Finite Group Databasebeta

← #13156306

Subgroups — #13156306

Order 1728.

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

normal subgroups only

This group has 680 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
1#11723494C211yesmaximal, characteristic
2C3×#63919C211yesmaximal, characteristic, complemented
3#11722078C211yesmaximal, characteristic, complemented
4#11722035C212yesmaximal
5C3×#63574C212yesmaximal, complemented
6#10590466—32nomaximal
7#10590433—31nomaximal
8C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
9C2×#6289—21no—
10C2×C3×#627—21no—
11C3×C4×C3⋊Dic3C2211yescharacteristic
12C2×C3×C3⋊(C3⋊C8)C2212yes—
13C32×C4⋊Dic3C2211yescharacteristic
14C32×C4⋊Dic3C2212yes—
15#88804—21no—
16C3×#3724—21no—
17#64102—31no—
18C3×#888—32no—
19#63919D311yescharacteristic, complemented
20#63894—32no—
21#63894—61no—
22#63636—32no—
23C3×#808—32no—
24C3×#808—32no—
25#63574D312yescomplemented
26#63572—31no—
27Dic3⋊(C3⋊C8)—32no—
28#63564—32no—
29#63564—32no—
30#63564—62no—
31C22×C3⋊(C3⋊D3)—21no—
32C2×C33×C4C2311yescharacteristic
33C4⋊(C3⋊(C3⋊D3))—21no—
34C4⋊(C3⋊(C3⋊D3))Q812yescomplemented
35C2×C3×C3⋊Dic3D411yescharacteristic
36C2×C3×C3⋊Dic3D411yescharacteristic
37C2×C32×Dic3—21no—
38C2×C32×Dic3—22no—
39#6289—42no—
40C3×#627—42no—
41C3×C3⋊(C3⋊C8)—22no—
42#5084—91no—
43#4870—92no—
44C2×C4⋊(C3⋊D3)—31no—
45C2×C4⋊(C3⋊D3)—32no—
46C2×C4⋊(C3⋊D3)—61no—
47C2×C4⋊(C3⋊D3)—62no—
48C2×C4⋊(C3⋊D3)—121no—
49C2×C4×C3⋊D3—61no—
50C2×C3×Dic6—62no—
51C2×#627—21no—
52C2×#624—62no—
53C2×#624—62no—
54C32×C4⋊C4—32no—
55C32×C4⋊C4—31no—
56#3752—61no—
57C4×C3⋊Dic3C2×D311yescharacteristic
58C2×C3⋊(C3⋊C8)C2×D312yes—
59C3×C4⋊Dic3—21no—
60C3×C4⋊Dic3—22no—
61C3×C4⋊Dic3C2×D312yes—
62C3×C4⋊Dic3C2×D312yes—
63C3×C4⋊Dic3C2×D312yes—
64C3×Dic3⋊C4—62no—
65C3×C4×Dic3—32no—
66C3×C4×Dic3—61no—
67C2×C3×C3⋊C8—32no—
68C2×C3×C3⋊C8—32no—
69C2×C3×C3⋊C8—62no—
70#3724—21no—
71#3722—62no—
72#3722—62no—
73C22×C33C2×D411yescharacteristic
74C2×C3⋊(C3⋊D3)—22no—
75C2×C3⋊(C3⋊D3)—41no—
76C2×C3⋊(C3⋊D3)—41no—
77C33×C4C2×Q811yescharacteristic
78C33×C4#4111yescharacteristic
79C3×C3⋊Dic3—21no—
80C3×C3⋊Dic3SD1612yes—
81C32×Dic3—41no—
82C32×Dic3—42no—
83C3×C4⋊Q8—91no—
84#893—92no—
85#893—181no—
86#888—32no—
87C4×D12—91no—
88C3×#106—92no—
89#821—92no—
90Dic3⋊C8—92no—
91#809—92no—
92#809—92no—
93#809—182no—
94#808—32no—
95#808—32no—
96C4⋊(C3⋊C8)—91no—
97C22×C3⋊D3—61no—
98C22×C3⋊D3—62no—
99C22×C3⋊D3—62no—
100C22×C3⋊D3—122no—
101C22×C3⋊D3—122no—
102C2×C32×C4—22no—
103C2×C32×C4—21no—
104C2×C32×C4—41no—
105C2×C32×C4—61no—
106C2×C32×C4—62no—
107C2×C32×C4C22×D312yes—
108C2×C32×C4C22×D311yescharacteristic
109C2×C3⋊Dic3#6711yescharacteristic
110C2×C3⋊Dic3#6711yescharacteristic
111C4⋊(C3⋊D3)—32no—
112C4⋊(C3⋊D3)—34no—
113C4⋊(C3⋊D3)—62no—
114C4⋊(C3⋊D3)—64no—
115C4⋊(C3⋊D3)—61no—
116C4⋊(C3⋊D3)—62no—
117C4⋊(C3⋊D3)—62no—
118C4⋊(C3⋊D3)—122no—
119C4⋊(C3⋊D3)—122no—
120C4⋊(C3⋊D3)—122no—
121C4×C3⋊D3—122no—
122C2×C3×Dic3—22no—
123C2×C3×Dic3—22no—
124C2×C3×Dic3—22no—
125C2×C3×Dic3—22no—
126C2×C3×Dic3—32no—
127C2×C3×Dic3—32no—
128C2×C3×Dic3—42no—
129C2×C3×Dic3—61no—
130C2×C3×Dic3—61no—
131C3×Dic6—124no—
132#627—42no—
133#624—124no—
134#624—124no—
135C3⋊(C3⋊C8)—22no—
136C3×C3⋊C8—62no—
137C3×C3⋊C8—62no—
138C3×C3⋊C8—122no—
139#476—271noSylow
140C2×C33C22⋊Q811yescharacteristic
141C2×C33C2×SD1611yescharacteristic
142C2×C33AGL(1,ℤ/8)11yescharacteristic
143C3⋊(C3⋊D3)—42no—
144C2×C3×Q8—181no—
145C2×D12—91no—
146C2×D12—92no—
147C2×D12—181no—
148C2×D12—182no—
149C2×D12—361no—
150C2×C4×D3—181no—
151C2×C4×D3—182no—
152C2×C4×D3—182no—
153C2×Dic6—62no—
154C2×C3×C8—92no—
155C3×C4⋊C4—32no—
156C3×C4⋊C4—32no—
157C3×C4⋊C4—32no—
158C3×C4⋊C4—61no—
159C3×C4⋊C4—62no—
160C3×C4⋊C4—181no—
161C3×C42—91no—
162(C2×D3)⋊C4—181no—
163(C2×D3)⋊C4—182no—
164(C2×D3)⋊C4—182no—
165C4⋊Dic3D3211yescharacteristic
166C4⋊Dic3D3212yes—
167Dic3⋊C4—62no—
168C4×Dic3—32no—
169C4×Dic3—61no—
170C2×C3⋊C8—32no—
171C2×C3⋊C8—32no—
172C2×C3⋊C8—62no—
173C22×C32—22no—
174C22×C32—21no—
175C22×C32—41no—
176C22×C32D3×D412yes—
177C22×C32C2×#6711yescharacteristic
178C2×C3⋊D3—62no—
179C2×C3⋊D3—64no—
180C2×C3⋊D3—64no—
181C2×C3⋊D3—124no—
182C2×C3⋊D3—124no—
183C2×C3⋊D3—121no—
184C2×C3⋊D3—121no—
185C2×C3⋊D3—122no—
186C2×C3⋊D3—122no—
187C2×C3⋊D3—122no—
188C2×C3⋊D3—122no—
189C2×C3⋊D3—242no—
190C2×C3⋊D3—242no—
191C2×C3⋊D3—242no—
192C2×C3⋊D3—242no—
193C32×C4—21no—
194C32×C4—22no—
195C32×C4—21no—
196C32×C4—22no—
197C32×C4—41no—
198C32×C4—41no—
199C32×C4—121no—
200C32×C4—122no—
201C32×C4#23512yes—
202C32×C4D3×Q812yes—
203C32×C4D3×Q811yescharacteristic
204C32×C4#23911yescharacteristic
205C3⋊Dic3—21no—
206C3⋊Dic3#21512yes—
207C3×Dic3—34no—
208C3×Dic3—42no—
209C3×Dic3—42no—
210C3×Dic3—42no—
211C3×Dic3—42no—
212C3×Dic3—62no—
213C3×Dic3—62no—
214C3×Dic3—82no—
215C3×Dic3—121no—
216C4⋊Q8—91no—
217C4×D4—271no—
218#106—92no—
219C4⋊C8—271no—
220#102—272no—
221C33#47611yescharacteristic, complemented, Sylow
222C22×D3—182no—
223C22×D3—182no—
224C22×D3—181no—
225C22×D3—362no—
226C22×D3—362no—
227C3×Q8—362no—
228C2×C3×C4—22no—
229C2×C3×C4—21no—
230C2×C3×C4—41no—
231C2×C3×C4—62no—
232C2×C3×C4—62no—
233C2×C3×C4—62no—
234C2×C3×C4—62no—
235C2×C3×C4—91no—
236C2×C3×C4—91no—
237C2×C3×C4—122no—
238C2×C3×C4C2×D3211yescharacteristic
239C2×C3×C4C2×D3212yes—
240C2×Dic3—22no—
241C2×Dic3—21no—
242C2×Dic3—32no—
243C2×Dic3—32no—
244C2×Dic3—61no—
245C2×Dic3—61no—
246D12—92no—
247D12—94no—
248D12—182no—
249D12—184no—
250D12—182no—
251D12—181no—
252D12—182no—
253D12—362no—
254D12—362no—
255D12—362no—
256C4×D3—364no—
257C4×D3—362no—
258C4×D3—364no—
259Dic6—124no—
260C3×C8—182no—
261C3⋊C8—62no—
262C3⋊C8—62no—
263C3⋊C8—122no—
264C2×C32—21no—
265C2×C32—22no—
266C2×C32—21no—
267C2×C32—22no—
268C2×C32—22no—
269C2×C32—21no—
270C2×C32—41no—
271C2×C32—41no—
272C2×C32—41no—
273C2×C32#89612yes—
274C2×C32D3×SD1612yes—
275C2×C32#91412yes—
276C2×C32#93211yescharacteristic
277C2×C32C2×#21511yescharacteristic
278C2×C32#94611yescharacteristic
279C3⋊D3—122no—
280C3⋊D3—124no—
281C3⋊D3—124no—
282C3⋊D3—244no—
283C3⋊D3—244no—
284C2×Q8—181no—
285C2×D4—271no—
286C22×C4—541no—
287C2×C8—92no—
288C4⋊C4—32no—
289C4⋊C4—31no—
290C4⋊C4—181no—
291C22⋊C4—541no—
292C42—91no—
293C22×C3—21no—
294C22×C3—22no—
295C22×C3—41no—
296C22×C3D3×#6712yes—
297C22×C3#381111yescharacteristic
298C2×D3—184no—
299C2×D3—182no—
300C2×D3—184no—
301C2×D3—364no—
302C2×D3—364no—
303C2×D3—362no—
304C2×D3—362no—
305C2×D3—361no—
306C2×D3—362no—
307C2×D3—362no—
308C2×D3—361no—
309C2×D3—722no—
310C2×D3—722no—
311C2×D3—722no—
312C2×D3—722no—
313C3×C4—21no—
314C3×C4—22no—
315C3×C4—22no—
316C3×C4—21no—
317C3×C4—41no—
318C3×C4—41no—
319C3×C4—92no—
320C3×C4—122no—
321C3×C4—122no—
322C3×C4—122no—
323C3×C4—122no—
324C3×C4—181no—
325C3×C4—242no—
326C3×C4#379711yescharacteristic
327C3×C4#379712yes—
328C3×C4#379811yescharacteristic
329C3×C4#379912yes—
330Dic3—34no—
331Dic3—41no—
332Dic3—42no—
333Dic3—62no—
334Dic3—62no—
335Dic3—121no—
336C32—22no—
337C32—21no—
338C32—41no—
339C32#487012yescomplemented
340C32#508411yescharacteristic, complemented
341C23—541no—
342Q8—362no—
343D4—272no—
344D4—541no—
345C2×C4—61no—
346C2×C4—62no—
347C2×C4—91no—
348C2×C4—91no—
349C2×C4—1082no—
350C2×C4D3311yescharacteristic
351C8—182no—
352C2×C3—21no—
353C2×C3—22no—
354C2×C3—21no—
355C2×C3—22no—
356C2×C3—21no—
357C2×C3—22no—
358C2×C3—41no—
359C2×C3—41no—
360C2×C3—41no—
361C2×C3#6386511yescharacteristic
362C2×C3#6388812yes—
363C2×C3#6393011yescharacteristic
364C2×C3#6393412yes—
365C2×C3D3×#21512yes—
366C2×C3#6394111yescharacteristic
367D3—364no—
368D3—364no—
369D3—362no—
370D3—724no—
371D3—724no—
372C22—542no—
373C22—1081no—
374C22—1081no—
375C22#8895211yescharacteristic
376C4—92no—
377C4—121no—
378C4—122no—
379C4—181no—
380C4#8895811yescharacteristic
381C4#8896511yescharacteristic
382C3—21no—
383C3—22no—
384C3—41no—
385C3#1059043311yescharacteristic, complemented
386C3#1059046612yescomplemented
387C2—1082no—
388C2#1172283811yescharacteristic
389C2#1172287311yescharacteristic
390C2#1172297711yescharacteristic
subgroupG/H
centerC22#88952
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×C4⋊(C3⋊(C3⋊D3))C22
hypocenterC33#476
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C3×#63919C2semidirect
3. #11722078C2semidirect
5. C3×#63574C2semidirect
19. #63919D3semidirect
25. #63574D3semidirect
34. C4⋊(C3⋊(C3⋊D3))Q8semidirect
221. C33#476semidirect
339. C32#4870semidirect
340. C32#5084semidirect
385. C3#10590433semidirect
386. C3#10590466semidirect

Maximal subgroups

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

subgroupindex
#117234942
C3×#639192
#117220782
#117220352
C3×#635742
#105904663
#105904333