Finite Group Databasebeta

← #13160071

Subgroups — #13160071

Order 1728.

1140 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1140 conjugacy classes of subgroups — too many to draw. Only its 210 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×#89021C211yesmaximal, characteristic
2C2×C32×ΓL(1,𝔽25)C211yesmaximal, characteristic, complemented
3D3×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
4#11722186C211yesmaximal, characteristic, complemented
5C3×#63648C211yesmaximal, characteristic, complemented
6#11722042C211yesmaximal, characteristic, complemented
7#11722034C211yesmaximal, characteristic, complemented
8#10592832—31nomaximal
9#10590670—34nomaximal
10C2×#6308C2211yescharacteristic
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×C4×D3C2211yescharacteristic
13C2×#6290—21no—
14C2×Dic3×C3⋊D3—21no—
15C2×D3×C3⋊Dic3—21no—
16#89021—22no—
17#89021C414yescomplemented
18C2×C33×C8C2211yescharacteristic
19C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
20C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
21C2×C32×C3⋊C8C2211yescharacteristic
22C32×ΓL(1,𝔽25)—22no—
23Dic3⋊(C3⋊Dic3)—21no—
24C2×#3828—31no—
25C2×#3795—34no—
26C2×C32×M16—31no—
27C2×C4⋊(C3⋊Dic3)—31no—
28C2×C3×ΓL(1,𝔽25)D314yescomplemented
29D3×C4⋊Dic3—34no—
30#63648—31no—
31#63648D311yescharacteristic, complemented
32#63643—31no—
33#63643—31no—
34#63643—34no—
35#63643—64no—
36C3×#821—34no—
37#63567—34no—
38#63563—34no—
39C22×C3×C3⋊D3—21no—
40C22×C32×D3D411yescharacteristic
41C2×C33×C4C2311yescharacteristic
42C2×C3⋊(C3⋊Dic3)—21no—
43C2×C3⋊(C3⋊Dic3)—21no—
44#6308—21no—
45#6308C2×C412yescomplemented
46C2×C3×C3⋊Dic3—21no—
47C3×C4⋊(C3⋊D3)—21no—
48C3×C4⋊(C3⋊D3)C2×C412yescomplemented
49C2×C32×Dic3D411yescharacteristic
50C32×C4×D3C2×C412yes—
51C32×C4×D3D412yes—
52#6290—24no—
53Dic3×C3⋊D3—24no—
54D3×C3⋊Dic3—42no—
55C33×C8—21no—
56C32×C3⋊C8—21no—
57#5133—94no—
58#4771—91no—
59C22×C32×C4—31no—
60C2×#638—61no—
61C22×C3⋊Dic3—61no—
62#3828—34no—
63#3828—62no—
64C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
65C2×C4×C3⋊D3—61no—
66C2×#634—31no—
67C2×#634—34no—
68C2×#634—64no—
69C2×C3×D12—34no—
70C2×C3×C4×D3C2×D314yes—
71C2×#625—64no—
72C2×D3×Dic3—64no—
73C2×D3×Dic3—64no—
74#3795—316no—
75#3795—68no—
76C32×M16—62no—
77C2×C32×C8—24no—
78C2×C32×C8—31no—
79C2×C32×C8C2×D314yes—
80C2×C32×C8C2×D311yescharacteristic
81C4⋊(C3⋊Dic3)—31no—
82C4⋊(C3⋊Dic3)—34no—
83C4⋊(C3⋊Dic3)—61no—
84C4⋊(C3⋊Dic3)—64no—
85C4⋊(C3⋊Dic3)C2×D311yescharacteristic
86C3×C4⋊Dic3—34no—
87C2×C3×C3⋊C8C2×D314yes—
88C3×ΓL(1,𝔽25)—28no—
89Dic3⋊Dic3—64no—
90C22×C33C2×D411yescharacteristic
91C2×C3×C3⋊D3—22no—
92C2×C3×C3⋊D3—41no—
93C2×C3×C3⋊D3—41no—
94C2×C32×D3—21no—
95C2×C32×D3—21no—
96C2×C32×D3C22⋊C412yes—
97C33×C4C22×C411yescharacteristic
98C33×C4C2×D411yescharacteristic
99C3⋊(C3⋊Dic3)—22no—
100C3⋊(C3⋊Dic3)—41no—
101C3×C3⋊Dic3—41no—
102C32×Dic3C22⋊C412yes—
103C2×#237—91no—
104C2×#235—94no—
105C2×C3×M16—34no—
106C2×C4⋊Dic3—94no—
107C2×ΓL(1,𝔽25)C3⋊D311yescharacteristic, complemented
108D3×C4⋊C4—91no—
109C3×#102—91no—
110#832—91no—
111#821—34no—
112#821—94no—
113#816—94no—
114#816—94no—
115#816—91no—
116#816—184no—
117#810—91no—
118C23×C32—31no—
119C22×C3⋊D3—21no—
120C22×C3×D3—64no—
121C22×C3×D3D1214yes—
122C2×C32×C4—24no—
123C2×C32×C4—31no—
124C2×C32×C4—32no—
125C2×C32×C4—32no—
126C2×C32×C4C22×D314yes—
127C2×C32×C4C22×D311yescharacteristic
128#638—64no—
129C2×C3⋊Dic3—21no—
130C2×C3⋊Dic3—61no—
131C2×C3⋊Dic3—61no—
132C2×C3⋊Dic3—64no—
133C2×C3⋊Dic3—64no—
134C2×C3⋊Dic3—124no—
135C2×C3⋊Dic3—124no—
136C2×C3⋊Dic3—122no—
137C4⋊(C3⋊D3)—21no—
138C4⋊(C3⋊D3)C4×D312yescomplemented
139C4×C3⋊D3—64no—
140#634—32no—
141#634—38no—
142#634—68no—
143#634—61no—
144#634—64no—
145#634—124no—
146C2×C3×Dic3—64no—
147C2×C3×Dic3D1214yes—
148C3×D12—38no—
149C3×D12—64no—
150C3×C4×D3C4×D318yes—
151C3×C4×D3#6718yes—
152#625—616no—
153D3×Dic3—616no—
154D3×Dic3—128no—
155C32×C8—21no—
156C32×C8—24no—
157C32×C8—44no—
158C32×C8—61no—
159C3×C3⋊C8—24no—
160#417—271noSylow
161C2×C33C2×C22⋊C411yescharacteristic
162C2×C33AGL(1,ℤ/8)11yescharacteristic
163C2×C33#13711yescharacteristic
164C3×C3⋊D3—42no—
165C32×D3—22no—
166C2×C3×D4—91no—
167C22×C3×C4—34no—
168C2×#67—184no—
169C2×#67—181no—
170C22×Dic3—184no—
171C22×Dic3—181no—
172#237—94no—
173#237—182no—
174#235—916no—
175#235—188no—
176C2×D12—34no—
177C2×C4×D3—184no—
178C2×C4×D3—181no—
179C2×C4×D3C2×C3⋊D311yescharacteristic
180C2×Dic6—94no—
181C2×Dic6—91no—
182C2×Dic6—184no—
183C3×M16—68no—
184C2×C3×C8—24no—
185C2×C3×C8—34no—
186C2×C3×C8D3214yes—
187C2×C3×C8C2×C3⋊D311yescharacteristic
188C3×C4⋊C4—91no—
189C4⋊Dic3—34no—
190C4⋊Dic3—94no—
191C4⋊Dic3—91no—
192C4⋊Dic3—184no—
193C4⋊Dic3—184no—
194Dic3⋊C4—181no—
195C2×C3⋊C8C2×C3⋊D311yescharacteristic
196ΓL(1,𝔽25)—22no—
197C22×C32—24no—
198C22×C32—32no—
199C22×C32—61no—
200C22×C32—61no—
201C22×C32C2×D1214yes—
202C22×C32D3×D411yescharacteristic
203C2×C3⋊D3—22no—
204C2×C3⋊D3—41no—
205C2×C3⋊D3—41no—
206C2×C3×D3—24no—
207C2×C3×D3—24no—
208C2×C3×D3—68no—
209C2×C3×D3—124no—
210C2×C3×D3—124no—
211C2×C3×D3(C2×D3)⋊C418yes—
212C32×C4—24no—
213C32×C4—24no—
214C32×C4—32no—
215C32×C4C2×C4×D314yes—
216C32×C4C2×C4×D311yescharacteristic
217C32×C4D3×D411yescharacteristic
218C32×C4C2×#6714yes—
219C3⋊Dic3—41no—
220C3⋊Dic3—62no—
221C3⋊Dic3—68no—
222C3⋊Dic3—128no—
223C3⋊Dic3—121no—
224C3⋊Dic3—124no—
225C3⋊Dic3—244no—
226C3×Dic3—124no—
227C3×Dic3(C2×D3)⋊C418yes—
228C2×#41—271no—
229C2×M16—31no—
230C2×C4⋊C4—271no—
231#103—271no—
232#103—271no—
233#102—91no—
234#102—271no—
235C33#41711yescharacteristic, complemented, Sylow
236C23×C3—34no—
237C23×C3—181no—
238C22×D3—64no—
239C22×D3C4⋊(C3⋊D3)11yescharacteristic
240C3×D4—92no—
241C3×D4—181no—
242C2×C3×C4—24no—
243C2×C3×C4—34no—
244C2×C3×C4—38no—
245C2×C3×C4—38no—
246C2×C3×C4—181no—
247C2×C3×C4C2×D3214yes—
248C2×C3×C4C22×C3⋊D311yescharacteristic
249#67—1816no—
250#67—184no—
251C2×Dic3—64no—
252C2×Dic3—184no—
253C2×Dic3—181no—
254C2×Dic3—184no—
255C2×Dic3—184no—
256C2×Dic3—181no—
257C2×Dic3—364no—
258C2×Dic3—364no—
259C2×Dic3—368no—
260C2×Dic3C4⋊(C3⋊D3)11yescharacteristic
261D12—38no—
262D12—64no—
263C4×D3—1816no—
264C4×D3—362no—
265C4×D3C4×C3⋊D312yes—
266C4×D3#63812yes—
267Dic6—98no—
268Dic6—92no—
269Dic6—188no—
270Dic6—184no—
271Dic6—181no—
272Dic6—364no—
273C3×C8—24no—
274C3×C8—21no—
275C3×C8—44no—
276C3×C8—64no—
277C3⋊C8—21no—
278C2×C32—24no—
279C2×C32—24no—
280C2×C32—24no—
281C2×C32—62no—
282C2×C32D3×C22⋊C411yescharacteristic
283C2×C32#90814yes—
284C2×C32#90914yes—
285C2×C32#91111yescharacteristic
286C2×C32#91511yescharacteristic
287C2×C32C2×(C2×D3)⋊C414yes—
288C3⋊D3—42no—
289C3×D3—28no—
290C3×D3—128no—
291#41—274no—
292#41—542no—
293C2×Q8—271no—
294C2×D4—91no—
295C2×D4—541no—
296C22×C4—31no—
297C22×C4—541no—
298C22×C4—541no—
299M16—62no—
300C2×C8—31no—
301C2×C8D3×C3⋊D311yescharacteristic
302C4⋊C4—91no—
303C4⋊C4—271no—
304C4⋊C4—541no—
305C22×C3—24no—
306C22×C3—38no—
307C22×C3—64no—
308C22×C3—64no—
309C22×C3—182no—
310C22×C3—361no—
311C22×C3—361no—
312C22×C3D3×D1214yes—
313C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
314C2×D3—21no—
315C2×D3—21no—
316C2×D3—68no—
317C2×D3—124no—
318C2×D3—124no—
319C2×D3#375212yes—
320C3×C4—24no—
321C3×C4—24no—
322C3×C4—38no—
323C3×C4—361no—
324C3×C4C4×D3214yes—
325C3×C4D3×#6714yes—
326C3×C4C2×C4×C3⋊D311yescharacteristic
327C3×C4C2×#63811yescharacteristic
328Dic3—124no—
329Dic3—182no—
330Dic3—188no—
331Dic3—368no—
332Dic3—364no—
333Dic3—361no—
334Dic3—724no—
335Dic3#375212yes—
336C32—24no—
337C32#477111yescharacteristic, complemented
338C32#513314yescomplemented
339C23—31no—
340C23—181no—
341Q8—272no—
342Q8—541no—
343D4—92no—
344D4—181no—
345D4—544no—
346C2×C4—31no—
347C2×C4—32no—
348C2×C4—32no—
349C2×C4—181no—
350C2×C4—544no—
351C2×C4—541no—
352C2×C4—541no—
353C2×C4—1082no—
354C2×C4C2×D3×C3⋊D311yescharacteristic
355C8—21no—
356C8—61no—
357C2×C3—24no—
358C2×C3—24no—
359C2×C3—24no—
360C2×C3—68no—
361C2×C3—362no—
362C2×C3#6379514yes—
363C2×C3#6380014yes—
364C2×C3D3×(C2×D3)⋊C414yes—
365C2×C3#6411711yescharacteristic
366C2×C3#6411811yescharacteristic
367C2×C3C2×#375211yescharacteristic
368D3—22no—
369D3—128no—
370C22—32no—
371C22—61no—
372C22—61no—
373C22—182no—
374C22—361no—
375C22—361no—
376C22D3×C4⋊(C3⋊D3)11yescharacteristic
377C4—32no—
378C4—361no—
379C4—542no—
380C4—1081no—
381C4C4×D3×C3⋊D311yescharacteristic
382C4D3×#63811yescharacteristic
383C3—24no—
384C3#1059067014yescomplemented
385C3#1059283211yescharacteristic, complemented
386C2—62no—
387C2—362no—
388C2#1172331611yescharacteristic
389C2#1172332611yescharacteristic
390C2D3×#375211yescharacteristic
subgroupG/H
centerC22D3×C4⋊(C3⋊D3)
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×#89021C2
cosocleC2×C33×C4C23
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#417
LCSG ▷ 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×C32×ΓL(1,𝔽25)C2semidirect
3. D3×C4⋊(C3⋊Dic3)C2semidirect
4. #11722186C2semidirect
5. C3×#63648C2semidirect
6. #11722042C2semidirect
7. #11722034C2semidirect
17. #89021C4semidirect
28. C2×C3×ΓL(1,𝔽25)D3semidirect
31. #63648D3semidirect
45. #6308C2×C4semidirect
48. C3×C4⋊(C3⋊D3)C2×C4semidirect
107. C2×ΓL(1,𝔽25)C3⋊D3semidirect
138. C4⋊(C3⋊D3)C4×D3semidirect
235. C33#417semidirect
337. C32#4771semidirect
338. C32#5133semidirect
384. C3#10590670semidirect
385. C3#10592832semidirect

Maximal subgroups

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

subgroupindex
C2×#890212
C2×C32×ΓL(1,𝔽25)2
D3×C4⋊(C3⋊Dic3)2
#117221862
C3×#636482
#117220422
#117220342
#105928323
#105906703