Finite Group Databasebeta

← #13155504

Subgroups — #13155504

Order 1728.

690 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 106 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#11723358C212yesmaximal
2C2×#88785C211yesmaximal, characteristic
3#11722081C211yesmaximal, characteristic, complemented
4C3×#63578C211yesmaximal, characteristic, complemented
5C3×#63577C212yesmaximal, complemented
6#10590972—32nomaximal
7#10590913—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#88785—22no—
15#88785C412yes—
16#88785C412yes—
17M16×C3⋊D3—32no—
18C2×C8×C3⋊D3—31no—
19#63818—32no—
20#63818—32no—
21#63818—62no—
22C2×#3710—32no—
23C2×#3710—61no—
24C3×#822—32no—
25C3×#822—32no—
26C3×#819—32no—
27#63578—31no—
28#63578D311yescharacteristic, complemented
29#63577—32no—
30#63577D312yescomplemented
31C22×C3⋊(C3⋊D3)Q811yescharacteristic
32C2×C33×C4C2311yescharacteristic
33C2×C3⋊(C3⋊Dic3)Q811yescharacteristic
34C4×C3⋊(C3⋊D3)C2×C411yescharacteristic
35C4×C3⋊(C3⋊D3)C2×C411yescharacteristic
36C4×C3⋊(C3⋊D3)D411yescharacteristic
37C4×C3⋊(C3⋊D3)D411yescharacteristic
38C3×C3⋊(C3⋊C8)—22no—
39C3×C3⋊(C3⋊C8)C2×C412yes—
40C32×C3⋊C8—22no—
41C32×C3⋊C8C2×C412yes—
42D3×#108—92no—
43D3×#108—91no—
44C2×C4×C3⋊D3—32no—
45C2×C4×C3⋊D3—31no—
46C2×C4×C3⋊D3—61no—
47C2×C4×C3⋊D3—62no—
48C2×C4×C3⋊D3—121no—
49C32×M16—32no—
50C2×C32×C8—31no—
51#3748C2×D312yes—
52C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
53#3743—62no—
54C8×C3⋊D3—32no—
55C8×C3⋊D3—32no—
56C8×C3⋊D3—62no—
57C3×#208—22no—
58C3×#208—32no—
59C3×#208—32no—
60C3×#208—62no—
61C3×#208C2×D312yes—
62C3×#208C2×D312yes—
63C2×C3×C3⋊C8—21no—
64C2×C3×C3⋊C8—32no—
65C2×C3×C3⋊C8—61no—
66C2×C3×C3⋊C8C2×D312yes—
67#3712—62no—
68#3712—62no—
69#3712—64no—
70#3710—34no—
71#3710—34no—
72#3710—62no—
73#3710—62no—
74#3710—62no—
75#3710—62no—
76#3710—64no—
77C22×C33C2×Q811yescharacteristic
78C2×C3⋊(C3⋊D3)—22no—
79C2×C3⋊(C3⋊D3)C4⋊C411yescharacteristic
80C2×C3⋊(C3⋊D3)C4⋊C411yescharacteristic
81C33×C4C22×C411yescharacteristic
82C33×C4C2×D411yescharacteristic
83C3⋊(C3⋊Dic3)C4⋊C411yescharacteristic
84C3⋊(C3⋊Dic3)C4⋊C411yescharacteristic
85D3×M16—92no—
86D3×M16—92no—
87D3×M16—92no—
88D3×M16—182no—
89C2×D3×C8—92no—
90C2×D3×C8—91no—
91C2×D3×C8—181no—
92C3×#108—92no—
93C3×#108—91no—
94#822—32no—
95#822—32no—
96#822—92no—
97#819—32no—
98#819—91no—
99C22×C3⋊D3—31no—
100C22×C3⋊D3—32no—
101C22×C3⋊D3—61no—
102C22×C3⋊D3—62no—
103C22×C3⋊D3—121no—
104C2×C32×C4—21no—
105C2×C32×C4—22no—
106C2×C32×C4—41no—
107C2×C32×C4C22×D312yes—
108C2×C32×C4C22×D311yescharacteristic
109C2×C3⋊Dic3—32no—
110C2×C3⋊Dic3—31no—
111C2×C3⋊Dic3—61no—
112C2×C3⋊Dic3—62no—
113C2×C3⋊Dic3—121no—
114C4×C3⋊D3—32no—
115C4×C3⋊D3—32no—
116C4×C3⋊D3—31no—
117C4×C3⋊D3—31no—
118C4×C3⋊D3—31no—
119C4×C3⋊D3—32no—
120C4×C3⋊D3—31no—
121C4×C3⋊D3—32no—
122C4×C3⋊D3—61no—
123C4×C3⋊D3—61no—
124C4×C3⋊D3—62no—
125C4×C3⋊D3—62no—
126C4×C3⋊D3—62no—
127C4×C3⋊D3—62no—
128C4×C3⋊D3—61no—
129C4×C3⋊D3—61no—
130C4×C3⋊D3—121no—
131C4×C3⋊D3—121no—
132C4×C3⋊D3—121no—
133C4×C3⋊D3—121no—
134C32×C8—32no—
135C32×C8—62no—
136C3⋊(C3⋊C8)—22no—
137C3⋊(C3⋊C8)C4×D312yes—
138C3×C3⋊C8—24no—
139C3×C3⋊C8—22no—
140C3×C3⋊C8—22no—
141C3×C3⋊C8—22no—
142C3×C3⋊C8—34no—
143C3×C3⋊C8—62no—
144C3×C3⋊C8—64no—
145C3×C3⋊C8—62no—
146C3×C3⋊C8—62no—
147C3×C3⋊C8C4×D314yes—
148C2×#108—271noSylow
149C2×C33—21no—
150C2×C33C2×C4⋊C411yescharacteristic
151C3⋊(C3⋊D3)—21no—
152C3⋊(C3⋊D3)#10812yescomplemented
153C2×C4×D3—92no—
154C2×C4×D3—91no—
155C2×C4×D3—181no—
156C2×C4×D3—182no—
157C2×C4×D3—361no—
158C3×M16—32no—
159C3×M16—32no—
160C3×M16—62no—
161C3×M16—92no—
162C2×C3×C8—32no—
163C2×C3×C8—61no—
164C2×C3×C8—91no—
165#208—32no—
166#208—32no—
167#208—62no—
168#208D3212yes—
169C2×C3⋊C8—32no—
170C2×C3⋊C8—61no—
171C2×C3⋊C8D3211yescharacteristic
172ΓL(1,𝔽25)—182no—
173ΓL(1,𝔽25)—182no—
174ΓL(1,𝔽25)—182no—
175ΓL(1,𝔽25)—184no—
176D3×C8—94no—
177D3×C8—94no—
178D3×C8—92no—
179D3×C8—92no—
180D3×C8—182no—
181D3×C8—182no—
182D3×C8—182no—
183D3×C8—182no—
184D3×C8—182no—
185D3×C8—184no—
186C22×C32—21no—
187C22×C32—22no—
188C22×C32—41no—
189C22×C32D3×Q811yescharacteristic
190C22×C32D3×Q812yes—
191C2×C3⋊D3—31no—
192C2×C3⋊D3—32no—
193C2×C3⋊D3—31no—
194C2×C3⋊D3—32no—
195C2×C3⋊D3—62no—
196C2×C3⋊D3—61no—
197C2×C3⋊D3—61no—
198C2×C3⋊D3—62no—
199C2×C3⋊D3—62no—
200C2×C3⋊D3—64no—
201C2×C3⋊D3—64no—
202C2×C3⋊D3—121no—
203C2×C3⋊D3—121no—
204C2×C3⋊D3—124no—
205C2×C3⋊D3—124no—
206C32×C4—22no—
207C32×C4—21no—
208C32×C4—21no—
209C32×C4—22no—
210C32×C4—41no—
211C32×C4—41no—
212C32×C4C2×C4×D311yescharacteristic
213C32×C4C2×C4×D312yes—
214C32×C4D3×D412yes—
215C32×C4D3×D411yescharacteristic
216C3⋊Dic3—31no—
217C3⋊Dic3—31no—
218C3⋊Dic3—32no—
219C3⋊Dic3—32no—
220C3⋊Dic3—62no—
221C3⋊Dic3—62no—
222C3⋊Dic3—61no—
223C3⋊Dic3—61no—
224C3⋊Dic3—121no—
225C3⋊Dic3—121no—
226C2×M16—272no—
227C22×C8—271no—
228#108—92no—
229#108—91no—
230#108—271no—
231C33C2×#10811yescharacteristic, complemented, Sylow
232C22×D3—92no—
233C22×D3—91no—
234C22×D3—181no—
235C22×D3—182no—
236C22×D3—361no—
237C2×C3×C4—21no—
238C2×C3×C4—22no—
239C2×C3×C4—41no—
240C2×C3×C4C2×D3212yes—
241C2×C3×C4C2×D3211yescharacteristic
242C2×Dic3—92no—
243C2×Dic3—91no—
244C2×Dic3—181no—
245C2×Dic3—182no—
246C2×Dic3—361no—
247C4×D3—92no—
248C4×D3—92no—
249C4×D3—91no—
250C4×D3—91no—
251C4×D3—91no—
252C4×D3—91no—
253C4×D3—92no—
254C4×D3—92no—
255C4×D3—181no—
256C4×D3—181no—
257C4×D3—182no—
258C4×D3—182no—
259C4×D3—182no—
260C4×D3—182no—
261C4×D3—181no—
262C4×D3—181no—
263C4×D3—361no—
264C4×D3—361no—
265C4×D3—361no—
266C4×D3—361no—
267C3×C8—34no—
268C3×C8—62no—
269C3×C8—64no—
270C3×C8—62no—
271C3×C8—62no—
272C3×C8—92no—
273C3×C8—182no—
274C3⋊C8—22no—
275C3⋊C8—34no—
276C3⋊C8—62no—
277C3⋊C8—64no—
278C3⋊C8—62no—
279C3⋊C8—62no—
280C3⋊C8#62412yes—
281C2×C32—21no—
282C2×C32—22no—
283C2×C32—22no—
284C2×C32—22no—
285C2×C32—21no—
286C2×C32—41no—
287C2×C32—42no—
288C2×C32—42no—
289C2×C32D3×C4⋊C411yescharacteristic
290C2×C32D3×C4⋊C412yes—
291C3⋊D3—34no—
292C3⋊D3—32no—
293C3⋊D3—62no—
294C3⋊D3—64no—
295C3⋊D3—61no—
296C3⋊D3—62no—
297C3⋊D3—62no—
298C3⋊D3—122no—
299C3⋊D3—122no—
300C3⋊D3—122no—
301C22×C4—271no—
302M16—32no—
303M16—92no—
304M16—542no—
305C2×C8—31no—
306C2×C8—91no—
307C2×C8—272no—
308C2×C8—272no—
309C2×C8—542no—
310C22×C3—21no—
311C22×C3—22no—
312C22×C3—41no—
313C22×C3#379712yes—
314C22×C3#379711yescharacteristic
315C2×D3—92no—
316C2×D3—91no—
317C2×D3—92no—
318C2×D3—91no—
319C2×D3—181no—
320C2×D3—182no—
321C2×D3—181no—
322C2×D3—182no—
323C2×D3—182no—
324C2×D3—184no—
325C2×D3—184no—
326C2×D3—361no—
327C2×D3—361no—
328C2×D3—364no—
329C2×D3—364no—
330C3×C4—21no—
331C3×C4—22no—
332C3×C4—22no—
333C3×C4—21no—
334C3×C4—41no—
335C3×C4—41no—
336C3×C4C4×D3212yes—
337C3×C4#380211yescharacteristic
338C3×C4C2×#62411yescharacteristic
339C3×C4#381112yes—
340Dic3—92no—
341Dic3—92no—
342Dic3—91no—
343Dic3—91no—
344Dic3—181no—
345Dic3—182no—
346Dic3—181no—
347Dic3—182no—
348Dic3—361no—
349Dic3—361no—
350C32—21no—
351C32—22no—
352C32—41no—
353C32D3×#10812yescomplemented
354C32D3×#10811yescharacteristic, complemented
355C23—271no—
356C2×C4—271no—
357C2×C4—271no—
358C2×C4—271no—
359C2×C4—271no—
360C2×C4—271no—
361C2×C4D3311yescharacteristic
362C8—32no—
363C8—62no—
364C8—92no—
365C8—182no—
366C2×C3—22no—
367C2×C3—21no—
368C2×C3—22no—
369C2×C3—21no—
370C2×C3—22no—
371C2×C3—42no—
372C2×C3—42no—
373C2×C3—41no—
374C2×C3#6388312yes—
375C2×C3#6390011yescharacteristic
376D3—92no—
377D3—94no—
378D3—182no—
379D3—184no—
380D3—182no—
381D3—181no—
382D3—182no—
383D3—362no—
384D3—362no—
385D3—362no—
386C22—271no—
387C22—271no—
388C22—542no—
389C22#8895811yescharacteristic
390C4—271no—
391C4—271no—
392C4D3×#62411yescharacteristic
393C4#8895411yescharacteristic
394C3—21no—
395C3—22no—
396C3—41no—
397C3#1059091311yescharacteristic, complemented
398C3#1059097212yescomplemented
399C2—21no—
400C2—272no—
401C2—541no—
402C2#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
hypocenterC33C2×#108
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
28. #63578D3semidirect
30. #63577D3semidirect
152. C3⋊(C3⋊D3)#108semidirect
231. C33C2×#108semidirect
353. C32D3×#108semidirect
354. C32D3×#108semidirect
397. C3#10590913semidirect
398. C3#10590972semidirect

Maximal subgroups

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

subgroupindex
#117233582
C2×#887852
#117220812
C3×#635782
C3×#635772
#105909723
#105909133