3—62no—
167C22×D3—64no—
168C3×Q8—62no—
169C3×Q8—91no—
170C3×Q8—181no—
171C3×Q8C2×#15711yescharacteristic
172C3×D4—21no—
173C3×D4—64no—
174C3×D4—62no—
175C3×D4—91no—
176C3×D4—181no—
177C3×D4—181no—
178C3×D4—181no—
179C3×D4—181no—
180C3×D4C2×#15711yescharacteristic
181C2×C3×C4—21no—
182C2×C3×C4—64no—
183C2×C3×C4—62no—
184C2×C3×C4—91no—
185C2×C3×C4—91no—
186C2×C3×C4—91no—
187C2×C3×C4—91no—
188C2×C3×C4—91no—
189C2×C3×C4—181no—
190C2×C3×C4—181no—
191C2×C3×C4—181no—
192C2×C3×C4—181no—
193C2×C3×C4—182no—
194C2×C3×C4—362no—
195C2×C3×C4C2×#15711yescharacteristic
196#67—64no—
197#67—64no—
198#67—64no—
199C2×Dic3—62no—
200C2×Dic3—64no—
201D12—62no—
202D12—64no—
203C4×D3—62no—
204C4×D3—62no—
205C4×D3—62no—
206C4×D3—62no—
207C4×D3—64no—
208C4×D3—64no—
209C4×D3—64no—
210Dic6—62no—
211Dic6—64no—
212C3×C8—182no—
213C2×C32—22no—
214C2×C32—42no—
215C2×C32—44no—
216C3×D3—62no—
217C3×D3—62no—
218C3×D3—122no—
219C3×D3—124no—
220#41—91no—
221#41—91no—
222#41—91no—
223#41—181no—
224#41—181no—
225#41#108811yescharacteristic, complemented
226C2×Q8—91no—
227C2×D4—91no—
228C2×D4—181no—
229C22×C4—91no—
230C22×C4—181no—
231C22×C4—181no—
232M16—181no—
233M16—181no—
234C2×C8—92no—
235C4⋊C4—92no—
236C4⋊C4—181no—
237C22×C3—21no—
238C22×C3—64no—
239C22×C3—62no—
240C22×C3—91no—
241C22×C3—91no—
242C22×C3—181no—
243C22×C3—181no—
244C22×C3—181no—
245C22×C3—361no—
246C22×C3—361no—
247C22×C3C22⋊#15711yescharacteristic
248C2×D3—62no—
249C2×D3—62no—
250C2×D3—64no—
251C2×D3—122no—
252C2×D3—122no—
253C2×D3—124no—
254C2×D3—124no—
255C3×C4—21no—
256C3×C4—64no—
257C3×C4—62no—
258C3×C4—62no—
259C3×C4—91no—
260C3×C4—91no—
261C3×C4—181no—
262C3×C4—362no—
263C3×C4C22⋊#15711yescharacteristic
264C3×C4C22×#15711yescharacteristic
265Dic3—62no—
266Dic3—62no—
267Dic3—64no—
268C32—22no—
269C23—91no—
270C23—181no—
271Q8—91no—
272Q8—181no—
273Q8C2×#108811yescharacteristic, complemented
274D4—21no—
275D4—91no—
276D4—181no—
277D4—181no—
278D4—181no—
279D4—181no—
280D4C2×#108811yescharacteristic, complemented
281C2×C4—21no—
282C2×C4—91no—
283C2×C4—91no—
284C2×C4—91no—
285C2×C4—91no—
286C2×C4—91no—
287C2×C4—181no—
288C2×C4—181no—
289C2×C4—181no—
290C2×C4—181no—
291C2×C4—182no—
292C2×C4—362no—
293C2×C4C2×#108811yescharacteristic
294C8—182no—
295C2×C3—21no—
296C2×C3—41no—
297C2×C3—62no—
298C2×C3—91no—
299C2×C3—91no—
300C2×C3—122no—
301C2×C3—124no—
302C2×C3—181no—
303C2×C3—361no—
304C2×C3C2×C22⋊#15711yescharacteristic
305D3—62no—
306D3—62no—
307D3—122no—
308D3—124no—
309C22—21no—
310C22—91no—
311C22—91no—
312C22—181no—
313C22—181no—
314C22—181no—
315C22—361no—
316C22—361no—
317C22C22⋊#108811yescharacteristic
318C4—21no—
319C4—91no—
320C4—91no—
321C4—181no—
322C4—362no—
323C4C22⋊#108811yescharacteristic
324C4C22×#108811yescharacteristic
325C3—62no—
326C3#1059384011yescharacteristic
327C2—21no—
328C2—41no—
329C2—91no—
330C2—91no—
331C2—181no—
332C2—361no—
333C2C2×C22⋊#108811yescharacteristic
subgroupG/H
center, socleC2×C3C2×C22⋊#157
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC3×C4C22×#157
Fitting#41×H(3,𝔽3)C4
prime radical#41×#272C2
cosocleC4×#272C23
hypercenterC3×#41#157
hypocenterH(3,𝔽3)#417
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#41

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊#1088C2semidirect
3. #11722445C2semidirect
4. #11720790C2semidirect
5. #11720788C2semidirect
7. #41×H(3,𝔽3)C4semidirect
8. #88748C4semidirect
11. #88744C4semidirect
15. #88740C4semidirect
28. Q8×H(3,𝔽3)C2×C4semidirect
30. D4×H(3,𝔽3)C2×C4semidirect
39. C4⋊#272C2×C4semidirect
48. #6229C2×C4semidirect
163. H(3,𝔽3)#417semidirect
225. #41#1088semidirect
273. Q8C2×#1088semidirect
280. D4C2×#1088semidirect

Maximal subgroups

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

subgroupindex
#41×#2722
C2×C4⋊#10882
#117224452
#117207902
#117207882
C3×#4179