Finite Group Databasebeta

← #13048713

Subgroups — #13048713

Order 1296.

533 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 533 conjugacy classes of subgroups — too many to draw. Only its 74 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#10619994C211yesmaximal, characteristic, complemented
2#10619977C211yesmaximal, characteristic, complemented
3C2×C3×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
4#10619887C211yesmaximal, characteristic, complemented
5#10619885C211yesmaximal, characteristic, complemented
6#10619883C211yesmaximal, characteristic, complemented
7#10619881C211yesmaximal, characteristic, complemented
8#89046—31nomaximal
9#89031—31nomaximal
10#88671—33nomaximal
11C22×C3×H(3,𝔽3)C2211yescharacteristic
12C2×C3⋊#272C2211yescharacteristic
13C2×C3⋊((C3⋊D3)⋊C3)C2211yescharacteristic
14C3⋊#1084C2211yescharacteristic
15C3⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16C3×(C3⋊Dic3)⋊C3C2211yescharacteristic
17C3×(C3⋊Dic3)⋊C3C2211yescharacteristic
18#6312D311yescharacteristic, complemented
19C3×#638—31no—
20C3×#638—31no—
21C3×#638—31no—
22C3×#638—61no—
23C2×C3×C3⋊Dic3D311yescharacteristic, complemented
24C2×C32×Dic3—31no—
25#6296—31no—
26#6295—31no—
27#6294—31no—
28#6294—31no—
29#6293—31no—
30#6292—31no—
31#6289—31no—
32Dic3×C3⋊D3—31no—
33#6235—33no—
34#6235—63no—
35#638⋊C3—33no—
36C2×(C3⋊Dic3)⋊C3D313yescomplemented
37#6200—33no—
38#6199—33no—
39((C3⋊D3)⋊C3)⋊C4—33no—
40#6196—33no—
41C2×C3×H(3,𝔽3)—21no—
42C2×C3×H(3,𝔽3)C2311yescharacteristic
43C3⋊#272—21no—
44C3⋊((C3⋊D3)⋊C3)—21no—
45#3828—91no—
46#3805—91no—
47#3804—93no—
48#3804—91no—
49#3804—91no—
50C22×C33—21no—
51C22×C33C2×D311yescharacteristic
52C22×C33C2×D311yescharacteristic
53C2×C3⋊(C3⋊D3)C2×D311yescharacteristic
54C2×C3×C3⋊D3—31no—
55C2×C3×C3⋊D3—31no—
56C2×C3×C3⋊D3—31no—
57C2×C3×C3⋊D3—61no—
58C3⋊(C3⋊Dic3)C2×D311yescharacteristic
59C3×C3⋊Dic3—31no—
60C3×C3⋊Dic3—31no—
61C3×C3⋊Dic3—31no—
62C3×C3⋊Dic3—61no—
63C3×C3⋊Dic3C2×D311yescharacteristic
64C3×C3⋊Dic3C2×D311yescharacteristic
65C32×Dic3—31no—
66C32×Dic3—31no—
67C22×H(3,𝔽3)—23no—
68C22×H(3,𝔽3)C2×D313yes—
69C2×#272—33no—
70C2×#272—63no—
71C2×(C3⋊D3)⋊C3—33no—
72#1084—33no—
73#1084—63no—
74(C3⋊Dic3)⋊C3—33no—
75(C3⋊Dic3)⋊C3C2×D313yes—
76(C3⋊Dic3)⋊C3C2×D313yes—
77C3×H(3,𝔽3)#4111yescharacteristic, complemented, Sylow
78C2×C32×C4—91no—
79#638—31no—
80#638—31no—
81#638—31no—
82#638—33no—
83#638—61no—
84#638—91no—
85#638—181no—
86C2×C3⋊Dic3C3⋊D311yescharacteristic, complemented
87C4⋊(C3⋊D3)—91no—
88C4×C3⋊D3—91no—
89C4×C3⋊D3—91no—
90#634—91no—
91C3×#67—91no—
92C3×#67—93no—
93C3×#67—91no—
94C3×#67—91no—
95C3×#67—91no—
96C3×#67—183no—
97C3×#67—181no—
98C3×#67—181no—
99C2×C3×Dic3—33no—
100C2×C3×Dic3—31no—
101C2×C3×Dic3—91no—
102#627—93no—
103#627—91no—
104#627—91no—
105#627—91no—
106#626—93no—
107#626—91no—
108#626—91no—
109#625—91no—
110#624—93no—
111#624—91no—
112#624—91no—
113D3×Dic3—93no—
114D3×Dic3—91no—
115D3×Dic3—91no—
116D3×Dic3—91no—
117D3×Dic3—91no—
118C2×C33—21no—
119C2×C33—21no—
120C2×C33—21no—
121C2×C33—41no—
122C2×C33C22×D311yescharacteristic
123C2×C33C22×D311yescharacteristic
124C3⋊(C3⋊D3)—21no—
125C3×C3⋊D3—61no—
126C3×C3⋊D3—61no—
127C3×C3⋊D3—61no—
128C3×C3⋊D3—121no—
129C2×H(3,𝔽3)—23no—
130C2×H(3,𝔽3)—23no—
131C2×H(3,𝔽3)—43no—
132C2×H(3,𝔽3)C22×D313yes—
133#272—63no—
134#272—123no—
135(C3⋊D3)⋊C3—63no—
136#237—271no—
137#237—271no—
138#235—271no—
139#235—273no—
140C22×C32—21no—
141C22×C32—23no—
142C22×C32—31no—
143C22×C32—31no—
144C22×C32—61no—
145C22×C32—61no—
146C22×C32—61no—
147C22×C32—121no—
148C22×C32D3211yescharacteristic
149C22×C32D3213yes—
150C22×C32C2×C3⋊D311yescharacteristic
151C2×C3⋊D3—31no—
152C2×C3⋊D3—31no—
153C2×C3⋊D3—31no—
154C2×C3⋊D3—33no—
155C2×C3⋊D3—61no—
156C2×C3⋊D3—91no—
157C2×C3⋊D3—181no—
158C2×C3×D3—91no—
159C2×C3×D3—93no—
160C2×C3×D3—91no—
161C2×C3×D3—91no—
162C2×C3×D3—91no—
163C2×C3×D3—183no—
164C2×C3×D3—181no—
165C2×C3×D3—181no—
166C32×C4—91no—
167C32×C4—91no—
168C3⋊Dic3—31no—
169C3⋊Dic3—31no—
170C3⋊Dic3—31no—
171C3⋊Dic3—33no—
172C3⋊Dic3—61no—
173C3⋊Dic3—91no—
174C3⋊Dic3—181no—
175C3⋊Dic3C2×C3⋊D311yescharacteristic
176C3⋊Dic3C2×C3⋊D311yescharacteristic
177C3×Dic3—33no—
178C3×Dic3—33no—
179C3×Dic3—31no—
180C3×Dic3—31no—
181C3×Dic3—91no—
182C3×Dic3—91no—
183C3×Dic3—91no—
184C3×Dic3—93no—
185C3×Dic3—91no—
186C3×Dic3—91no—
187C3×Dic3—91no—
188C3×Dic3—183no—
189C3×Dic3—181no—
190C3×Dic3—181no—
191C33—21no—
192C33#23511yescharacteristic, complemented
193C33#23711yescharacteristic, complemented
194H(3,𝔽3)—23no—
195H(3,𝔽3)#23513yescomplemented
196C3×D4—271no—
197C3×D4—271no—
198C2×C3×C4—93no—
199C2×C3×C4—91no—
200#67—93no—
201#67—91no—
202#67—91no—
203#67—91no—
204#67—91no—
205#67—181no—
206#67—181no—
207#67—183no—
208#67—273no—
209#67—271no—
210#67—271no—
211C2×Dic3—31no—
212C2×Dic3—91no—
213C2×Dic3—271no—
214C2×Dic3—271no—
215D12—271no—
216D12—273no—
217C4×D3—271no—
218C4×D3—271no—
219C4×D3—271no—
220C4×D3—273no—
221C4×D3—271no—
222C4×D3—273no—
223Dic6—271no—
224Dic6—271no—
225Dic6—271no—
226Dic6—273no—
227C2×C32—21no—
228C2×C32—23no—
229C2×C32—21no—
230C2×C32—21no—
231C2×C32—23no—
232C2×C32—31no—
233C2×C32—31no—
234C2×C32—43no—
235C2×C32—41no—
236C2×C32—61no—
237C2×C32—61no—
238C2×C32—61no—
239C2×C32—61no—
240C2×C32—61no—
241C2×C32—121no—
242C2×C32—121no—
243C2×C32—121no—
244C2×C32—121no—
245C2×C32—122no—
246C2×C32C2×D3211yescharacteristic
247C2×C32C2×D3213yes—
248C2×C32C22×C3⋊D311yescharacteristic
249C3⋊D3—61no—
250C3⋊D3—61no—
251C3⋊D3—61no—
252C3⋊D3—63no—
253C3⋊D3—121no—
254C3⋊D3—181no—
255C3⋊D3—361no—
256C3×D3—181no—
257C3×D3—181no—
258C3×D3—183no—
259C3×D3—181no—
260C3×D3—181no—
261C3×D3—363no—
262C3×D3—361no—
263C3×D3—361no—
264#41—811noSylow
265C22×C3—21no—
266C22×C3—31no—
267C22×C3—33no—
268C22×C3—61no—
269C22×C3—63no—
270C22×C3—271no—
271C22×C3—271no—
272C22×C3((C3⋊D3)⋊C3)⋊C211yescharacteristic
273C22×C3D3×C3⋊D311yescharacteristic
274C2×D3—93no—
275C2×D3—91no—
276C2×D3—91no—
277C2×D3—91no—
278C2×D3—91no—
279C2×D3—181no—
280C2×D3—181no—
281C2×D3—183no—
282C2×D3—273no—
283C3×C4—93no—
284C3×C4—93no—
285C3×C4—91no—
286C3×C4—91no—
287C3×C4—271no—
288C3×C4—271no—
289Dic3—31no—
290Dic3—31no—
291Dic3—91no—
292Dic3—91no—
293Dic3—93no—
294Dic3—91no—
295Dic3—91no—
296Dic3—91no—
297Dic3—91no—
298Dic3—181no—
299Dic3—181no—
300Dic3—183no—
301Dic3—273no—
302C32—21no—
303C32—23no—
304C32—31no—
305C32—31no—
306C32—61no—
307C32—61no—
308C32—61no—
309C32—121no—
310C32#380411yescharacteristic
311C32#380413yescomplemented
312C32#382811yescharacteristic, complemented
313Q8—811no—
314D4—271no—
315D4—271no—
316D4—811no—
317C2×C4—91no—
318C2×C4—811no—
319C2×C4—811no—
320C2×C3—21no—
321C2×C3—21no—
322C2×C3—21no—
323C2×C3—31no—
324C2×C3—33no—
325C2×C3—41no—
326C2×C3—63no—
327C2×C3—61no—
328C2×C3—61no—
329C2×C3—63no—
330C2×C3—121no—
331C2×C3—123no—
332C2×C3—541no—
333C2×C3—541no—
334C2×C3C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
335C2×C3C2×D3×C3⋊D311yescharacteristic
336D3—183no—
337D3—181no—
338D3—181no—
339D3—181no—
340D3—181no—
341D3—361no—
342D3—361no—
343D3—363no—
344D3—543no—
345C22—271no—
346C22—271no—
347C22((C3⋊D3)⋊C3)⋊D311yescharacteristic
348C4—91no—
349C4—91no—
350C4—271no—
351C4—271no—
352C3—21no—
353C3—33no—
354C3—31no—
355C3—61no—
356C3—63no—
357C3#8867111yescharacteristic, complemented
358C3#8903111yescharacteristic
359C2—21no—
360C2—541no—
361C2—541no—
362C2C2×((C3⋊D3)⋊C3)⋊D311yescharacteristic
subgroupG/H
centerC2C2×((C3⋊D3)⋊C3)⋊D3
commutator, cosocleC2×C3×H(3,𝔽3)C23
FrattiniC2×C3C2×D3×C3⋊D3
FittingC22×C3×H(3,𝔽3)C22
socleC2×C32C2×D32
hypercenterC22((C3⋊D3)⋊C3)⋊D3
hypocenterC3×H(3,𝔽3)#41
LCSG ▷ C2×C3×H(3,𝔽3) ▷ C3×H(3,𝔽3)
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. #10619994C2semidirect
2. #10619977C2semidirect
3. C2×C3×(C3⋊Dic3)⋊C3C2semidirect
4. #10619887C2semidirect
5. #10619885C2semidirect
6. #10619883C2semidirect
7. #10619881C2semidirect
18. #6312D3semidirect
23. C2×C3×C3⋊Dic3D3semidirect
36. C2×(C3⋊Dic3)⋊C3D3semidirect
77. C3×H(3,𝔽3)#41semidirect
86. C2×C3⋊Dic3C3⋊D3semidirect
192. C33#235semidirect
193. C33#237semidirect
195. H(3,𝔽3)#235semidirect
311. C32#3804semidirect
312. C32#3828semidirect
357. C3#88671semidirect

Maximal subgroups

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

subgroupindex
#106199942
#106199772
C2×C3×(C3⋊Dic3)⋊C32
#106198872
#106198852
#106198832
#106198812
#890463
#890313
#886713