Finite Group Databasebeta

← #13159351

Subgroups — #13159351

Order 1728.

1016 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1016 conjugacy classes of subgroups — too many to draw. Only its 146 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#11723938C211yesmaximal, characteristic, complemented
2#11723531C211yesmaximal, characteristic
3C2×#88794C211yesmaximal, characteristic, complemented
4C2×#88788C211yesmaximal, characteristic, complemented
5C3×#63634C211yesmaximal, characteristic, complemented
6C32×#809C211yesmaximal, characteristic, complemented
7#11722042C211yesmaximal, characteristic, complemented
8#10592479—31nomaximal
9#10590506—34nomaximal
10C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
11C2×#6308C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×#6291—21no—
14C33×C4⋊C4C2211yescharacteristic
15#88859—21no—
16C4⋊(C3⋊(C3⋊Dic3))—21no—
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20(C2×D3)⋊(C3⋊Dic3)—21no—
21#88794—22no—
22#88788—22no—
23#64147—31no—
24C2×#3754—31no—
25C2×#3743—31no—
26#64106—31no—
27#64106—34no—
28#64106—64no—
29#63916—34no—
30C2×#3716—34no—
31C2×#3712—34no—
32C32×#102—31no—
33#63643—31no—
34#63634D311yescharacteristic, complemented
35C3×#809D314yescomplemented
36C3×#807—34no—
37#63567—34no—
38C22×C3⋊(C3⋊D3)D411yescharacteristic
39C22×C32×D3—21no—
40C2×C33×C4—21no—
41C2×C33×C4C2311yescharacteristic
42C2×C3⋊(C3⋊Dic3)—21no—
43C2×C3⋊(C3⋊Dic3)D411yescharacteristic
44C4×C3⋊(C3⋊D3)—21no—
45C4×C3⋊(C3⋊D3)—21no—
46#6308—21no—
47#6308—21no—
48C2×C3×C3⋊Dic3—21no—
49C32×D12—21no—
50C32×D12—21no—
51#6291—42no—
52C3×C3⋊(C3⋊C8)D412yescomplemented
53C32×C3⋊C8—21no—
54#4886—91no—
55#4780—94no—
56C2×C32×D4—31no—
57C2×#638—61no—
58C2×C4×C3⋊D3—31no—
59C2×C4×C3⋊D3—34no—
60C2×C4×C3⋊D3—64no—
61C2×#634—31no—
62C2×#634—34no—
63C2×#634—64no—
64C2×C3×D12C2×D314yes—
65C2×#626—64no—
66C2×C32×C8—31no—
67C32×C4⋊C4—24no—
68C32×C4⋊C4C2×D314yes—
69C32×C4⋊C4C2×D311yescharacteristic
70#3757—61no—
71#3754—62no—
72#3752—61no—
73#3752—64no—
74#3752—68no—
75C4⋊(C3⋊Dic3)—61no—
76C4⋊(C3⋊Dic3)—64no—
77C4⋊(C3⋊Dic3)—68no—
78C4⋊(C3⋊Dic3)C2×D311yescharacteristic
79C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
80#3743—62no—
81C3×C4⋊Dic3—34no—
82C2×C3×C3⋊C8—34no—
83C2×C3×C3⋊C8C2×D314yes—
84(C2×D3)⋊Dic3—64no—
85#3716—68no—
86#3712—68no—
87C22×C33C2×D411yescharacteristic
88C2×C3⋊(C3⋊D3)—21no—
89C2×C3⋊(C3⋊D3)—21no—
90C2×C3⋊(C3⋊D3)—21no—
91C2×C32×D3—41no—
92C2×C32×D3—41no—
93C2×C32×D3—41no—
94C33×C4—41no—
95C33×C4C2×D411yescharacteristic
96C33×C4#4111yescharacteristic
97C3⋊(C3⋊Dic3)—21no—
98C3⋊(C3⋊Dic3)—41no—
99C3×C3⋊Dic3—41no—
100#938—94no—
101C2×#214—94no—
102C2×#204—91no—
103C2×ΓL(1,𝔽25)—94no—
104C2×ΓL(1,𝔽25)—91no—
105#897—94no—
106#897—91no—
107#897—184no—
108#895—91no—
109C3×#107—91no—
110C3×#102—34no—
111#816—94no—
112#810—91no—
113#809C3⋊D311yescharacteristic, complemented
114#807—34no—
115C23×C32—61no—
116C22×C3⋊D3—31no—
117C22×C3⋊D3—34no—
118C22×C3⋊D3—64no—
119C22×C3×D3—24no—
120C32×D4—61no—
121C32×D4—61no—
122C2×C32×C4—28no—
123C2×C32×C4—24no—
124C2×C32×C4—21no—
125C2×C32×C4—24no—
126C2×C32×C4C22×D314yes—
127C2×C32×C4C22×D311yescharacteristic
128#638—122no—
129C2×C3⋊Dic3—21no—
130C2×C3⋊Dic3—31no—
131C2×C3⋊Dic3—34no—
132C2×C3⋊Dic3—64no—
133C2×C3⋊Dic3—68no—
134C2×C3⋊Dic3—64no—
135C2×C3⋊Dic3—61no—
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—61no—
143#634—61no—
144#634—64no—
145#634—64no—
146#634—124no—
147#634—124no—
148C2×C3×Dic3—64no—
149C3×D12—24no—
150C3×D12—24no—
151#626—128no—
152C32×C8—61no—
153C3⋊(C3⋊C8)D1212yescomplemented
154C3×C3⋊C8—24no—
155C3×C3⋊C8—38no—
156#468—271noSylow
157C2×C33#12111yescharacteristic
158C2×C33AGL(1,ℤ/8)11yescharacteristic
159C2×C33#13711yescharacteristic
160C3⋊(C3⋊D3)—41no—
161C32×D3—81no—
162C2×C3×D4—34no—
163C2×#67—184no—
164C2×D12—181no—
165C2×D12C2×C3⋊D311yescharacteristic
166C2×C4×D3—94no—
167C2×C4×D3—91no—
168C2×C4×D3—184no—
169C2×Dic6—94no—
170C2×Dic6—91no—
171C2×Dic6—184no—
172C2×C3×C8—34no—
173C2×C3×C8—91no—
174C3×C4⋊C4—24no—
175C3×C4⋊C4—91no—
176C3×C4⋊C4D3214yes—
177C3×C4⋊C4C2×C3⋊D311yescharacteristic
178#217—184no—
179#214—188no—
180(C2×D3)⋊C4—181no—
181(C2×D3)⋊C4—184no—
182(C2×D3)⋊C4—181no—
183(C2×D3)⋊C4—188no—
184C4⋊Dic3—34no—
185C4⋊Dic3—184no—
186C4⋊Dic3—181no—
187C4⋊Dic3—188no—
188C2×C3⋊C8—34no—
189C2×C3⋊C8C2×C3⋊D311yescharacteristic
190#204—182no—
191ΓL(1,𝔽25)—182no—
192ΓL(1,𝔽25)—188no—
193C22×C32—24no—
194C22×C32—121no—
195C22×C32—121no—
196C22×C32—121no—
197C22×C32D3×D414yes—
198C22×C32D3×D411yescharacteristic
199C2×C3⋊D3—68no—
200C2×C3⋊D3—61no—
201C2×C3⋊D3—61no—
202C2×C3⋊D3—64no—
203C2×C3⋊D3—61no—
204C2×C3⋊D3—64no—
205C2×C3⋊D3—64no—
206C2×C3⋊D3—124no—
207C2×C3⋊D3—124no—
208C2×C3×D3—44no—
209C2×C3×D3—44no—
210C2×C3×D3—44no—
211C32×C4—24no—
212C32×C4—24no—
213C32×C4—48no—
214C32×C4—44no—
215C32×C4—41no—
216C32×C4C2×D1211yescharacteristic
217C32×C4#23514yes—
218C32×C4D3×D414yes—
219C32×C4#23911yescharacteristic
220C3⋊Dic3—41no—
221C3⋊Dic3—61no—
222C3⋊Dic3—64no—
223C3⋊Dic3—124no—
224C3⋊Dic3—128no—
225C3⋊Dic3—124no—
226C3⋊Dic3—121no—
227C3×Dic3—124no—
228C2×SD16—271no—
229C2×M16—271no—
230C22⋊Q8—271no—
231#121—271no—
232#107—91no—
233#103—271no—
234#102—31no—
235C33#46811yescharacteristic, complemented, Sylow
236C23×C3—64no—
237C22×D3—21no—
238C22×D3—94no—
239C22×D3—91no—
240C22×D3—184no—
241C3×D4—64no—
242C3×D4—64no—
243C2×C3×C4—28no—
244C2×C3×C4—24no—
245C2×C3×C4—21no—
246C2×C3×C4—24no—
247C2×C3×C4—181no—
248C2×C3×C4C2×D3214yes—
249C2×C3×C4C22×C3⋊D311yescharacteristic
250#67—368no—
251C2×Dic3—64no—
252C2×Dic3—94no—
253C2×Dic3—91no—
254C2×Dic3—184no—
255C2×Dic3—188no—
256C2×Dic3—184no—
257C2×Dic3—181no—
258D12—21no—
259D12—21no—
260D12—362no—
261C4×D3—184no—
262C4×D3—184no—
263C4×D3—181no—
264C4×D3—181no—
265C4×D3—364no—
266C4×D3—364no—
267Dic6—184no—
268Dic6—184no—
269Dic6—181no—
270Dic6—181no—
271Dic6—364no—
272Dic6—364no—
273C3×C8—64no—
274C3×C8—92no—
275C3⋊C8—21no—
276C3⋊C8—38no—
277C2×C32—24no—
278C2×C32—24no—
279C2×C32—24no—
280C2×C32—241no—
281C2×C32#88414yes—
282C2×C32#89511yescharacteristic
283C2×C32#91114yes—
284C2×C32#91111yescharacteristic
285C2×C32#91514yes—
286C2×C32#91811yescharacteristic
287C3⋊D3—128no—
288C3⋊D3—121no—
289C3⋊D3—124no—
290C3×D3—84no—
291C2×Q8—271no—
292C2×D4—31no—
293C2×D4—541no—
294C22×C4—271no—
295SD16—542no—
296M16—542no—
297C2×C8—31no—
298C2×C8—91no—
299C4⋊C4—91no—
300C4⋊C4—541no—
301C4⋊C4D3×C3⋊D311yescharacteristic
302C22⋊C4—541no—
303C22⋊C4—541no—
304C22×C3—24no—
305C22×C3—124no—
306C22×C3—124no—
307C22×C3—124no—
308C22×C3#381114yes—
309C22×C3D4×C3⋊D311yescharacteristic
310C2×D3—41no—
311C2×D3—41no—
312C2×D3—41no—
313C2×D3—188no—
314C2×D3—181no—
315C2×D3—181no—
316C2×D3—181no—
317C2×D3—184no—
318C2×D3—184no—
319C2×D3—184no—
320C2×D3—364no—
321C2×D3—364no—
322C3×C4—24no—
323C3×C4—24no—
324C3×C4—48no—
325C3×C4—41no—
326C3×C4—44no—
327C3×C4—361no—
328C3×C4#379914yes—
329C3×C4D3×D1214yes—
330C3×C4#382811yescharacteristic
331C3×C4D4×C3⋊D311yescharacteristic
332Dic3—124no—
333Dic3—181no—
334Dic3—184no—
335Dic3—364no—
336Dic3—368no—
337Dic3—361no—
338Dic3—364no—
339C32—24no—
340C32#478014yescomplemented
341C32#488611yescharacteristic, complemented
342C23—61no—
343C23—271no—
344Q8—541no—
345Q8—541no—
346D4—61no—
347D4—61no—
348D4—1082no—
349C2×C4—21no—
350C2×C4—181no—
351C2×C4—271no—
352C2×C4—541no—
353C2×C4—541no—
354C2×C4—541no—
355C2×C4C2×D3×C3⋊D311yescharacteristic
356C8—61no—
357C8—92no—
358C2×C3—24no—
359C2×C3—24no—
360C2×C3—24no—
361C2×C3—244no—
362C2×C3#6391014yes—
363C2×C3#6392714yes—
364C2×C3#6395114yes—
365C2×C3#6409311yescharacteristic
366C2×C3#6412011yescharacteristic
367C2×C3#6412411yescharacteristic
368D3—81no—
369D3—368no—
370D3—364no—
371D3—361no—
372C22—121no—
373C22—121no—
374C22—121no—
375C22—541no—
376C22—541no—
377C22—541no—
378C22#8903911yescharacteristic
379C4—41no—
380C4—361no—
381C4—541no—
382C4—1081no—
383C4#8902211yescharacteristic
384C4C3⋊D3×D1211yescharacteristic
385C3—24no—
386C3#1059050614yescomplemented
387C3#1059247911yescharacteristic, complemented
388C2—241no—
389C2—1081no—
390C2#1172352511yescharacteristic
391C2#1172354811yescharacteristic
392C2#1172358811yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radical#11723531C2
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#468
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #11723938C2semidirect
3. C2×#88794C2semidirect
4. C2×#88788C2semidirect
5. C3×#63634C2semidirect
6. C32×#809C2semidirect
7. #11722042C2semidirect
34. #63634D3semidirect
35. C3×#809D3semidirect
52. C3×C3⋊(C3⋊C8)D4semidirect
113. #809C3⋊D3semidirect
153. C3⋊(C3⋊C8)D12semidirect
235. C33#468semidirect
340. C32#4780semidirect
341. C32#4886semidirect
386. C3#10590506semidirect
387. C3#10592479semidirect

Maximal subgroups

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

subgroupindex
#117239382
#117235312
C2×#887942
C2×#887882
C3×#636342
C32×#8092
#117220422
#105924793
#105905063