Finite Group Databasebeta

← #13167374

Subgroups — #13167374

Order 1728.

374 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 55 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#11724337C211yesmaximal, characteristic
2#11723652C211yesmaximal, characteristic, complemented
3#11723651C211yesmaximal, characteristic, complemented
4C3×#63767C211yesmaximal, characteristic, complemented
5#11722787C211yesmaximal, characteristic, complemented
6#11722785C211yesmaximal, characteristic, complemented
7#11722785C211yesmaximal, characteristic, complemented
8#10593298—31nomaximal
9#89042C2211yescharacteristic
10#89042C2211yescharacteristic
11C3×C4×#157C2211yescharacteristic
12C3×#3787C2211yescharacteristic
13#88943C411yescharacteristic, complemented
14#88942C411yescharacteristic, complemented
15#88942C411yescharacteristic, complemented
16#88941C411yescharacteristic, complemented
17C3×#3743—21no—
18C3×C8×C3⋊D3—21no—
19C3×C8×C3⋊D3C2211yescharacteristic
20#88809—21no—
21#88809C2211yescharacteristic
22#88809C2211yescharacteristic
23#88808—21no—
24#64223—31no—
25#63805—31no—
26#63804—61no—
27#63803—31no—
28#63803—61no—
29#63767D311yescharacteristic, complemented
30#63733—31no—
31#63731—31no—
32#63731—31no—
33C2×C3×#157—21no—
34C3×C4×C3⋊D3—21no—
35C3×C4×C3⋊D3—21no—
36C3×C4×C3⋊D3C2311yescharacteristic
37#6296C2×C411yescharacteristic
38#6296C2×C411yescharacteristic
39#6295—21no—
40#6295—21no—
41#6295C2×C411yescharacteristic
42#6295C2×C411yescharacteristic
43C3×#622C2×C411yescharacteristic
44C3×#622C2×C411yescharacteristic
45C33×C8D411yescharacteristic
46C3×C3⋊(C3⋊C8)—21no—
47C3×C3⋊(C3⋊C8)—21no—
48C3×C3⋊(C3⋊C8)D411yescharacteristic
49#4717—91nomaximal
50#3798—31no—
51#3798—31no—
52#3798—61no—
53#3798—61no—
54C4×#157C2×D311yescharacteristic
55#3787C2×D311yescharacteristic
56#3776—31no—
57#3775—31no—
58#3775—31no—
59#3774—31no—
60#3743—21no—
61C8×C3⋊D3—21no—
62C8×C3⋊D3C2×D311yescharacteristic
63C3×ΓL(1,𝔽25)—61no—
64C3×ΓL(1,𝔽25)—61no—
65C3×ΓL(1,𝔽25)—121no—
66C3×D3×C8—61no—
67C3×D3×C8—61no—
68C3×D3×C8—61no—
69C3×D3×C8—61no—
70C3×D3×C8—121no—
71#3712—31no—
72#3712—31no—
73#3712—61no—
74#3711—61no—
75#3711—61no—
76#3711—61no—
77#3711—61no—
78#3710—61no—
79D3×C3⋊C8—61no—
80D3×C3⋊C8—61no—
81C2×C3×C3⋊D3—21no—
82C2×C3×C3⋊D3—21no—
83C2×C3×C3⋊D3#4111yescharacteristic
84C3×#157—41no—
85C33×C4C2×D411yescharacteristic
86C3×C3⋊Dic3—21no—
87C3×C3⋊Dic3—21no—
88C3×C3⋊Dic3C22×C411yescharacteristic
89#907—181no—
90#904—91no—
91#901—91no—
92#901—91no—
93#901—181no—
94C3×#97—91no—
95#819—91no—
96#805—91no—
97#805—91no—
98C2×#157—21no—
99C4×C3⋊D3—21no—
100C4×C3⋊D3—21no—
101C4×C3⋊D3C22×D311yescharacteristic
102C3×C4×D3—61no—
103C3×C4×D3—61no—
104C3×C4×D3—61no—
105C3×C4×D3—61no—
106C3×C4×D3—61no—
107C3×C4×D3—61no—
108C3×C4×D3—121no—
109C3×C4×D3—121no—
110#627—31no—
111#627—31no—
112#627—61no—
113#627—61no—
114#626—61no—
115#626—61no—
116#626—61no—
117#626—61no—
118#626—61no—
119#626—61no—
120#625—31no—
121#625—31no—
122#625—61no—
123#625—61no—
124#622C4×D311yescharacteristic
125#622C4×D311yescharacteristic
126C32×C8—21no—
127C32×C8—21no—
128C32×C8—41no—
129C32×C8—41no—
130C32×C8D1211yescharacteristic
131C3⋊(C3⋊C8)—21no—
132C3⋊(C3⋊C8)—21no—
133C3⋊(C3⋊C8)D1211yescharacteristic
134C3×C3⋊C8—61no—
135C3×C3⋊C8—61no—
136C3×C3⋊C8—61no—
137C3×C3⋊C8—61no—
138C3×C3⋊C8—61no—
139C3×C3⋊C8—61no—
140C3×C3⋊C8—121no—
141C3×C3⋊C8—121no—
142#444—271noSylow
143C2×C33C4×D411yescharacteristic
144C3×C3⋊D3—21no—
145C3×C3⋊D3—41no—
146C3×C3⋊D3—41no—
147#235—91no—
148#235—91no—
149#235—181no—
150#235—181no—
151C3×M16—181no—
152C2×C3×C8—91no—
153C2×C3×C8—91no—
154C2×C3×C8—181no—
155C3×C42—91no—
156#208—91no—
157#208—91no—
158#208—181no—
159C2×C3⋊C8—181no—
160Dic12—91no—
161D24—91no—
162#204—91no—
163#204—91no—
164ΓL(1,𝔽25)—61no—
165ΓL(1,𝔽25)—61no—
166ΓL(1,𝔽25)—121no—
167ΓL(1,𝔽25)—181no—
168ΓL(1,𝔽25)—181no—
169ΓL(1,𝔽25)—181no—
170ΓL(1,𝔽25)—181no—
171D3×C8—61no—
172D3×C8—61no—
173D3×C8—61no—
174D3×C8—61no—
175D3×C8—121no—
176D3×C8—181no—
177D3×C8—181no—
178C2×C3⋊D3—21no—
179C2×C3⋊D3—21no—
180C2×C3⋊D3#23511yescharacteristic
181C2×C3×D3—61no—
182C2×C3×D3—61no—
183C2×C3×D3—61no—
184C2×C3×D3—61no—
185C2×C3×D3—61no—
186C2×C3×D3—61no—
187C2×C3×D3—121no—
188C2×C3×D3—121no—
189#157—41no—
190C32×C4—21no—
191C32×C4—21no—
192C32×C4—41no—
193C32×C4—41no—
194C32×C4C2×D1211yescharacteristic
195C3⋊Dic3—21no—
196C3⋊Dic3—21no—
197C3⋊Dic3C2×C4×D311yescharacteristic
198C3×Dic3—61no—
199C3×Dic3—61no—
200C3×Dic3—61no—
201C3×Dic3—61no—
202C3×Dic3—61no—
203C3×Dic3—61no—
204C3×Dic3—121no—
205C3×Dic3—121no—
206#135—271no—
207#131—271no—
208#131—271no—
209#108—271no—
210#104—271no—
211#104—271no—
212#97—91no—
213C33#44411yescharacteristic, complemented, Sylow
214C2×C3×C4—91no—
215C2×C3×C4—181no—
216C2×C3×C4—181no—
217C2×C3×C4—181no—
218#67—181no—
219#67—181no—
220#67—181no—
221#67—181no—
222#67—181no—
223#67—181no—
224D12—91no—
225D12—91no—
226D12—181no—
227D12—181no—
228C4×D3—61no—
229C4×D3—61no—
230C4×D3—61no—
231C4×D3—61no—
232C4×D3—61no—
233C4×D3—61no—
234C4×D3—121no—
235C4×D3—121no—
236Dic6—91no—
237Dic6—91no—
238Dic6—181no—
239Dic6—181no—
240C3×C8—21no—
241C3×C8—21no—
242C3×C8—41no—
243C3×C8—41no—
244C3×C8—91no—
245C3×C8—91no—
246C3×C8—91no—
247C3×C8—181no—
248C3×C8—181no—
249C3×C8GO+(4,𝔽2)11yescharacteristic
250C3⋊C8—61no—
251C3⋊C8—61no—
252C3⋊C8—61no—
253C3⋊C8—61no—
254C3⋊C8—61no—
255C3⋊C8—61no—
256C3⋊C8—121no—
257C3⋊C8—121no—
258C2×C32—21no—
259C2×C32—21no—
260C2×C32—41no—
261C2×C32—41no—
262C2×C32C4×D1211yescharacteristic
263C3⋊D3—21no—
264C3⋊D3—41no—
265C3⋊D3—41no—
266C3×D3—121no—
267C3×D3—121no—
268C3×D3—121no—
269C3×D3—121no—
270C3×D3—122no—
271C3×D3—121no—
272C3×D3—121no—
273C3×D3—241no—
274#41—271no—
275#41—271no—
276Q16—271no—
277SD16—271no—
278SD16—271no—
279D8—271no—
280M16—181no—
281M16—271no—
282M16—271no—
283M16—541no—
284C2×C8—91no—
285C2×C8—91no—
286C2×C8—181no—
287C2×C8—541no—
288C42—91no—
289C22×C3—91no—
290C22×C3—181no—
291C22×C3—181no—
292C2×D3—61no—
293C2×D3—61no—
294C2×D3—61no—
295C2×D3—61no—
296C2×D3—61no—
297C2×D3—61no—
298C2×D3—121no—
299C2×D3—121no—
300C3×C4—21no—
301C3×C4—21no—
302C3×C4—41no—
303C3×C4—41no—
304C3×C4—91no—
305C3×C4—181no—
306C3×C4—181no—
307C3×C4—361no—
308C3×C4C2×GO+(4,𝔽2)11yescharacteristic
309Dic3—61no—
310Dic3—61no—
311Dic3—61no—
312Dic3—61no—
313Dic3—61no—
314Dic3—61no—
315Dic3—121no—
316Dic3—121no—
317C32—21no—
318C32—21no—
319C32—41no—
320C32—41no—
321C32#471711yescharacteristic, complemented
322Q8—271no—
323Q8—271no—
324D4—271no—
325D4—271no—
326D4—541no—
327D4—541no—
328C2×C4—91no—
329C2×C4—181no—
330C2×C4—181no—
331C2×C4—181no—
332C8—91no—
333C8—91no—
334C8—91no—
335C8—181no—
336C8—181no—
337C8#157⋊D311yescharacteristic
338C2×C3—21no—
339C2×C3—21no—
340C2×C3—41no—
341C2×C3—41no—
342C2×C3—181no—
343C2×C3—361no—
344C2×C3—361no—
345C2×C3C4×GO+(4,𝔽2)11yescharacteristic
346D3—121no—
347D3—121no—
348D3—121no—
349D3—121no—
350D3—122no—
351D3—121no—
352D3—121no—
353D3—241no—
354C22—91no—
355C22—181no—
356C22—181no—
357C4—91no—
358C4—181no—
359C4—181no—
360C4—361no—
361C4C2×#157⋊D311yescharacteristic
362C3—21no—
363C3—21no—
364C3—41no—
365C3—41no—
366C3#1059329811yescharacteristic, complemented
367C2—181no—
368C2—361no—
369C2—361no—
370C2C4×#157⋊D311yescharacteristic
subgroupG/H
center, FrattiniC4C2×#157⋊D3
commutatorC3×C3⋊Dic3C22×C4
FittingC33×C8D4
socleC2×C33C4×D4
prime radical#11724337C2
cosocleC3×C4×C3⋊D3C23
hypercenterC8#157⋊D3
hypocenterC33#444
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
2. #11723652C2semidirect
3. #11723651C2semidirect
4. C3×#63767C2semidirect
5. #11722787C2semidirect
6. #11722785C2semidirect
7. #11722785C2semidirect
13. #88943C4semidirect
14. #88942C4semidirect
15. #88942C4semidirect
16. #88941C4semidirect
29. #63767D3semidirect
213. C33#444semidirect
321. C32#4717semidirect
366. C3#10593298semidirect

Maximal subgroups

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

subgroupindex
#117243372
#117236522
#117236512
C3×#637672
#117227872
#117227852
#105932983
#47179