Finite Group Databasebeta

← #13161276

Subgroups — #13161276

Order 1728.

956 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 956 conjugacy classes of subgroups — too many to draw. Only its 154 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#11723987C211yesmaximal, characteristic
2C2×C32×#216C211yesmaximal, characteristic, complemented
3#11723456C211yesmaximal, characteristic
4C2×#88788C211yesmaximal, characteristic
5C3×#63662C211yesmaximal, characteristic, complemented
6#11722052C211yesmaximal, characteristic, complemented
7#11722038C211yesmaximal, characteristic, complemented
8#10592888—31nomaximal
9#10591600—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13#88859—21no—
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15#88857—21no—
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×#216—22no—
19C2×C32×C3⋊C8C2211yescharacteristic
20#88807—21no—
21#88804—21no—
22#88788—22no—
23C2×C32×Q16—31no—
24#64155—31no—
25#64155—34no—
26#64155—31no—
27#64155—64no—
28C2×#3743—31no—
29C2×C3×#216D314yescomplemented
30#63871—34no—
31C2×#3712—34no—
32#63662—31no—
33#63662D311yescharacteristic, complemented
34#63644—31no—
35C3×#835—34no—
36#63573—34no—
37#63566—34no—
38C22×C3⋊(C3⋊D3)D411yescharacteristic
39C33×Q8—21no—
40C33×Q8—21no—
41C2×C33×C4—21no—
42C2×C33×C4C2311yescharacteristic
43C2×C3⋊(C3⋊Dic3)—21no—
44C2×C3⋊(C3⋊Dic3)D411yescharacteristic
45C4×C3⋊(C3⋊D3)—21no—
46C4×C3⋊(C3⋊D3)—21no—
47C2×C3×C3⋊Dic3—21no—
48C2×C32×Dic3—21no—
49C32×Dic6—21no—
50C32×Dic6—21no—
51C3×C3⋊(C3⋊C8)—21no—
52C32×C3⋊C8D412yes—
53#5189—94no—
54#4816—91no—
55C2×C32×Q8—24no—
56C2×C32×Q8—31no—
57C2×C32×Q8C2×D314yes—
58C2×C32×Q8C2×D311yescharacteristic
59C2×C4×C3⋊D3—31no—
60C2×C4×C3⋊D3—34no—
61C2×C4×C3⋊D3—64no—
62C2×C3×Dic6C2×D314yes—
63C32×Q16—62no—
64C2×C32×C8—31no—
65#3752—61no—
66#3752—61no—
67#3752—64no—
68#3752—68no—
69C4⋊(C3⋊Dic3)—31no—
70C4⋊(C3⋊Dic3)—34no—
71C4⋊(C3⋊Dic3)—64no—
72C4⋊(C3⋊Dic3)C2×D311yescharacteristic
73#3750—61no—
74#3750—61no—
75#3750—64no—
76#3750—68no—
77C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
78#3743—62no—
79C3×#216—28no—
80C3×C4⋊Dic3—34no—
81C2×C3×C3⋊C8—34no—
82C2×C3×C3⋊C8C2×D314yes—
83Dic3⋊Dic3—64no—
84#3722—64no—
85#3712—68no—
86C22×C33C2×D411yescharacteristic
87C2×C3⋊(C3⋊D3)—21no—
88C2×C3⋊(C3⋊D3)—21no—
89C2×C3⋊(C3⋊D3)—21no—
90C33×C4—41no—
91C33×C4C2×D411yescharacteristic
92C33×C4#4111yescharacteristic
93C3⋊(C3⋊Dic3)—21no—
94C3⋊(C3⋊Dic3)—41no—
95C3×C3⋊Dic3—41no—
96C32×Dic3—41no—
97C2×C3×Q16—34no—
98#946—94no—
99#946—94no—
100#946—91no—
101#946—184no—
102C2×#216C3⋊D311yescharacteristic, complemented
103C2×ΓL(1,𝔽25)—94no—
104C2×ΓL(1,𝔽25)—91no—
105#897—91no—
106C3×#103—91no—
107#835—34no—
108#835—94no—
109#817—94no—
110#816—91no—
111#808—91no—
112C22×C3⋊D3—31no—
113C22×C3⋊D3—34no—
114C22×C3⋊D3—64no—
115C32×Q8—21no—
116C32×Q8—21no—
117C32×Q8—24no—
118C32×Q8—24no—
119C32×Q8—44no—
120C32×Q8—44no—
121C32×Q8—61no—
122C32×Q8—61no—
123C2×C32×C4—28no—
124C2×C32×C4—24no—
125C2×C32×C4—21no—
126C2×C32×C4—24no—
127C2×C32×C4—61no—
128C2×C32×C4C22×D314yes—
129C2×C32×C4C22×D311yescharacteristic
130C2×C3⋊Dic3—21no—
131C2×C3⋊Dic3—31no—
132C2×C3⋊Dic3—34no—
133C2×C3⋊Dic3—64no—
134C2×C3⋊Dic3—68no—
135C2×C3⋊Dic3—64no—
136C2×C3⋊Dic3—61no—
137C4×C3⋊D3—61no—
138C4×C3⋊D3—61no—
139C4×C3⋊D3—64no—
140C4×C3⋊D3—64no—
141C4×C3⋊D3—124no—
142C4×C3⋊D3—124no—
143C2×C3×Dic3—24no—
144C2×C3×Dic3—64no—
145C3×Dic6—24no—
146C3×Dic6—24no—
147C32×C8—32no—
148C3⋊(C3⋊C8)—21no—
149C3×C3⋊C8—64no—
150C3×C3⋊C8#6718yes—
151#470—271noSylow
152C2×C33#12111yescharacteristic
153C2×C33#13711yescharacteristic
154C2×C33#13711yescharacteristic
155C3⋊(C3⋊D3)—41no—
156C2×C3×Q8—24no—
157C2×C3×Q8—34no—
158C2×C3×Q8D3214yes—
159C2×C3×Q8C2×C3⋊D311yescharacteristic
160C2×C4×D3—94no—
161C2×C4×D3—91no—
162C2×C4×D3—184no—
163C2×Dic6C2×C3⋊D311yescharacteristic
164C3×Q16—68no—
165C2×C3×C8—34no—
166C2×C3×C8—91no—
167C3×C4⋊C4—91no—
168#216—22no—
169(C2×D3)⋊C4—181no—
170(C2×D3)⋊C4—184no—
171(C2×D3)⋊C4—184no—
172(C2×D3)⋊C4—181no—
173(C2×D3)⋊C4—188no—
174C4⋊Dic3—34no—
175C4⋊Dic3—94no—
176C4⋊Dic3—91no—
177C4⋊Dic3—184no—
178C4⋊Dic3—181no—
179Dic3⋊C4—184no—
180Dic3⋊C4—184no—
181Dic3⋊C4—181no—
182Dic3⋊C4—188no—
183C2×C3⋊C8—34no—
184C2×C3⋊C8C2×C3⋊D311yescharacteristic
185ΓL(1,𝔽25)—188no—
186ΓL(1,𝔽25)—182no—
187C22×C32—24no—
188C22×C32D3×D414yes—
189C22×C32D3×D411yescharacteristic
190C2×C3⋊D3—68no—
191C2×C3⋊D3—61no—
192C2×C3⋊D3—61no—
193C2×C3⋊D3—64no—
194C2×C3⋊D3—61no—
195C2×C3⋊D3—64no—
196C2×C3⋊D3—64no—
197C2×C3⋊D3—124no—
198C2×C3⋊D3—124no—
199C32×C4—24no—
200C32×C4—24no—
201C32×C4—48no—
202C32×C4—41no—
203C32×C4—44no—
204C32×C4—121no—
205C32×C4#23511yescharacteristic
206C32×C4D3×D411yescharacteristic
207C32×C4#23714yes—
208C32×C4C2×#6714yes—
209C3⋊Dic3—41no—
210C3⋊Dic3—61no—
211C3⋊Dic3—64no—
212C3⋊Dic3—124no—
213C3⋊Dic3—128no—
214C3⋊Dic3—124no—
215C3⋊Dic3—121no—
216C3×Dic3—44no—
217C3×Dic3—124no—
218C2×Q16—31no—
219C2×M16—271no—
220C22⋊Q8—271no—
221C22⋊Q8—271no—
222#106—271no—
223#103—91no—
224#103—271no—
225C33#47011yescharacteristic, complemented, Sylow
226C22×D3—94no—
227C22×D3—91no—
228C22×D3—184no—
229C3×Q8—24no—
230C3×Q8—24no—
231C3×Q8—21no—
232C3×Q8—21no—
233C3×Q8—44no—
234C3×Q8—44no—
235C3×Q8—64no—
236C3×Q8—64no—
237C2×C3×C4—28no—
238C2×C3×C4—24no—
239C2×C3×C4—21no—
240C2×C3×C4—24no—
241C2×C3×C4—64no—
242C2×C3×C4—181no—
243C2×C3×C4C2×D3214yes—
244C2×C3×C4C22×C3⋊D311yescharacteristic
245C2×Dic3—21no—
246C2×Dic3—64no—
247C2×Dic3—94no—
248C2×Dic3—91no—
249C2×Dic3—184no—
250C2×Dic3—188no—
251C2×Dic3—184no—
252C2×Dic3—181no—
253C4×D3—184no—
254C4×D3—184no—
255C4×D3—181no—
256C4×D3—181no—
257C4×D3—364no—
258C4×D3—364no—
259Dic6—21no—
260Dic6—21no—
261C3×C8—38no—
262C3×C8—181no—
263C3⋊C8—64no—
264C3⋊C8#63812yes—
265C2×C32—24no—
266C2×C32—24no—
267C2×C32—24no—
268C2×C32#88411yescharacteristic
269C2×C32#91511yescharacteristic
270C2×C32#91814yes—
271C2×C32#91814yes—
272C2×C32#91811yescharacteristic
273C2×C32#93814yes—
274C3⋊D3—128no—
275C3⋊D3—124no—
276C3⋊D3—121no—
277C2×Q8—31no—
278C2×Q8D3×C3⋊D311yescharacteristic
279C22×C4—271no—
280Q16—62no—
281M16—542no—
282C2×C8—31no—
283C2×C8—91no—
284C4⋊C4—91no—
285C4⋊C4—271no—
286C4⋊C4—541no—
287C4⋊C4—541no—
288C22⋊C4—541no—
289C22⋊C4—541no—
290C22×C3—24no—
291C22×C3#381114yes—
292C22×C3D4×C3⋊D311yescharacteristic
293C2×D3—188no—
294C2×D3—181no—
295C2×D3—181no—
296C2×D3—181no—
297C2×D3—184no—
298C2×D3—184no—
299C2×D3—184no—
300C2×D3—364no—
301C2×D3—364no—
302C3×C4—24no—
303C3×C4—24no—
304C3×C4—48no—
305C3×C4—41no—
306C3×C4—44no—
307C3×C4—124no—
308C3×C4—361no—
309C3×C4#380414yes—
310C3×C4D3×#6714yes—
311C3×C4#383011yescharacteristic
312C3×C4C2×#63811yescharacteristic
313Dic3—41no—
314Dic3—124no—
315Dic3—181no—
316Dic3—184no—
317Dic3—364no—
318Dic3—368no—
319Dic3—361no—
320Dic3—364no—
321C32—24no—
322C32#481611yescharacteristic, complemented
323C32#518914yescomplemented
324C23—271no—
325Q8—21no—
326Q8—21no—
327Q8—61no—
328Q8—61no—
329C2×C4—21no—
330C2×C4—61no—
331C2×C4—181no—
332C2×C4—271no—
333C2×C4—541no—
334C2×C4—541no—
335C2×C4—541no—
336C2×C4C2×D3×C3⋊D311yescharacteristic
337C8—32no—
338C8—181no—
339C2×C3—24no—
340C2×C3—24no—
341C2×C3—24no—
342C2×C3#6394514yes—
343C2×C3#6395114yes—
344C2×C3#6395814yes—
345C2×C3#6412711yescharacteristic
346C2×C3#6412711yescharacteristic
347C2×C3#6414711yescharacteristic
348D3—368no—
349D3—364no—
350D3—361no—
351C22—541no—
352C22—541no—
353C22—541no—
354C22#8903911yescharacteristic
355C4—41no—
356C4—121no—
357C4—361no—
358C4—541no—
359C4—1081no—
360C4#8902811yescharacteristic
361C4D3×#63811yescharacteristic
362C3—24no—
363C3#1059160014yescomplemented
364C3#1059288811yescharacteristic, complemented
365C2—1081no—
366C2#1172357811yescharacteristic
367C2#1172359011yescharacteristic
368C2#1172359911yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC22×C3⋊(C3⋊D3)D4
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#470
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. C2×C32×#216C2semidirect
5. C3×#63662C2semidirect
6. #11722052C2semidirect
7. #11722038C2semidirect
29. C2×C3×#216D3semidirect
33. #63662D3semidirect
102. C2×#216C3⋊D3semidirect
225. C33#470semidirect
322. C32#4816semidirect
323. C32#5189semidirect
363. C3#10591600semidirect
364. C3#10592888semidirect

Maximal subgroups

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

subgroupindex
#117239872
C2×C32×#2162
#117234562
C2×#887882
C3×#636622
#117220522
#117220382
#105928883
#105916003