Finite Group Databasebeta

← #72252733

Subgroups — #72252733

Order 3456.

7252 conjugacy classes of subgroups; only the 673 normal subgroups are shown.

normal subgroups only

This group has 7252 conjugacy classes of subgroups — too many to draw. Only its 673 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#13164865C2110yesmaximal, complemented
2C2×#11722577C215yesmaximal, complemented
3C3×#87536C313yesmaximal, complemented
4C32×H(7,𝔽2)C311yesmaximal, characteristic, complemented
5H(3,𝔽3)×2−1+4—41nomaximal
62−1+4⋊H(3,𝔽3)C22110yescomplemented
7#11722577C22110yescomplemented
8C2×#41⋊H(3,𝔽3)C22115yescomplemented
9C32×#585—35no—
10C32×#585—31no—
11C32×#585C2×C3110yescomplemented
12C2×C32×H(5,𝔽2)—310no—
13C2×C32×H(5,𝔽2)C2×C315yescomplemented
14C3×#5945C2×C3130yescomplemented
15C2×C3×#994C2×C3115yescomplemented
16#41×H(3,𝔽3)—410no—
17C2×Q8×H(3,𝔽3)—45no—
18#41⋊H(3,𝔽3)C23110yescomplemented
19C2×Q8⋊H(3,𝔽3)C2315yescomplemented
20C3×H(7,𝔽2)—31no—
21C3×H(7,𝔽2)C3211yescharacteristic
22#87536—33no—
23C32×2−1+4—35no—
24C32×2−1+4—310no—
25C32×2−1+4—43no—
26C32×2−1+4A411yescharacteristic
27C32×2−1+4C22×C3110yes—
28C32×H(5,𝔽2)—320no—
29C32×H(5,𝔽2)—330no—
30C32×H(5,𝔽2)—310no—
31C32×H(5,𝔽2)—330no—
32C32×H(5,𝔽2)C22×C3110yes—
33C2×C32×#41—330no—
34C2×C32×#41—320no—
35C2×C32×#41—310no—
36C2×C32×#41—35no—
37C2×C32×#41C22×C3115yes—
38C22×C32×D4—315no—
39C22×C32×D4—320no—
40C3×2−1+4⋊C3C22×C3130yescomplemented
41C3×#994C22×C3130yescomplemented
42C2×C3×#41⋊C3C22×C3145yescomplemented
43Q8×H(3,𝔽3)—410no—
44D4×H(3,𝔽3)—410no—
45C2×C4×H(3,𝔽3)—415no—
46Q8⋊H(3,𝔽3)C2411yescharacteristic
47C3×#585—35no—
48C3×#585—315no—
49C3×#585—33no—
50C3×#585—31no—
51C3×#585—310no—
52C3×#585C2×C32110yes—
53C2×C3×H(5,𝔽2)—310no—
54C2×C3×H(5,𝔽2)—330no—
55C2×C3×H(5,𝔽2)—35no—
56C2×C3×H(5,𝔽2)C2×C3215yes—
57#5945—330no—
58C2×#994—315no—
59C24×C32—310no—
60C32×#41—360no—
61C32×#41—310no—
62C32×#41—320no—
63C32×#41—310no—
64C32×#41—320no—
65C32×#41—330no—
66C32×#41—330no—
67C32×#41—430no—
68C32×#41C2×A4110yes—
69C32×#41C23×C3110yes—
70C2×C32×Q8—310no—
71C2×C32×Q8—310no—
72C2×C32×Q8—415no—
73C2×C32×Q8C2×A415yes—
74C2×C32×Q8C23×C315yes—
75C2×C32×D4—360no—
76C2×C32×D4—360no—
77C2×C32×D4—315no—
78C2×C32×D4—330no—
79C2×C32×D4—330no—
80C2×C32×D4—315no—
81C22×C32×C4—320no—
82C22×C32×C4—315no—
83C3×#41⋊C3C23×C3130yescomplemented
84C2×C3×SL(2,𝔽3)C23×C3115yescomplemented
85H(7,𝔽2)H(3,𝔽3)11yescharacteristic, complemented, Sylow
86C22×H(3,𝔽3)—45no—
87C4×H(3,𝔽3)—410no—
88C3×2−1+4—315no—
89C3×2−1+4—330no—
90C3×2−1+4—35no—
91C3×2−1+4—310no—
92C3×2−1+4—31no—
93C3×2−1+4—310no—
94C3×2−1+4—123no—
95C3×2−1+4C3×A411yescharacteristic
96C3×2−1+4C22×C32110yes—
97C3×H(5,𝔽2)—320no—
98C3×H(5,𝔽2)—360no—
99C3×H(5,𝔽2)—390no—
100C3×H(5,𝔽2)—330no—
101C3×H(5,𝔽2)—330no—
102C3×H(5,𝔽2)—390no—
103C3×H(5,𝔽2)—310no—
104C3×H(5,𝔽2)—330no—
105C3×H(5,𝔽2)—310no—
106C3×H(5,𝔽2)C22×C32110yes—
107C2×C3×#41—330no—
108C2×C3×#41—390no—
109C2×C3×#41—360no—
110C2×C3×#41—320no—
111C2×C3×#41—330no—
112C2×C3×#41—315no—
113C2×C3×#41—310no—
114C2×C3×#41—35no—
115C2×C3×#41—315no—
116C2×C3×#41C22×C32115yes—
117C22×C3×D4—360no—
118C22×C3×D4—320no—
119C22×C3×D4—345no—
120C22×C3×D4—315no—
1212−1+4⋊C3—330no—
122#994—330no—
123C2×#41⋊C3—345no—
124C23×C32—320no—
125C23×C32—315no—
126C23×C32—2410no—
127C32×Q8—35no—
128C32×Q8—310no—
129C32×Q8—430no—
130C32×Q8C22×A4110yes—
131C32×Q8C24×C311yescharacteristic
132C32×D4—310no—
133C32×D4—330no—
134C32×D4—320no—
135C32×D4—330no—
136C32×D4—430no—
137C32×D4C22×A4110yes—
138C2×C32×C4—320no—
139C2×C32×C4—35no—
140C2×C32×C4—330no—
141C2×C32×C4—310no—
142C2×C32×C4—445no—
143C2×C32×C4C22×A4115yes—
144C3×SL(2,𝔽3)C24×C313yes—
145#585—35no—
146#585—31no—
147#585C2×H(3,𝔽3)110yescomplemented
148C2×H(5,𝔽2)—310no—
149C2×H(5,𝔽2)C2×H(3,𝔽3)15yescomplemented
150C2×H(3,𝔽3)—41no—
151C2×H(3,𝔽3)—85no—
152C24×C3—330no—
153C24×C3—310no—
154C3×#41—3180no—
155C3×#41—360no—
156C3×#41—330no—
157C3×#41—360no—
158C3×#41—330no—
159C3×#41—360no—
160C3×#41—390no—
161C3×#41—390no—
162C3×#41—310no—
163C3×#41—320no—
164C3×#41—310no—
165C3×#41—320no—
166C3×#41—330no—
167C3×#41—330no—
168C3×#41—310no—
169C3×#41—310no—
170C3×#41—1230no—
171C3×#41C2×C3×A4110yes—
172C3×#41C23×C32110yes—
173C2×C3×Q8—310no—
174C2×C3×Q8—330no—
175C2×C3×Q8—330no—
176C2×C3×Q8—310no—
177C2×C3×Q8—35no—
178C2×C3×Q8—35no—
179C2×C3×Q8—1215no—
180C2×C3×Q8C2×C3×A415yes—
181C2×C3×Q8C23×C3215yes—
182C2×C3×D4—315no—
183C2×C3×D4—345no—
184C2×C3×D4—3180no—
185C2×C3×D4—3180no—
186C2×C3×D4—360no—
187C2×C3×D4—360no—
188C2×C3×D4—390no—
189C2×C3×D4—390no—
190C2×C3×D4—345no—
191C2×C3×D4—330no—
192C2×C3×D4—330no—
193C2×C3×D4—315no—
194C22×C3×C4—360no—
195C22×C3×C4—320no—
196C22×C3×C4—345no—
197C22×C3×C4—315no—
198#41⋊C3—330no—
199C2×SL(2,𝔽3)—315no—
200C22×C32—310no—
201C22×C32—415no—
202C22×C32—1215no—
203C22×C32—1220no—
204C22×C32C23×A415yes—
205C32×C4—35no—
206C32×C4—31no—
207C32×C4—430no—
208C32×C4C23×A4110yes—
2092−1+4—35no—
2102−1+4—310no—
2112−1+4C22⋊H(3,𝔽3)11yescharacteristic
2122−1+4C22×H(3,𝔽3)110yes—
213H(5,𝔽2)—320no—
214H(5,𝔽2)—330no—
215H(5,𝔽2)—310no—
216H(5,𝔽2)—330no—
217H(5,𝔽2)C22×H(3,𝔽3)110yes—
218C2×#41—330no—
219C2×#41—320no—
220C2×#41—310no—
221C2×#41—35no—
222C2×#41C22×H(3,𝔽3)115yes—
223C22×D4—320no—
224C22×D4—315no—
225H(3,𝔽3)—41noSylow
226C23×C3—360no—
227C23×C3—320no—
228C23×C3—345no—
229C23×C3—315no—
230C23×C3—2410no—
231C23×C3—2430no—
232C3×Q8—35no—
233C3×Q8—315no—
234C3×Q8—330no—
235C3×Q8—310no—
236C3×Q8—310no—
237C3×Q8—31no—
238C3×Q8—1230no—
239C3×Q8C22×C3×A4110yes—
240C3×Q8C24×C3211yescharacteristic
241C3×D4—310no—
242C3×D4—330no—
243C3×D4—390no—
244C3×D4—330no—
245C3×D4—360no—
246C3×D4—390no—
247C3×D4—320no—
248C3×D4—330no—
249C3×D4—310no—
250C3×D4—1230no—
251C3×D4C22×C3×A4110yes—
252C2×C3×C4—360no—
253C2×C3×C4—320no—
254C2×C3×C4—315no—
255C2×C3×C4—330no—
256C2×C3×C4—390no—
257C2×C3×C4—35no—
258C2×C3×C4—330no—
259C2×C3×C4—310no—
260C2×C3×C4—315no—
261C2×C3×C4—1245no—
262C2×C3×C4C22×C3×A4115yes—
263SL(2,𝔽3)—33no—
264C2×C32—25no—
265C2×C32—43no—
266C2×C32—610no—
267C2×C32—815no—
268C2×C32C24×A411yescharacteristic
269C24—310no—
270#41—360no—
271#41—310no—
272#41—320no—
273#41—310no—
274#41—320no—
275#41—330no—
276#41—330no—
277#41C2×C22⋊H(3,𝔽3)110yes—
278#41C23×H(3,𝔽3)110yes—
279C2×Q8—310no—
280C2×Q8—310no—
281C2×Q8C2×C22⋊H(3,𝔽3)15yes—
282C2×Q8C23×H(3,𝔽3)15yes—
283C2×D4—315no—
284C2×D4—360no—
285C2×D4—360no—
286C2×D4—330no—
287C2×D4—330no—
288C2×D4—315no—
289C22×C4—320no—
290C22×C4—315no—
291C22×C3—330no—
292C22×C3—310no—
293C22×C3—35no—
294C22×C3—1215no—
295C22×C3—1245no—
296C22×C3—1260no—
297C22×C3—1220no—
298C22×C3—1215no—
299C22×C3C23×C3×A415yes—
300C3×C4—33no—
301C3×C4—315no—
302C3×C4—31no—
303C3×C4—35no—
304C3×C4—310no—
305C3×C4—1230no—
306C3×C4C23×C3×A4110yes—
307C32—43no—
308C32#8753611yescharacteristic, complemented
309C23—320no—
310C23—315no—
311C23—2410no—
312Q8—35no—
313Q8—310no—
314Q8C22×C22⋊H(3,𝔽3)110yes—
315Q8C24×H(3,𝔽3)11yescharacteristic
316D4—310no—
317D4—330no—
318D4—320no—
319D4—330no—
320D4C22×C22⋊H(3,𝔽3)110yes—
321C2×C4—320no—
322C2×C4—35no—
323C2×C4—310no—
324C2×C4—330no—
325C2×C4C22×C22⋊H(3,𝔽3)115yes—
326C2×C3—25no—
327C2×C3—31no—
328C2×C3—65no—
329C2×C3—630no—
330C2×C3—610no—
331C2×C3—123no—
332C2×C3—2415no—
333C2×C3C24×C3×A411yescharacteristic
334C22—310no—
335C22—1215no—
336C22—1220no—
337C22C23×C22⋊H(3,𝔽3)15yes—
338C4—31no—
339C4—35no—
340C4C23×C22⋊H(3,𝔽3)110yes—
341C3—31no—
342C3—123no—
343C3C3×#8753611yescharacteristic
344C2—25no—
345C2—610no—
346C2C24×C22⋊H(3,𝔽3)11yescharacteristic
subgroupG/H
center, Frattini, socleC2×C3C24×C3×A4
commutator, cosocleC3×Q8C24×C32
FittingC32×H(7,𝔽2)C3
hypercenterC32×2−1+4A4
hypocenterQ8C24×H(3,𝔽3)
LCSG ▷ C3×Q8 ▷ Q8
UCSC1 ◁ C2×C3 ◁ C32×2−1+4

Complements

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

NH ≅ G/Ntype
1. #13164865C2semidirect
2. C2×#11722577C2semidirect
3. C3×#87536C3semidirect
4. C32×H(7,𝔽2)C3semidirect
6. 2−1+4⋊H(3,𝔽3)C22semidirect
7. #11722577C22semidirect
8. C2×#41⋊H(3,𝔽3)C22semidirect
11. C32×#585C2×C3semidirect
13. C2×C32×H(5,𝔽2)C2×C3semidirect
14. C3×#5945C2×C3semidirect
15. C2×C3×#994C2×C3semidirect
18. #41⋊H(3,𝔽3)C23semidirect
19. C2×Q8⋊H(3,𝔽3)C23semidirect
40. C3×2−1+4⋊C3C22×C3semidirect
41. C3×#994C22×C3semidirect
42. C2×C3×#41⋊C3C22×C3semidirect
83. C3×#41⋊C3C23×C3semidirect
84. C2×C3×SL(2,𝔽3)C23×C3semidirect
85. H(7,𝔽2)H(3,𝔽3)semidirect
147. #585C2×H(3,𝔽3)semidirect
149. C2×H(5,𝔽2)C2×H(3,𝔽3)semidirect
308. C32#87536semidirect

Maximal subgroups

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

subgroupindex
#131648652
C2×#117225772
C3×#875363
C32×H(7,𝔽2)3
H(3,𝔽3)×2−1+44