Finite Group Databasebeta

← #13167016

Subgroups — #13167016

Order 1728.

812 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 812 conjugacy classes of subgroups — too many to draw. Only its 103 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×C3×#3802C211yesmaximal, characteristic
2C2×#88988C211yesmaximal, characteristic, complemented
3#11723689C211yesmaximal, characteristic, complemented
4#11722773C214yesmaximal, complemented
5#10593365—31nomaximal
6C22×C3×D32—21no—
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C2×C32×D12—21no—
9C2×#6294—21no—
10C2×C3×#625C2212yes—
11C3×#3802—22no—
12C3×#3802C414yescomplemented
13C2×C3⋊#622C2212yescomplemented
14#88988—22no—
15C3×C4⋊(C3⋊Dic3)—21no—
16#88815C2212yescomplemented
17C2×C3×D3×D4—61no—
18C2×#3802D311yescharacteristic
19C2×#3787—31no—
20#63900—31no—
21D3×C4⋊Dic3—61no—
22#63739—34no—
23C22×C3×C3⋊D3D411yescharacteristic
24C22×C32×D3—22no—
25C2×C3×D32—24no—
26C2×C3×D32—42no—
27C2×C3×D32—42no—
28C2×C33×C4D411yescharacteristic
29C2×C3×C3⋊Dic3—22no—
30C2×C3×C3⋊Dic3C2311yescharacteristic
31C3×C4×C3⋊D3C2×C412yes—
32C3×C4×C3⋊D3D412yes—
33C32×D12—24no—
34#6294—42no—
35C3×#625—22no—
36C3×#625C2×C414yescomplemented
37C3⋊#622—22no—
38C2×#832—91nomaximal
39C23×C3×D3—62no—
40C22×D32—21no—
41C2×C32×D4—61no—
42C2×C4×C3⋊D3C2×D311yescharacteristic
43C2×C3×#67—62no—
44C3×D3×D4—64no—
45C3×D3×D4—64no—
46C2×C3×D12—21no—
47C2×C3×D12—21no—
48C2×C3×D12—41no—
49C2×C3×C4×D3—61no—
50C2×C3×C4×D3—61no—
51C2×#625C2×D312yes—
52C2×#624—61no—
53C2×D3×Dic3—62no—
54#3802—22no—
55#3802Dic314yescomplemented
56C2×#622—32no—
57#3787—62no—
58C4⋊(C3⋊Dic3)—21no—
59C3×C4⋊Dic3—61no—
60C3×C4⋊Dic3—61no—
61C3×C4⋊Dic3—121no—
62#3724—32no—
63Dic3⋊Dic3—62no—
64C22×C33C2×D411yescharacteristic
65C2×C3×C3⋊D3C22⋊C412yes—
66C2×C3×C3⋊D3D812yes—
67C2×C3×C3⋊D3SD1612yes—
68C2×C32×D3—24no—
69C2×C32×D3—42no—
70C2×C32×D3—42no—
71C3×D32—48no—
72C33×C4C22⋊C412yes—
73C3×C3⋊Dic3—42no—
74C3×C3⋊Dic3C22×C411yescharacteristic
75C3×C3⋊Dic3C2×D411yescharacteristic
76C22×C3×D4—91no—
77C2×D3×D4—61no—
78C2×C4⋊Dic3—91no—
79C22×C3⋊C8—91no—
80D3×C4⋊C4—181no—
81#832—94no—
82C23×C32—62no—
83C22×C3⋊D3#6711yescharacteristic
84C22×C3×D3—22no—
85C22×C3×D3—22no—
86C22×C3×D3—42no—
87C22×C3×D3—61no—
88C22×C3×D3—61no—
89C22×C3×D3—68no—
90C22×C3×D3—124no—
91C22×C3×D3—124no—
92C2×D32—24no—
93C2×D32—42no—
94C2×D32—42no—
95C32×D4—64no—
96C2×C32×C4—21no—
97C2×C32×C4—21no—
98C2×C32×C4—41no—
99C2×C32×C4—41no—
100C2×C32×C4#6711yescharacteristic
101C2×C3⋊Dic3—22no—
102C2×C3⋊Dic3C22×D311yescharacteristic
103C4×C3⋊D3C2×Dic312yes—
104C4×C3⋊D3#6712yes—
105C3×#67—64no—
106C3×#67—64no—
107C2×C3×Dic3—61no—
108C2×C3×Dic3—61no—
109C2×C3×Dic3—62no—
110C2×C3×Dic3—62no—
111C2×C3×Dic3—122no—
112C3×D12—24no—
113C3×D12—24no—
114C3×D12—44no—
115C3×C4×D3—62no—
116C3×C4×D3—62no—
117C3×C4×D3—62no—
118C3×C4×D3—62no—
119#625—22no—
120#625C2×Dic314yescomplemented
121#624—122no—
122D3×Dic3—124no—
123#622—62no—
124C2×#102—271noSylow
125C2×C33C2×C22⋊C411yescharacteristic
126C2×C33C2×D811yescharacteristic
127C2×C33C2×SD1611yescharacteristic
128C3×C3⋊D3#10214yescomplemented
129C32×D3—44no—
130C24×C3—181no—
131C23×D3—62no—
132C2×C3×D4—61no—
133C2×C3×D4—61no—
134C2×C3×D4—94no—
135C2×C3×D4—92no—
136C2×C3×D4—121no—
137C2×C3×D4—182no—
138C22×C3×C4—91no—
139C2×#67—62no—
140C22×Dic3—181no—
141D3×D4—64no—
142D3×D4—64no—
143C2×D12—21no—
144C2×C4×D3—61no—
145C2×C4×D3—61no—
146C2×C4×D3—182no—
147C3×C4⋊C4—181no—
148C4⋊Dic3—61no—
149C4⋊Dic3—61no—
150C4⋊Dic3—92no—
151C4⋊Dic3—121no—
152Dic3⋊C4—182no—
153C2×C3⋊C8—92no—
154C2×C3⋊C8—182no—
155C22×C32—21no—
156C22×C32—21no—
157C22×C32—41no—
158C22×C32—41no—
159C22×C32—64no—
160C22×C32—122no—
161C22×C32—122no—
162C22×C32C2×#6711yescharacteristic
163C2×C3⋊D3#21312yes—
164C2×C3⋊D3#21412yes—
165C2×C3⋊D3#21712yes—
166C2×C3×D3—24no—
167C2×C3×D3—24no—
168C2×C3×D3—44no—
169C2×C3×D3—42no—
170C2×C3×D3—42no—
171C2×C3×D3—42no—
172C2×C3×D3—42no—
173C2×C3×D3—62no—
174C2×C3×D3—62no—
175C2×C3×D3—62no—
176C2×C3×D3—62no—
177C2×C3×D3—62no—
178C2×C3×D3—62no—
179C2×C3×D3—82no—
180C2×C3×D3—82no—
181C2×C3×D3—1216no—
182D32—48no—
183C32×C4—22no—
184C32×C4—22no—
185C32×C4—42no—
186C32×C4—42no—
187C32×C4#21712yes—
188C3⋊Dic3—42no—
189C3⋊Dic3C22×Dic311yescharacteristic
190C3⋊Dic3C2×#6711yescharacteristic
191C3×Dic3—61no—
192C3×Dic3—61no—
193C3×Dic3—61no—
194C3×Dic3—61no—
195C3×Dic3—122no—
196C3×Dic3—122no—
197C3×Dic3—242no—
198C22×D4—91no—
199C22×C8—271no—
200C2×C4⋊C4—271no—
201#102—274no—
202C33C2×#10211yescharacteristic, complemented, Sylow
203C23×C3—62no—
204C23×C3—62no—
205C23×C3—91no—
206C23×C3—122no—
207C23×C3—184no—
208C23×C3—362no—
209C23×C3—362no—
210C22×D3—22no—
211C22×D3—61no—
212C22×D3—61no—
213C22×D3—68no—
214C22×D3—124no—
215C22×D3—124no—
216C3×D4—64no—
217C3×D4—64no—
218C3×D4—94no—
219C3×D4—124no—
220C3×D4—182no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—91no—
226C2×C3×C4—92no—
227C2×C3×C4—92no—
228C2×C3×C4—182no—
229C2×C3×C4GO+(4,𝔽2)11yescharacteristic
230#67—64no—
231#67—64no—
232C2×Dic3—61no—
233C2×Dic3—61no—
234C2×Dic3—62no—
235C2×Dic3—62no—
236C2×Dic3—122no—
237C2×Dic3—362no—
238D12—24no—
239C4×D3—62no—
240C4×D3—62no—
241C4×D3—62no—
242C4×D3—62no—
243C4×D3—364no—
244C3⋊C8—182no—
245C2×C32—21no—
246C2×C32—21no—
247C2×C32—21no—
248C2×C32—21no—
249C2×C32—21no—
250C2×C32—21no—
251C2×C32—41no—
252C2×C32—41no—
253C2×C32—41no—
254C2×C32—41no—
255C2×C32—41no—
256C2×C32—41no—
257C2×C32—124no—
258C2×C32C2×#21311yescharacteristic
259C2×C32C2×#21411yescharacteristic
260C2×C32C2×#21711yescharacteristic
261C3⋊D3#83214yescomplemented
262C3×D3—44no—
263C3×D3—44no—
264C3×D3—64no—
265C3×D3—64no—
266C3×D3—84no—
267C24—181no—
268C2×D4—61no—
269C2×D4—94no—
270C2×D4—92no—
271C2×D4—182no—
272C22×C4—91no—
273C22×C4—541no—
274C2×C8—272no—
275C2×C8—542no—
276C4⋊C4—181no—
277C4⋊C4—272no—
278C22×C3—21no—
279C22×C3—21no—
280C22×C3—41no—
281C22×C3—41no—
282C22×C3—64no—
283C22×C3—64no—
284C22×C3—92no—
285C22×C3—92no—
286C22×C3—92no—
287C22×C3—124no—
288C22×C3—122no—
289C22×C3—122no—
290C22×C3—122no—
291C22×C3—122no—
292C22×C3—242no—
293C22×C3—242no—
294C22×C3—368no—
295C22×C3C2×GO+(4,𝔽2)11yescharacteristic
296C2×D3—24no—
297C2×D3—42no—
298C2×D3—42no—
299C2×D3—62no—
300C2×D3—62no—
301C2×D3—62no—
302C2×D3—62no—
303C2×D3—62no—
304C2×D3—62no—
305C2×D3—1216no—
306C3×C4—22no—
307C3×C4—22no—
308C3×C4—42no—
309C3×C4—42no—
310C3×C4—91no—
311C3×C4—91no—
312C3×C4—362no—
313C3×C4D32⋊C412yes—
314Dic3—61no—
315Dic3—61no—
316Dic3—61no—
317Dic3—61no—
318Dic3—122no—
319Dic3—122no—
320Dic3—242no—
321C32—21no—
322C32—21no—
323C32—41no—
324C32—41no—
325C32C2×#83211yescharacteristic, complemented
326C23—62no—
327C23—91no—
328C23—184no—
329C23—362no—
330C23—362no—
331D4—64no—
332D4—94no—
333D4—182no—
334C2×C4—91no—
335C2×C4—92no—
336C2×C4—92no—
337C2×C4—182no—
338C2×C4—1082no—
339C2×C4D32⋊D311yescharacteristic
340C8—542no—
341C2×C3—21no—
342C2×C3—21no—
343C2×C3—21no—
344C2×C3—21no—
345C2×C3—21no—
346C2×C3—21no—
347C2×C3—41no—
348C2×C3—41no—
349C2×C3—41no—
350C2×C3—41no—
351C2×C3—41no—
352C2×C3—41no—
353C2×C3—94no—
354C2×C3—124no—
355C2×C3—124no—
356C2×C3—244no—
357C2×C3#6422511yescharacteristic
358C2×C3#6422711yescharacteristic
359C2×C3C2×D32⋊C411yescharacteristic
360D3—44no—
361D3—64no—
362D3—64no—
363C22—64no—
364C22—92no—
365C22—92no—
366C22—92no—
367C22—122no—
368C22—122no—
369C22—368no—
370C22C2×D32⋊D311yescharacteristic
371C4—91no—
372C4—91no—
373C4—362no—
374C4D32⋊Dic312yes—
375C3—21no—
376C3—21no—
377C3—41no—
378C3—41no—
379C3#1059336511yescharacteristic, complemented
380C2—94no—
381C2—124no—
382C2#1172430611yescharacteristic
383C2#1172431011yescharacteristic
384C2C2×D32⋊Dic311yescharacteristic
subgroupG/H
center, FrattiniC22C2×D32⋊D3
commutatorC3×C3⋊Dic3C22×C4
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC2×C3×#3802C2
cosocleC2×C3×C3⋊Dic3C23
hypercenterC2×C4D32⋊D3
hypocenterC33C2×#102
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×#88988C2semidirect
3. #11723689C2semidirect
4. #11722773C2semidirect
12. C3×#3802C4semidirect
13. C2×C3⋊#622C22semidirect
16. #88815C22semidirect
36. C3×#625C2×C4semidirect
55. #3802Dic3semidirect
120. #625C2×Dic3semidirect
128. C3×C3⋊D3#102semidirect
202. C33C2×#102semidirect
261. C3⋊D3#832semidirect
325. C32C2×#832semidirect
379. C3#10593365semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38022
C2×#889882
#117236892
#117227732
#105933653
C2×#8329