Finite Group Databasebeta

← #13167506

Subgroups — #13167506

Order 1728.

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

normal subgroups only

This group has 766 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
1C2×C4⋊#1113C211yesmaximal, characteristic
2C2×#88942C212yesmaximal, complemented
3#11723671C211yesmaximal, characteristic, complemented
4C3×#63778C211yesmaximal, characteristic, complemented
5#11722793C212yesmaximal, complemented
6#10593354—31nomaximal
7C22×#1113—21no—
8C2×C3×C4⋊(C3⋊D3)—21no—
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C2×#6296C2211yescharacteristic, complemented
11C2×#6295C2212yes—
12C4⋊#1113—24no—
13C2×C3×#622C2212yescomplemented
14#88942—24no—
15C3×#3750—21no—
16#88815C2211yescharacteristic, complemented
17#88814—21no—
18C2×#3802—31no—
19C2×D3×D12—61no—
20C2×#3775—32no—
21#63865—31no—
22#63851—61no—
23#63778D311yescharacteristic, complemented
24#63739—32no—
25C22×C3×C3⋊D3—22no—
26C22×C3×C3⋊D3D411yescharacteristic
27C2×#1113—24no—
28C2×#1113—42no—
29C2×#1113—42no—
30C2×C33×C4D411yescharacteristic
31C2×C3×C3⋊Dic3—21no—
32C2×C3×C3⋊Dic3—21no—
33C2×C3×C3⋊Dic3C2311yescharacteristic
34C3×C4⋊(C3⋊D3)—24no—
35C3×C4×C3⋊D3—22no—
36#6296—22no—
37#6295—22no—
38#6295D414yes—
39C3×#622—22no—
40#4731—91nomaximal
41C22×D32—62no—
42C22×D32—61no—
43C2×C4⋊(C3⋊D3)—21no—
44C2×C4×C3⋊D3C2×D311yescharacteristic
45C2×C3×D12—61no—
46C2×C3×D12—61no—
47C2×C3×D12—121no—
48C2×C3×C4×D3—61no—
49C2×C3×C4×D3—61no—
50C2×#627—31no—
51C2×#627—61no—
52C2×#626—62no—
53C2×#625—32no—
54#3802—64no—
55D3×D12—68no—
56C2×#622C2×D312yescomplemented
57#3775—64no—
58#3750—21no—
59C3×Dic3⋊C4—61no—
60C3×Dic3⋊C4—61no—
61C3×Dic3⋊C4—121no—
62#3724—31no—
63Dic3⋊Dic3—61no—
64#3722—61no—
65(C2×D3)⋊Dic3—61no—
66(C2×D3)⋊Dic3—61no—
67C22×C33C2×D411yescharacteristic
68C2×C3×C3⋊D3—21no—
69C2×C3×C3⋊D3—21no—
70C2×C3×C3⋊D3—24no—
71C2×C3×C3⋊D3—42no—
72C2×C3×C3⋊D3—42no—
73C2×C3×C3⋊D3SD1612yescomplemented
74#1113—48no—
75C33×C4—21no—
76C3×C3⋊Dic3—41no—
77C3×C3⋊Dic3—41no—
78C3×C3⋊Dic3C2×D412yes—
79C2×D3×D4—181no—
80C22×D12—91no—
81#924—91no—
82C2×#204—92no—
83#896—181no—
84C3×C22⋊C8—91no—
85#821—92no—
86C22×C3⋊D3—22no—
87C22×C3⋊D3D1211yescharacteristic
88C22×C3×D3—61no—
89C22×C3×D3—61no—
90C22×C3×D3—62no—
91C22×C3×D3—62no—
92C22×C3×D3—122no—
93C2×D32—68no—
94C2×D32—64no—
95C2×D32—124no—
96C2×D32—124no—
97C2×D32—122no—
98C2×D32—122no—
99C2×C32×C4—21no—
100C2×C32×C4—21no—
101C2×C32×C4—41no—
102C2×C32×C4—41no—
103C2×C32×C4D1211yescharacteristic
104C2×C3⋊Dic3—21no—
105C2×C3⋊Dic3—21no—
106C2×C3⋊Dic3C22×D311yescharacteristic
107C4⋊(C3⋊D3)—24no—
108C4×C3⋊D3—22no—
109C2×C3×Dic3—61no—
110C2×C3×Dic3—61no—
111C2×C3×Dic3—61no—
112C2×C3×Dic3—61no—
113C2×C3×Dic3—61no—
114C2×C3×Dic3—61no—
115C2×C3×Dic3—121no—
116C2×C3×Dic3—121no—
117C3×D12—64no—
118C3×D12—64no—
119C3×D12—124no—
120C3×C4×D3—64no—
121C3×C4×D3—122no—
122#627—62no—
123#627—122no—
124#626—64no—
125#626—64no—
126#625—34no—
127#625—62no—
128#622—22no—
129#450—271noSylow
130C2×C33#12011yescharacteristic
131C2×C33C2×SD1611yescharacteristic
132C2×C33AGL(1,ℤ/8)11yescharacteristic
133C3×C3⋊D3—22no—
134C3×C3⋊D3—44no—
135C23×D3—181no—
136C23×D3—182no—
137C2×C3×D4—181no—
138C22×C3×C4—91no—
139C2×#67—182no—
140D3×D4—188no—
141C2×D12—61no—
142C2×D12—61no—
143C2×D12—92no—
144C2×D12—121no—
145C2×D12—184no—
146C2×C4×D3—61no—
147C2×C4×D3—61no—
148C2×Dic6—91no—
149C2×Dic6—181no—
150C2×C3×C8—92no—
151C3×C4⋊C4—181no—
152#217—181no—
153(C2×D3)⋊C4—181no—
154(C2×D3)⋊C4—181no—
155C4⋊Dic3—91no—
156Dic3⋊C4—61no—
157Dic3⋊C4—61no—
158Dic3⋊C4—121no—
159Dic3⋊C4—181no—
160#204—184no—
161C22×C32—21no—
162C22×C32—21no—
163C22×C32—41no—
164C22×C32—41no—
165C22×C32C2×D1211yescharacteristic
166C2×C3⋊D3—21no—
167C2×C3⋊D3—21no—
168C2×C3⋊D3—24no—
169C2×C3⋊D3—42no—
170C2×C3⋊D3—42no—
171C2×C3⋊D3#20412yescomplemented
172C2×C3×D3—62no—
173C2×C3×D3—62no—
174C2×C3×D3—62no—
175C2×C3×D3—62no—
176C2×C3×D3—64no—
177C2×C3×D3—64no—
178C2×C3×D3—121no—
179C2×C3×D3—121no—
180C2×C3×D3—124no—
181C2×C3×D3—122no—
182C2×C3×D3—122no—
183C2×C3×D3—122no—
184C2×C3×D3—122no—
185C2×C3×D3—242no—
186C2×C3×D3—242no—
187D32—1216no—
188D32—128no—
189C32×C4—22no—
190C32×C4—21no—
191C32×C4—41no—
192C32×C4—42no—
193C32×C4—81no—
194C3⋊Dic3—41no—
195C3⋊Dic3—41no—
196C3⋊Dic3D3×D412yes—
197C3×Dic3—62no—
198C3×Dic3—62no—
199C3×Dic3—121no—
200C3×Dic3—121no—
201C3×Dic3—122no—
202C3×Dic3—121no—
203C3×Dic3—121no—
204C3×Dic3—241no—
205C22×D4—271no—
206C2×SD16—272no—
207C22⋊Q8—271no—
208#102—272no—
209C22⋊C8—91no—
210C33#45011yescharacteristic, complemented, Sylow
211C23×C3—91no—
212C23×C3—182no—
213C22×D3—61no—
214C22×D3—61no—
215C22×D3—62no—
216C22×D3—62no—
217C22×D3—122no—
218C22×D3—184no—
219C22×D3—188no—
220C22×D3—364no—
221C22×D3—364no—
222C22×D3—362no—
223C22×D3—362no—
224C3×D4—184no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—91no—
230C2×C3×C4—182no—
231C2×C3×C4—181no—
232C2×C3×C4—181no—
233C2×C3×C4GO+(4,𝔽2)11yescharacteristic
234#67—184no—
235#67—184no—
236C2×Dic3—61no—
237C2×Dic3—61no—
238C2×Dic3—61no—
239C2×Dic3—61no—
240C2×Dic3—61no—
241C2×Dic3—61no—
242C2×Dic3—121no—
243C2×Dic3—121no—
244D12—64no—
245D12—64no—
246D12—94no—
247D12—124no—
248D12—182no—
249C4×D3—64no—
250C4×D3—122no—
251Dic6—182no—
252Dic6—362no—
253C3×C8—182no—
254C2×C32—21no—
255C2×C32—21no—
256C2×C32—21no—
257C2×C32—21no—
258C2×C32—21no—
259C2×C32—21no—
260C2×C32—41no—
261C2×C32—41no—
262C2×C32—41no—
263C2×C32—41no—
264C2×C32—41no—
265C2×C32—41no—
266C2×C32#88211yescharacteristic
267C2×C32C2×#20411yescharacteristic
268C2×C32#90811yescharacteristic
269C3⋊D3—22no—
270C3⋊D3—44no—
271C3×D3—64no—
272C3×D3—122no—
273C3×D3—124no—
274C3×D3—124no—
275C3×D3—244no—
276C24—541no—
277C2×Q8—271no—
278C2×D4—181no—
279C2×D4—272no—
280C2×D4—544no—
281C22×C4—91no—
282SD16—544no—
283C2×C8—92no—
284C4⋊C4—181no—
285C4⋊C4—271no—
286C22⋊C4—541no—
287C22×C3—21no—
288C22×C3—21no—
289C22×C3—41no—
290C22×C3—41no—
291C22×C3—92no—
292C22×C3—184no—
293C22×C3—181no—
294C22×C3—181no—
295C22×C3—362no—
296C22×C3—362no—
297C22×C3C2×GO+(4,𝔽2)11yescharacteristic
298C2×D3—62no—
299C2×D3—62no—
300C2×D3—62no—
301C2×D3—62no—
302C2×D3—64no—
303C2×D3—64no—
304C2×D3—121no—
305C2×D3—121no—
306C2×D3—124no—
307C2×D3—122no—
308C2×D3—122no—
309C2×D3—122no—
310C2×D3—122no—
311C2×D3—242no—
312C2×D3—242no—
313C2×D3—3616no—
314C2×D3—368no—
315C3×C4—22no—
316C3×C4—21no—
317C3×C4—41no—
318C3×C4—42no—
319C3×C4—81no—
320C3×C4—92no—
321C3×C4—361no—
322C3×C4—361no—
323Dic3—62no—
324Dic3—62no—
325Dic3—121no—
326Dic3—121no—
327Dic3—122no—
328Dic3—121no—
329Dic3—121no—
330Dic3—241no—
331C32—21no—
332C32—21no—
333C32—41no—
334C32—41no—
335C32#473111yescharacteristic, complemented
336C23—91no—
337C23—182no—
338C23—544no—
339C23—1082no—
340C23—1082no—
341Q8—542no—
342D4—184no—
343D4—274no—
344D4—542no—
345C2×C4—91no—
346C2×C4—182no—
347C2×C4—181no—
348C2×C4—181no—
349C2×C4#157⋊D311yescharacteristic
350C8—182no—
351C2×C3—21no—
352C2×C3—21no—
353C2×C3—21no—
354C2×C3—21no—
355C2×C3—21no—
356C2×C3—21no—
357C2×C3—41no—
358C2×C3—41no—
359C2×C3—41no—
360C2×C3—41no—
361C2×C3—41no—
362C2×C3—41no—
363C2×C3—182no—
364C2×C3—364no—
365C2×C3#6422411yescharacteristic
366C2×C3#6422711yescharacteristic
367C2×C3#6424111yescharacteristic
368D3—64no—
369D3—122no—
370D3—124no—
371D3—124no—
372D3—244no—
373C22—92no—
374C22—181no—
375C22—181no—
376C22—184no—
377C22—362no—
378C22—362no—
379C22—1088no—
380C22C2×#157⋊D311yescharacteristic
381C4—21no—
382C4—92no—
383C4—361no—
384C4—361no—
385C3—21no—
386C3—21no—
387C3—41no—
388C3—41no—
389C3#1059335411yescharacteristic, complemented
390C2—182no—
391C2—364no—
392C2#1172433911yescharacteristic
393C2#1172434011yescharacteristic
394C2#1172435411yescharacteristic
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutator, cosocleC2×C3×C3⋊Dic3C23
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC2×C4⋊#1113C2
hypercenterC2×C4#157⋊D3
hypocenterC33#450
LCSG ▷ C2×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×#88942C2semidirect
3. #11723671C2semidirect
4. C3×#63778C2semidirect
5. #11722793C2semidirect
10. C2×#6296C22semidirect
13. C2×C3×#622C22semidirect
16. #88815C22semidirect
23. #63778D3semidirect
56. C2×#622C2×D3semidirect
73. C2×C3×C3⋊D3SD16semidirect
171. C2×C3⋊D3#204semidirect
210. C33#450semidirect
335. C32#4731semidirect
389. C3#10593354semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊#11132
C2×#889422
#117236712
C3×#637782
#117227932
#105933543
#47319