Finite Group Databasebeta

← #13160057

Subgroups — #13160057

Order 1728.

1088 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 1088 conjugacy classes of subgroups — too many to draw. Only its 160 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×#89019C211yesmaximal, characteristic
2C2×C3×#3746C211yesmaximal, characteristic, complemented
3#11723449C211yesmaximal, characteristic
4C2×#88798C211yesmaximal, characteristic, complemented
5#11722190C211yesmaximal, characteristic
6C32×#816C211yesmaximal, characteristic, complemented
7#11722039C211yesmaximal, characteristic, complemented
8#10592508—31nomaximal
9#10590950—34nomaximal
10C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×#6293—21no—
15C2×#6289—21no—
16#89019—22no—
17#89019—22no—
18C2×C33×C8C2211yescharacteristic
19C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
20C3×#3746—22no—
21C32×C4⋊Dic3C2211yescharacteristic
22Dic3⋊(C3⋊Dic3)—21no—
23#88804—21no—
24#88798—22no—
25C2×Q8×C3⋊D3—31no—
26C2×#3797—34no—
27C2×#3756—31no—
28C2×#3746D311yescharacteristic, complemented
29#64106—31no—
30C2×C3×Dic12—34no—
31#63871—34no—
32C2×#3718—34no—
33C32×#103—31no—
34#63647—31no—
35#63647—34no—
36#63647—64no—
37#63637—31no—
38C3×#816D314yescomplemented
39#63565—34no—
40C22×C3⋊(C3⋊D3)D411yescharacteristic
41C2×C33×C4C2311yescharacteristic
42C2×C3⋊(C3⋊Dic3)D411yescharacteristic
43C4×C3⋊(C3⋊D3)—21no—
44C4×C3⋊(C3⋊D3)—21no—
45C2×C3×C3⋊Dic3—21no—
46C2×C3×C3⋊Dic3—21no—
47C3×#634—21no—
48C3×#634—21no—
49C3×#634—21no—
50C3×#634D412yes—
51C2×C32×Dic3—21no—
52C2×C32×Dic3—21no—
53C32×Dic6—21no—
54C32×Dic6D412yes—
55#6293—24no—
56#6289—24no—
57C33×C8—21no—
58C3⋊(C3⋊(C3⋊C8))—21no—
59#5186—91no—
60#4809—94no—
61C2×C32×Q8—31no—
62Q8×C3⋊D3—62no—
63Q8×C3⋊D3—62no—
64C2×C4×C3⋊D3—31no—
65C2×C4×C3⋊D3—34no—
66C2×C4×C3⋊D3—61no—
67C2×C4×C3⋊D3—64no—
68C2×#634—61no—
69C2×#634C2×D311yescharacteristic
70C2×#634C2×D311yescharacteristic
71C2×C3×Dic6—34no—
72C2×C3×Dic6—34no—
73C2×C3×Dic6C2×D314yes—
74C2×#627—64no—
75C2×#624—64no—
76#3797—68no—
77#3797—68no—
78C2×C32×C8—24no—
79C2×C32×C8C2×D314yes—
80C2×C32×C8C2×D311yescharacteristic
81C32×C4⋊C4—31no—
82#3756—62no—
83#3752—61no—
84C4⋊(C3⋊Dic3)—61no—
85C2×C3⋊(C3⋊C8)—31no—
86C2×C3⋊(C3⋊C8)—34no—
87C2×C3⋊(C3⋊C8)—64no—
88#3746—22no—
89C3×C4⋊Dic3C2×D314yes—
90C3×Dic12—68no—
91Dic3⋊Dic3—64no—
92#3722—64no—
93#3718—68no—
94C22×C33C2×D411yescharacteristic
95C2×C3⋊(C3⋊D3)—21no—
96C2×C3⋊(C3⋊D3)—21no—
97C2×C3⋊(C3⋊D3)Q1612yescomplemented
98C33×C4C2×D411yescharacteristic
99C33×C4C2×D411yescharacteristic
100C3⋊(C3⋊Dic3)—21no—
101C3×C3⋊Dic3—22no—
102C3×C3⋊Dic3—41no—
103C32×Dic3—22no—
104C32×Dic3—41no—
105C2×D3×Q8—94no—
106C2×D3×Q8—91no—
107C2×C3×Q16—91no—
108#946—91no—
109C2×#216—94no—
110C2×#216—91no—
111C2×Dic12—34no—
112#897—94no—
113C3×#103—34no—
114#835—91no—
115(C2×D3)⋊C8—94no—
116(C2×D3)⋊C8—91no—
117(C2×D3)⋊C8—184no—
118#816C3⋊D311yescharacteristic, complemented
119#810—94no—
120C22×C3⋊D3—31no—
121C22×C3⋊D3—34no—
122C22×C3⋊D3—64no—
123C32×Q8—32no—
124C32×Q8—61no—
125C2×C32×C4—24no—
126C2×C32×C4—61no—
127C2×C32×C4—61no—
128C2×C32×C4C22×D314yes—
129C2×C32×C4C22×D311yescharacteristic
130C2×C3⋊Dic3—21no—
131C2×C3⋊Dic3—21no—
132C2×C3⋊Dic3—31no—
133C2×C3⋊Dic3—34no—
134C2×C3⋊Dic3—64no—
135C4×C3⋊D3—64no—
136C4×C3⋊D3—61no—
137C4×C3⋊D3—61no—
138C4×C3⋊D3—64no—
139C4×C3⋊D3—64no—
140C4×C3⋊D3—124no—
141C4×C3⋊D3—124no—
142#634—21no—
143#634—21no—
144#634—21no—
145#634—64no—
146#634#6712yes—
147C2×C3×Dic3—24no—
148C2×C3×Dic3—24no—
149C2×C3×Dic3—64no—
150C2×C3×Dic3—64no—
151C3×Dic6—24no—
152C3×Dic6—38no—
153C3×Dic6—64no—
154C3×Dic6—64no—
155C3×Dic6—64no—
156C3×Dic6D1218yes—
157#627—616no—
158#624—616no—
159C32×C8—21no—
160C32×C8—24no—
161C32×C8—44no—
162C3⋊(C3⋊C8)—61no—
163C3⋊(C3⋊C8)—64no—
164C3⋊(C3⋊C8)—124no—
165#451—271noSylow
166C2×C33#12011yescharacteristic
167C2×C33C2×Q1611yescharacteristic
168C2×C33#13711yescharacteristic
169C3⋊(C3⋊D3)—22no—
170C2×C3×Q8—34no—
171C2×C3×Q8—91no—
172C2×C3×Q8—91no—
173D3×Q8—182no—
174D3×Q8—188no—
175D3×Q8—188no—
176D3×Q8—182no—
177C2×C4×D3—94no—
178C2×C4×D3—91no—
179C2×C4×D3—184no—
180C2×C4×D3—181no—
181C2×C4×D3—184no—
182C2×Dic6—34no—
183C2×Dic6—34no—
184C2×Dic6—181no—
185C2×Dic6—184no—
186C2×Dic6C2×C3⋊D311yescharacteristic
187C3×Q16—182no—
188C2×C3×C8—24no—
189C2×C3×C8D3214yes—
190C2×C3×C8C2×C3⋊D311yescharacteristic
191C3×C4⋊C4—34no—
192#216—188no—
193#216—182no—
194(C2×D3)⋊C4—181no—
195(C2×D3)⋊C4—184no—
196C4⋊Dic3—184no—
197C4⋊Dic3C2×C3⋊D311yescharacteristic
198Dic3⋊C4—181no—
199C2×C3⋊C8—94no—
200C2×C3⋊C8—91no—
201C2×C3⋊C8—184no—
202Dic12—68no—
203C22×C32—24no—
204C22×C32D3×D414yes—
205C22×C32D3×D411yescharacteristic
206C2×C3⋊D3—32no—
207C2×C3⋊D3—38no—
208C2×C3⋊D3—68no—
209C2×C3⋊D3—61no—
210C2×C3⋊D3—61no—
211C2×C3⋊D3—64no—
212C2×C3⋊D3—64no—
213C2×C3⋊D3—124no—
214C2×C3⋊D3—124no—
215C32×C4—24no—
216C32×C4—24no—
217C32×C4—62no—
218C32×C4—121no—
219C32×C4C2×D1214yes—
220C32×C4D3×D414yes—
221C32×C4D3×D411yescharacteristic
222C32×C4C2×#6711yescharacteristic
223C3⋊Dic3—22no—
224C3⋊Dic3—41no—
225C3⋊Dic3—61no—
226C3⋊Dic3—64no—
227C3⋊Dic3—124no—
228C3×Dic3—28no—
229C3×Dic3—44no—
230C3×Dic3—68no—
231C3×Dic3—124no—
232C22×Q8—271no—
233C2×Q16—91no—
234C2×Q16—271no—
235C22⋊Q8—271no—
236#103—31no—
237#103—271no—
238C22⋊C8—271no—
239C33#45111yescharacteristic, complemented, Sylow
240C22×D3—94no—
241C22×D3—91no—
242C22×D3—184no—
243C3×Q8—38no—
244C3×Q8—64no—
245C3×Q8—92no—
246C3×Q8—181no—
247C3×Q8—181no—
248C3×Q8—181no—
249C2×C3×C4—24no—
250C2×C3×C4—64no—
251C2×C3×C4—64no—
252C2×C3×C4—181no—
253C2×C3×C4—181no—
254C2×C3×C4C2×D3214yes—
255C2×C3×C4C22×C3⋊D311yescharacteristic
256C2×Dic3—21no—
257C2×Dic3—21no—
258C2×Dic3—64no—
259C2×Dic3—64no—
260C2×Dic3—94no—
261C2×Dic3—91no—
262C2×Dic3—184no—
263C4×D3—184no—
264C4×D3—1816no—
265C4×D3—184no—
266C4×D3—184no—
267C4×D3—181no—
268C4×D3—181no—
269C4×D3—364no—
270C4×D3—364no—
271Dic6—21no—
272Dic6—38no—
273Dic6—64no—
274Dic6—64no—
275Dic6—64no—
276Dic6—184no—
277Dic6—1816no—
278Dic6C4⋊(C3⋊D3)12yes—
279C3×C8—24no—
280C3×C8—21no—
281C3×C8—44no—
282C3⋊C8—184no—
283C3⋊C8—181no—
284C3⋊C8—364no—
285C2×C32—24no—
286C2×C32—24no—
287C2×C32—24no—
288C2×C32#88214yes—
289C2×C32#91514yes—
290C2×C32D3×Q1614yes—
291C2×C32D3×Q1611yescharacteristic
292C2×C32#91811yescharacteristic
293C2×C32#93711yescharacteristic
294C3⋊D3—62no—
295C3⋊D3—68no—
296C3⋊D3—128no—
297C2×Q8—31no—
298C2×Q8—91no—
299C2×Q8—91no—
300C2×Q8—541no—
301C2×Q8—542no—
302C2×Q8—542no—
303C22×C4—271no—
304C22×C4—541no—
305Q16—182no—
306Q16—542no—
307C2×C8—271no—
308C2×C8D3×C3⋊D311yescharacteristic
309C4⋊C4—31no—
310C4⋊C4—541no—
311C22⋊C4—541no—
312C22×C3—24no—
313C22×C3#380214yes—
314C22×C3D4×C3⋊D311yescharacteristic
315C2×D3—98no—
316C2×D3—92no—
317C2×D3—188no—
318C2×D3—184no—
319C2×D3—181no—
320C2×D3—181no—
321C2×D3—184no—
322C2×D3—364no—
323C2×D3—364no—
324C3×C4—24no—
325C3×C4—24no—
326C3×C4—68no—
327C3×C4—124no—
328C3×C4—182no—
329C3×C4—361no—
330C3×C4D3×D1214yes—
331C3×C4D3×#6714yes—
332C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
333C3×C4D4×C3⋊D311yescharacteristic
334Dic3—22no—
335Dic3—41no—
336Dic3—68no—
337Dic3—124no—
338Dic3—181no—
339Dic3—184no—
340Dic3—364no—
341C32—24no—
342C32#480914yescomplemented
343C32#518611yescharacteristic, complemented
344C23—271no—
345Q8—32no—
346Q8—61no—
347Q8—92no—
348Q8—181no—
349Q8—181no—
350Q8—181no—
351Q8—544no—
352C2×C4—61no—
353C2×C4—61no—
354C2×C4—181no—
355C2×C4—181no—
356C2×C4—271no—
357C2×C4—544no—
358C2×C4—541no—
359C2×C4—541no—
360C2×C4C2×D3×C3⋊D311yescharacteristic
361C8—21no—
362C8—541no—
363C2×C3—24no—
364C2×C3—24no—
365C2×C3—24no—
366C2×C3#6380214yes—
367C2×C3#6381114yes—
368C2×C3#6392214yes—
369C2×C3#6409111yescharacteristic
370C2×C3#6412411yescharacteristic
371C2×C3Q16×C3⋊D311yescharacteristic
372D3—188no—
373D3—182no—
374D3—368no—
375C22—272no—
376C22—541no—
377C22—541no—
378C22#8902611yescharacteristic
379C4—62no—
380C4—121no—
381C4—182no—
382C4—361no—
383C4—541no—
384C4D3×C4⋊(C3⋊D3)11yescharacteristic
385C4C3⋊D3×#6711yescharacteristic
386C3—24no—
387C3#1059095014yescomplemented
388C3#1059250811yescharacteristic, complemented
389C2—542no—
390C2#1172333011yescharacteristic
391C2#1172334411yescharacteristic
392C2#1172354111yescharacteristic
subgroupG/H
centerC22#89026
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC22×C3⋊(C3⋊D3)D4
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#451
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3746C2semidirect
4. C2×#88798C2semidirect
6. C32×#816C2semidirect
7. #11722039C2semidirect
28. C2×#3746D3semidirect
38. C3×#816D3semidirect
97. C2×C3⋊(C3⋊D3)Q16semidirect
118. #816C3⋊D3semidirect
239. C33#451semidirect
342. C32#4809semidirect
343. C32#5186semidirect
387. C3#10590950semidirect
388. C3#10592508semidirect

Maximal subgroups

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

subgroupindex
C2×#890192
C2×C3×#37462
#117234492
C2×#887982
#117221902
C32×#8162
#117220392
#105925083
#105909503