Finite Group Databasebeta

← #13155510

Subgroups — #13155510

Order 1728.

666 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 666 conjugacy classes of subgroups — too many to draw. Only its 104 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×#88788C211yesmaximal, characteristic
2#11723358C212yesmaximal
3#11722081C211yesmaximal, characteristic, complemented
4C3×#63578C211yesmaximal, characteristic, complemented
5C3×#63577C212yesmaximal, complemented
6#10590974—32nomaximal
7#10590914—31nomaximal
8C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
9C3×#3748C2212yes—
10C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
11C32×#208C2212yes—
12C2×C32×C3⋊C8C2211yescharacteristic
13#88788—22no—
14#88788C414yes—
15#88785—22no—
16M16×C3⋊D3—32no—
17C2×#3743—31no—
18C2×#3712—32no—
19C2×#3712—61no—
20#63818—32no—
21#63818—32no—
22#63818—62no—
23C3×#822—32no—
24C3×#822—32no—
25C3×#819—32no—
26#63578—31no—
27#63578D311yescharacteristic, complemented
28#63577—32no—
29#63577D312yescomplemented
30C22×C3⋊(C3⋊D3)Q811yescharacteristic
31C2×C33×C4C2311yescharacteristic
32C2×C3⋊(C3⋊Dic3)Q811yescharacteristic
33C4×C3⋊(C3⋊D3)C2×C412yes—
34C4×C3⋊(C3⋊D3)D412yes—
35C3×C3⋊(C3⋊C8)—22no—
36C3×C3⋊(C3⋊C8)C2×C412yes—
37C32×C3⋊C8—22no—
38C32×C3⋊C8C2×C412yes—
39#4893—92no—
40#4893—91no—
41C2×C4×C3⋊D3—31no—
42C2×C4×C3⋊D3—32no—
43C2×C4×C3⋊D3—62no—
44C2×C4×C3⋊D3—61no—
45C2×C4×C3⋊D3—121no—
46C32×M16—32no—
47C2×C32×C8—31no—
48#3748C2×D312yes—
49C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
50#3743—34no—
51#3743—62no—
52C8×C3⋊D3—62no—
53C3×#208—22no—
54C3×#208—32no—
55C3×#208—32no—
56C3×#208—62no—
57C3×#208C2×D312yes—
58C3×#208C2×D312yes—
59C2×C3×C3⋊C8—21no—
60C2×C3×C3⋊C8—32no—
61C2×C3×C3⋊C8—61no—
62C2×C3×C3⋊C8C2×D312yes—
63#3712—34no—
64#3712—34no—
65#3712—62no—
66#3712—62no—
67#3712—64no—
68#3712—64no—
69#3710—62no—
70#3710—62no—
71#3710—64no—
72C22×C33C2×Q811yescharacteristic
73C2×C3⋊(C3⋊D3)—41no—
74C2×C3⋊(C3⋊D3)C4⋊C412yes—
75C33×C4C22×C411yescharacteristic
76C33×C4C2×D411yescharacteristic
77C3⋊(C3⋊Dic3)C4⋊C412yes—
78D3×M16—92no—
79D3×M16—92no—
80D3×M16—92no—
81D3×M16—182no—
82C2×ΓL(1,𝔽25)—92no—
83C2×ΓL(1,𝔽25)—91no—
84C2×ΓL(1,𝔽25)—181no—
85C3×#108—92no—
86C3×#108—91no—
87#822—32no—
88#822—32no—
89#822—92no—
90#819—32no—
91#819—91no—
92C22×C3⋊D3—31no—
93C22×C3⋊D3—32no—
94C22×C3⋊D3—62no—
95C22×C3⋊D3—61no—
96C22×C3⋊D3—121no—
97C2×C32×C4—21no—
98C2×C32×C4—22no—
99C2×C32×C4—41no—
100C2×C32×C4C22×D312yes—
101C2×C32×C4C22×D311yescharacteristic
102C2×C3⋊Dic3—31no—
103C2×C3⋊Dic3—32no—
104C2×C3⋊Dic3—61no—
105C2×C3⋊Dic3—62no—
106C2×C3⋊Dic3—121no—
107C4×C3⋊D3—32no—
108C4×C3⋊D3—32no—
109C4×C3⋊D3—32no—
110C4×C3⋊D3—32no—
111C4×C3⋊D3—32no—
112C4×C3⋊D3—32no—
113C4×C3⋊D3—62no—
114C4×C3⋊D3—62no—
115C4×C3⋊D3—62no—
116C4×C3⋊D3—62no—
117C4×C3⋊D3—62no—
118C4×C3⋊D3—62no—
119C4×C3⋊D3—122no—
120C4×C3⋊D3—122no—
121C32×C8—32no—
122C32×C8—62no—
123C3⋊(C3⋊C8)—22no—
124C3⋊(C3⋊C8)C4×D312yes—
125C3×C3⋊C8—24no—
126C3×C3⋊C8—22no—
127C3×C3⋊C8—22no—
128C3×C3⋊C8—22no—
129C3×C3⋊C8—34no—
130C3×C3⋊C8—62no—
131C3×C3⋊C8—64no—
132C3×C3⋊C8—62no—
133C3×C3⋊C8—62no—
134C3×C3⋊C8C4×D314yes—
135#430—271noSylow
136C2×C33—21no—
137C2×C33C2×C4⋊C411yescharacteristic
138C3⋊(C3⋊D3)—22no—
139C2×C4×D3—92no—
140C2×C4×D3—91no—
141C2×C4×D3—181no—
142C2×C4×D3—182no—
143C2×C4×D3—361no—
144C3×M16—32no—
145C3×M16—32no—
146C3×M16—62no—
147C3×M16—92no—
148C2×C3×C8—32no—
149C2×C3×C8—61no—
150C2×C3×C8—91no—
151#208—32no—
152#208—32no—
153#208—62no—
154#208D3212yes—
155C2×C3⋊C8—32no—
156C2×C3⋊C8—61no—
157C2×C3⋊C8D3211yescharacteristic
158ΓL(1,𝔽25)—94no—
159ΓL(1,𝔽25)—94no—
160ΓL(1,𝔽25)—94no—
161ΓL(1,𝔽25)—184no—
162ΓL(1,𝔽25)—182no—
163ΓL(1,𝔽25)—182no—
164ΓL(1,𝔽25)—182no—
165ΓL(1,𝔽25)—184no—
166D3×C8—182no—
167D3×C8—182no—
168D3×C8—182no—
169D3×C8—184no—
170C22×C32—21no—
171C22×C32—22no—
172C22×C32—41no—
173C22×C32D3×Q811yescharacteristic
174C22×C32D3×Q812yes—
175C2×C3⋊D3—32no—
176C2×C3⋊D3—32no—
177C2×C3⋊D3—32no—
178C2×C3⋊D3—62no—
179C2×C3⋊D3—62no—
180C2×C3⋊D3—62no—
181C2×C3⋊D3—122no—
182C2×C3⋊D3—122no—
183C2×C3⋊D3—121no—
184C2×C3⋊D3—124no—
185C2×C3⋊D3—122no—
186C2×C3⋊D3—124no—
187C32×C4—22no—
188C32×C4—21no—
189C32×C4—21no—
190C32×C4—22no—
191C32×C4—41no—
192C32×C4—41no—
193C32×C4C2×C4×D311yescharacteristic
194C32×C4C2×C4×D312yes—
195C32×C4D3×D412yes—
196C32×C4D3×D411yescharacteristic
197C3⋊Dic3—32no—
198C3⋊Dic3—32no—
199C3⋊Dic3—32no—
200C3⋊Dic3—62no—
201C3⋊Dic3—62no—
202C3⋊Dic3—62no—
203C3⋊Dic3—122no—
204C2×M16—271no—
205C2×M16—272no—
206#108—92no—
207#108—91no—
208#108—271no—
209C33#43011yescharacteristic, complemented, Sylow
210C22×D3—92no—
211C22×D3—91no—
212C22×D3—182no—
213C22×D3—181no—
214C22×D3—361no—
215C2×C3×C4—21no—
216C2×C3×C4—22no—
217C2×C3×C4—41no—
218C2×C3×C4C2×D3212yes—
219C2×C3×C4C2×D3211yescharacteristic
220C2×Dic3—92no—
221C2×Dic3—91no—
222C2×Dic3—181no—
223C2×Dic3—182no—
224C2×Dic3—361no—
225C4×D3—92no—
226C4×D3—92no—
227C4×D3—92no—
228C4×D3—92no—
229C4×D3—92no—
230C4×D3—92no—
231C4×D3—182no—
232C4×D3—182no—
233C4×D3—182no—
234C4×D3—182no—
235C4×D3—182no—
236C4×D3—182no—
237C4×D3—362no—
238C4×D3—362no—
239C3×C8—34no—
240C3×C8—62no—
241C3×C8—64no—
242C3×C8—62no—
243C3×C8—62no—
244C3×C8—92no—
245C3×C8—182no—
246C3⋊C8—22no—
247C3⋊C8—34no—
248C3⋊C8—62no—
249C3⋊C8—64no—
250C3⋊C8—62no—
251C3⋊C8—62no—
252C3⋊C8#62412yes—
253C2×C32—21no—
254C2×C32—22no—
255C2×C32—22no—
256C2×C32—22no—
257C2×C32—21no—
258C2×C32—41no—
259C2×C32—42no—
260C2×C32—42no—
261C2×C32D3×C4⋊C411yescharacteristic
262C2×C32D3×C4⋊C412yes—
263C3⋊D3—62no—
264C3⋊D3—64no—
265C3⋊D3—62no—
266C3⋊D3—62no—
267C3⋊D3—122no—
268C3⋊D3—122no—
269C3⋊D3—124no—
270C22×C4—271no—
271M16—32no—
272M16—92no—
273M16—274no—
274M16—542no—
275C2×C8—31no—
276C2×C8—91no—
277C2×C8—542no—
278C22×C3—21no—
279C22×C3—22no—
280C22×C3—41no—
281C22×C3#379712yes—
282C22×C3#379711yescharacteristic
283C2×D3—92no—
284C2×D3—92no—
285C2×D3—92no—
286C2×D3—182no—
287C2×D3—182no—
288C2×D3—182no—
289C2×D3—362no—
290C2×D3—361no—
291C2×D3—362no—
292C2×D3—364no—
293C2×D3—362no—
294C2×D3—364no—
295C3×C4—21no—
296C3×C4—22no—
297C3×C4—22no—
298C3×C4—21no—
299C3×C4—41no—
300C3×C4—41no—
301C3×C4C4×D3212yes—
302C3×C4#380211yescharacteristic
303C3×C4C2×#62411yescharacteristic
304C3×C4#381112yes—
305Dic3—92no—
306Dic3—92no—
307Dic3—92no—
308Dic3—182no—
309Dic3—182no—
310Dic3—182no—
311Dic3—362no—
312C32—21no—
313C32—22no—
314C32—41no—
315C32#489312yescomplemented
316C32#489311yescharacteristic, complemented
317C23—271no—
318C2×C4—271no—
319C2×C4—272no—
320C2×C4—272no—
321C2×C4D3311yescharacteristic
322C8—32no—
323C8—62no—
324C8—92no—
325C8—182no—
326C2×C3—22no—
327C2×C3—21no—
328C2×C3—22no—
329C2×C3—22no—
330C2×C3—21no—
331C2×C3—42no—
332C2×C3—42no—
333C2×C3—41no—
334C2×C3#6388312yes—
335C2×C3#6390011yescharacteristic
336D3—184no—
337D3—182no—
338D3—182no—
339D3—182no—
340D3—362no—
341D3—362no—
342D3—364no—
343C22—272no—
344C22—1081no—
345C22#8895811yescharacteristic
346C4—272no—
347C4D3×#62411yescharacteristic
348C4#8895411yescharacteristic
349C3—22no—
350C3—21no—
351C3—41no—
352C3#1059091411yescharacteristic, complemented
353C3#1059097412yescomplemented
354C2—21no—
355C2—542no—
356C2#1172291211yescharacteristic
subgroupG/H
centerC4#88954
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC2×C33C2×C4⋊C4
prime radicalC22×C3⋊(C3⋊D3)Q8
hypocenterC33#430
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
3. #11722081C2semidirect
4. C3×#63578C2semidirect
5. C3×#63577C2semidirect
27. #63578D3semidirect
29. #63577D3semidirect
209. C33#430semidirect
315. C32#4893semidirect
316. C32#4893semidirect
352. C3#10590914semidirect
353. C3#10590974semidirect

Maximal subgroups

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

subgroupindex
C2×#887882
#117233582
#117220812
C3×#635782
C3×#635772
#105909743
#105909143