Finite Group Databasebeta

← #13165188

Subgroups — #13165188

Order 1728.

374 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 60 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
1C3×#64223C211yesmaximal, characteristic
2#11722823C211yesmaximal, characteristic, complemented
3#11722821C211yesmaximal, characteristic, complemented
4#11722787C211yesmaximal, characteristic, complemented
5#11722762C211yesmaximal, characteristic, complemented
6#11722762C211yesmaximal, characteristic, complemented
7#11722705C211yesmaximal, characteristic, complemented
8#10593298—31nomaximal
9C3×#3798C2211yescharacteristic
10C3×#3798C2211yescharacteristic
11C4×C3⋊#157C2211yescharacteristic
12C3×#3787C2211yescharacteristic
13C3×#3776C411yescharacteristic, complemented
14C3×#3775C411yescharacteristic, complemented
15C3×#3775C411yescharacteristic, complemented
16C3×#3774C411yescharacteristic, complemented
17#88809C2211yescharacteristic
18#88809C2211yescharacteristic
19#88786—21no—
20C3⋊D3×C3⋊C8C2211yescharacteristic
21D3×C3⋊(C3⋊C8)—21no—
22C3×#3711—21no—
23C3×D3×C3⋊C8—21no—
24#64223D311yescharacteristic
25#63815—61no—
26#63814—61no—
27#63805—31no—
28#63803—31no—
29#63767—31no—
30#63733—31no—
31#63731—31no—
32#63731—31no—
33C2×C3⋊#157—21no—
34C3×C4×C3⋊D3C2311yescharacteristic
35C32×C4×D3—21no—
36C32×C4×D3—21no—
37C3×#627C2×C411yescharacteristic
38C3×#627C2×C411yescharacteristic
39C3×#626—21no—
40C3×#626—21no—
41C3×#625C2×C411yescharacteristic
42C3×#625C2×C411yescharacteristic
43C3×#622C2×C411yescharacteristic
44C3×#622C2×C411yescharacteristic
45C3⋊(C3⋊(C3⋊C8))D411yescharacteristic
46C3×C3⋊(C3⋊C8)—21no—
47C3×C3⋊(C3⋊C8)—21no—
48C32×C3⋊C8D411yescharacteristic
49#5197—91nomaximal
50C3×#235—61no—
51C3×#235—61no—
52#3798C2×D311yescharacteristic
53#3798C2×D311yescharacteristic
54C4×#157—31no—
55#3787C2×D311yescharacteristic
56#3776Dic311yescharacteristic, complemented
57#3775Dic311yescharacteristic, complemented
58#3775Dic311yescharacteristic, complemented
59#3774Dic311yescharacteristic, complemented
60#3748—61no—
61C2×C3⋊(C3⋊C8)—61no—
62C8×C3⋊D3—31no—
63C3×#208—61no—
64C2×C3×C3⋊C8—61no—
65C3×ΓL(1,𝔽25)—61no—
66C3×D3×C8—61no—
67#3712—31no—
68#3712—31no—
69#3711—21no—
70#3711—61no—
71#3711—61no—
72#3711—61no—
73#3711—61no—
74#3711—121no—
75D3×C3⋊C8—21no—
76D3×C3⋊C8—61no—
77D3×C3⋊C8—61no—
78D3×C3⋊C8—61no—
79D3×C3⋊C8—61no—
80D3×C3⋊C8—121no—
81C2×C3×C3⋊D3#4111yescharacteristic
82C2×C32×D3—21no—
83C2×C32×D3—21no—
84C3⋊#157—41no—
85C33×C4C2×D411yescharacteristic
86C3×C3⋊Dic3C22×C411yescharacteristic
87C32×Dic3—21no—
88C32×Dic3—21no—
89C3×#135—91no—
90#948—91no—
91#948—91no—
92#907—181no—
93#901—181no—
94#837—91no—
95#837—91no—
96#819—91no—
97#815—91no—
98C2×#157—61no—
99C2×C32×C4—61no—
100C2×C32×C4—61no—
101C4×C3⋊D3C22×D311yescharacteristic
102C3×#67—61no—
103C3×#67—61no—
104C3×#67—61no—
105C3×#67—61no—
106C3×D12—61no—
107C3×D12—61no—
108C3×C4×D3—21no—
109C3×C4×D3—21no—
110C3×C4×D3—21no—
111C3×C4×D3—21no—
112C3×C4×D3—41no—
113C3×C4×D3—41no—
114C3×C4×D3—61no—
115C3×C4×D3—61no—
116C3×Dic6—61no—
117C3×Dic6—61no—
118#627C2×Dic311yescharacteristic
119#627C2×Dic311yescharacteristic
120#626—21no—
121#626—21no—
122#625C2×Dic311yescharacteristic
123#625C2×Dic311yescharacteristic
124#622C2×Dic311yescharacteristic
125#622C2×Dic311yescharacteristic
126C32×C8—31no—
127C3⋊(C3⋊C8)—21no—
128C3⋊(C3⋊C8)—21no—
129C3⋊(C3⋊C8)—31no—
130C3⋊(C3⋊C8)—61no—
131C3⋊(C3⋊C8)—61no—
132C3⋊(C3⋊C8)—121no—
133C3⋊(C3⋊C8)—121no—
134C3×C3⋊C8—21no—
135C3×C3⋊C8—21no—
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—
145C32×D3—41no—
146C32×D3—41no—
147C3×#41—91no—
148C3×#41—91no—
149#235—61no—
150#235—61no—
151C3×Q16—91no—
152C3×SD16—91no—
153C3×SD16—91no—
154C3×D8—91no—
155C3×M16—181no—
156C2×C3×C8—91no—
157C2×C3×C8—181no—
158C4×Dic3—91no—
159#208—61no—
160#208—91no—
161#208—91no—
162#208—181no—
163#208—181no—
164#208—361no—
165C2×C3⋊C8—61no—
166C2×C3⋊C8—91no—
167C2×C3⋊C8—181no—
168C2×C3⋊C8—181no—
169C2×C3⋊C8—361no—
170ΓL(1,𝔽25)—61no—
171ΓL(1,𝔽25)—181no—
172ΓL(1,𝔽25)—181no—
173ΓL(1,𝔽25)—181no—
174D3×C8—61no—
175D3×C8—181no—
176D3×C8—181no—
177D3×C8—181no—
178C22×C32—61no—
179C22×C32—61no—
180C2×C3⋊D3#23711yescharacteristic
181C2×C3×D3—21no—
182C2×C3×D3—21no—
183C2×C3×D3—21no—
184C2×C3×D3—21no—
185C2×C3×D3—41no—
186C2×C3×D3—41no—
187C2×C3×D3—61no—
188C2×C3×D3—61no—
189#157—121no—
190C32×C4—21no—
191C32×C4—21no—
192C32×C4—41no—
193C32×C4—41no—
194C32×C4—61no—
195C32×C4—61no—
196C32×C4D3×D411yescharacteristic
197C3⋊Dic3C22×Dic311yescharacteristic
198C3×Dic3—21no—
199C3×Dic3—21no—
200C3×Dic3—21no—
201C3×Dic3—21no—
202C3×Dic3—41no—
203C3×Dic3—41no—
204C3×Dic3—61no—
205C3×Dic3—61no—
206#135—91no—
207#131—271no—
208#131—271no—
209#108—271no—
210#104—271no—
211#104—271no—
212#97—271no—
213C33#44411yescharacteristic, complemented, Sylow
214C3×Q8—91no—
215C3×Q8—91no—
216C3×D4—91no—
217C3×D4—91no—
218C3×D4—181no—
219C3×D4—181no—
220C2×C3×C4—61no—
221C2×C3×C4—61no—
222C2×C3×C4—61no—
223C2×C3×C4—61no—
224C2×C3×C4—91no—
225C2×C3×C4—121no—
226C2×C3×C4—121no—
227#67—61no—
228#67—61no—
229#67—61no—
230#67—61no—
231C2×Dic3—181no—
232D12—61no—
233D12—61no—
234C4×D3—21no—
235C4×D3—21no—
236C4×D3—61no—
237C4×D3—61no—
238Dic6—61no—
239Dic6—61no—
240C3×C8—61no—
241C3×C8—61no—
242C3×C8—91no—
243C3×C8—91no—
244C3×C8—181no—
245C3×C8—181no—
246C3⋊C8—61no—
247C3⋊C8—61no—
248C3⋊C8—61no—
249C3⋊C8—61no—
250C3⋊C8—91no—
251C3⋊C8—121no—
252C3⋊C8—121no—
253C3⋊C8—181no—
254C3⋊C8—181no—
255C3⋊C8—361no—
256C3⋊C8—361no—
257C3⋊C8GO+(4,𝔽2)11yescharacteristic
258C2×C32—21no—
259C2×C32—21no—
260C2×C32—41no—
261C2×C32—41no—
262C2×C32—121no—
263C2×C32—121no—
264C2×C32D4×Dic311yescharacteristic
265C3⋊D3—21no—
266C3×D3—41no—
267C3×D3—41no—
268C3×D3—42no—
269C3×D3—41no—
270C3×D3—41no—
271C3×D3—81no—
272C3×D3—121no—
273C3×D3—121no—
274#41—91no—
275#41—91no—
276Q16—91no—
277SD16—91no—
278SD16—91no—
279D8—91no—
280M16—181no—
281M16—271no—
282M16—271no—
283M16—541no—
284C2×C8—91no—
285C2×C8—181no—
286C2×C8—271no—
287C2×C8—541no—
288C42—271no—
289C22×C3—61no—
290C22×C3—61no—
291C22×C3—61no—
292C22×C3—61no—
293C22×C3—91no—
294C22×C3—121no—
295C22×C3—121no—
296C2×D3—21no—
297C2×D3—21no—
298C2×D3—61no—
299C2×D3—61no—
300C3×C4—21no—
301C3×C4—21no—
302C3×C4—41no—
303C3×C4—41no—
304C3×C4—61no—
305C3×C4—61no—
306C3×C4—61no—
307C3×C4—61no—
308C3×C4—91no—
309C3×C4—121no—
310C3×C4—121no—
311C3×C4C2×GO+(4,𝔽2)11yescharacteristic
312Dic3—21no—
313Dic3—21no—
314Dic3—61no—
315Dic3—61no—
316Dic3—361no—
317C32—21no—
318C32—21no—
319C32—41no—
320C32—41no—
321C32#519711yescharacteristic, complemented
322Q8—91no—
323Q8—91no—
324D4—91no—
325D4—91no—
326D4—181no—
327D4—181no—
328C2×C4—61no—
329C2×C4—61no—
330C2×C4—91no—
331C2×C4—541no—
332C8—31no—
333C8—91no—
334C8—91no—
335C8—181no—
336C8—181no—
337C8—271no—
338C2×C3—21no—
339C2×C3—21no—
340C2×C3—41no—
341C2×C3—41no—
342C2×C3—121no—
343C2×C3—121no—
344C2×C3—122no—
345C2×C3—121no—
346C2×C3—121no—
347C2×C3—181no—
348C2×C3—241no—
349C2×C3C4×GO+(4,𝔽2)11yescharacteristic
350D3—41no—
351D3—41no—
352D3—121no—
353D3—121no—
354C22—61no—
355C22—61no—
356C22—91no—
357C4—61no—
358C4—61no—
359C4—91no—
360C4—1081no—
361C4D3×GO+(4,𝔽2)11yescharacteristic
362C3—21no—
363C3—21no—
364C3—41no—
365C3—41no—
366C3#1059329811yescharacteristic, complemented
367C2—121no—
368C2—121no—
369C2—181no—
370C2Dic3×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
center, Frattini, hypercenterC4D3×GO+(4,𝔽2)
commutatorC3×C3⋊Dic3C22×C4
FittingC33×C4C2×D4
socleC2×C33C4×D4
prime radicalC3×#64223C2
cosocleC3×C4×C3⋊D3C23
hypocenterC33#444
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
2. #11722823C2semidirect
3. #11722821C2semidirect
4. #11722787C2semidirect
5. #11722762C2semidirect
6. #11722762C2semidirect
7. #11722705C2semidirect
13. C3×#3776C4semidirect
14. C3×#3775C4semidirect
15. C3×#3775C4semidirect
16. C3×#3774C4semidirect
56. #3776Dic3semidirect
57. #3775Dic3semidirect
58. #3775Dic3semidirect
59. #3774Dic3semidirect
213. C33#444semidirect
321. C32#5197semidirect
366. C3#10593298semidirect

Maximal subgroups

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

subgroupindex
C3×#642232
#117228232
#117228212
#117227872
#117227622
#117227052
#105932983
#51979