Finite Group Databasebeta

← #11922920

Subgroups — #11922920

Order 1152.

500 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 65 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×#520⋊C3C211yesmaximal, characteristic, complemented
2#10593661C211yesmaximal, characteristic
3#10593659C211yesmaximal, characteristic, complemented
4#10593629C211yesmaximal, characteristic, complemented
5#10593628C211yesmaximal, characteristic, complemented
6#10593625C211yesmaximal, characteristic, complemented
7C4×C3⋊#226C211yesmaximal, characteristic, complemented
8#85371—33nomaximal
9#82569—31nomaximal
10C2×C3×#41⋊C3C2211yescharacteristic
11C2×C3⋊GL(2,𝔽3)C2211yescharacteristic
12C2×C3⋊#226C2211yescharacteristic
13#64107—41nomaximal
14C3×C4×SL(2,𝔽3)C2211yescharacteristic
15C3×C4×SL(2,𝔽3)C2211yescharacteristic
16C3⋊#859C2211yescharacteristic
17C3⋊#859C2211yescharacteristic
18C3×#520D311yescharacteristic, complemented
19#5534—31no—
20#520⋊C3D313yescomplemented
21#5427—33no—
22#5425—33no—
23#5395—33no—
24#5394—33no—
25#5391—33no—
26C4×#226—33no—
27#5239—31no—
28#5238—31no—
29C4×#216—31no—
30#5028—31no—
31#5026—31no—
32#5021—31no—
33#5018—31no—
34#5016—31no—
35C4×#213—31no—
36#5007—31no—
37#5000—31no—
38#4970—31no—
39#4967—31no—
40C3×#41⋊C3—21no—
41C2×C3×SL(2,𝔽3)C2311yescharacteristic
42C3⋊GL(2,𝔽3)—21no—
43C3⋊#226—21no—
44C32×C4⋊C4—41no—
45#3752—41no—
46#3752—41no—
47#3752—41no—
48C4⋊(C3⋊Dic3)—41no—
49#3750—41no—
50C4×C3⋊Dic3—41no—
51#3124—91noSylow
52C2×C3×#41C2×D311yescharacteristic
53C2×#41⋊C3C2×D313yes—
54C2×GL(2,𝔽3)—33no—
55C2×#226—33no—
56C3×C4×Q8C2×D311yescharacteristic
57C3×C4×Q8C2×D311yescharacteristic
58C3×C4×D4—31no—
59C3×C4×D4—31no—
60C3×#117—31no—
61C2×C3×C4⋊C4—31no—
62C2×#216—31no—
63C2×#215—31no—
64C2×#214—31no—
65C2×#213—31no—
66#930—31no—
67#924—31no—
68#898—61no—
69#898—123no—
70#896—61no—
71#885—61no—
72#883—61no—
73#875—31no—
74C4×D12—31no—
75#869—31no—
76C4×Dic6—31no—
77C4×SL(2,𝔽3)C2×D313yes—
78C4×SL(2,𝔽3)C2×D313yes—
79#859—33no—
80#859—33no—
81#835—31no—
82#835—31no—
83#832—31no—
84#832—31no—
85#830—31no—
86#830—31no—
87#810—31no—
88#810—31no—
89#809—31no—
90#809—31no—
91#807—31no—
92#807—31no—
93#807—31no—
94#807—31no—
95C4⋊(C3⋊C8)—31no—
96C4⋊(C3⋊C8)—31no—
97#803—31no—
98C4×C3⋊C8—31no—
99C22×C3⋊D3—41no—
100C2×C32×C4—41no—
101C2×C32×C4—41no—
102C2×C32×C4—41no—
103C2×C3⋊Dic3—41no—
104C2×C3⋊Dic3—41no—
105C2×C3⋊Dic3—41no—
106C3×SL(2,𝔽3)#4111yescharacteristic
107#542—91no—
108#520C3⋊D311yescharacteristic, complemented
109#484—91no—
110#480—91no—
111#477—91no—
112#474—91no—
113#464—91no—
114#463—91no—
115#452—91no—
116#449—91no—
117#440—91no—
118#440—91no—
119C4×Q16—91no—
120C4×D8—91no—
121#433—91no—
122C3×#41—21no—
123C3×#41—61no—
124C2×C3×Q8C22×D311yescharacteristic
125C2×C3×D4—31no—
126C22×C3×C4—31no—
127C22×C3×C4—61no—
128C2×#67—61no—
129C2×D12—31no—
130C2×C4×D3—61no—
131C2×Dic6—31no—
132C2×Dic6—61no—
133#41⋊C3—23no—
134C2×SL(2,𝔽3)C22×D313yes—
135GL(2,𝔽3)—63no—
136#226—63no—
137C3×C4⋊C4—31no—
138C3×C4⋊C4—31no—
139C3×C4⋊C4—31no—
140C3×C4⋊C4—43no—
141C3×C4⋊C4S411yescharacteristic
142C3×C22⋊C4—31no—
143C3×C22⋊C4—31no—
144C3×C42—31no—
145C3×C42—31no—
146#217—61no—
147#216—61no—
148#216—61no—
149#215—61no—
150#215—61no—
151#214—62no—
152#213—62no—
153(C2×D3)⋊C4—61no—
154(C2×D3)⋊C4—61no—
155(C2×D3)⋊C4—61no—
156(C2×D3)⋊C4—61no—
157(C2×D3)⋊C4—123no—
158(C2×D3)⋊C4—123no—
159(C2×D3)⋊C4—123no—
160C4⋊Dic3—31no—
161C4⋊Dic3—31no—
162C4⋊Dic3—61no—
163C4⋊Dic3—123no—
164Dic3⋊C4—61no—
165Dic3⋊C4—61no—
166Dic3⋊C4—123no—
167C4×Dic3—61no—
168C4×Dic3—123no—
169C2×C3⋊C8—31no—
170C2×C3⋊C8—31no—
171C2×C3⋊C8—31no—
172C2×C3⋊C8—31no—
173C22×C32—41no—
174C2×C3⋊D3—81no—
175C2×C3⋊D3—81no—
176C2×C3⋊D3—81no—
177C32×C4—81no—
178C32×C4—81no—
179C32×C4—81no—
180C3⋊Dic3—81no—
181C3⋊Dic3—81no—
182C3⋊Dic3—81no—
183C2×#41C2×C3⋊D311yescharacteristic
184C2×Q16—91no—
185C2×SD16—91no—
186C2×SD16—91no—
187C2×D8—91no—
188C4⋊Q8—91no—
189#126—181no—
190#124—91no—
191#124—181no—
192#123—181no—
193C22⋊Q8—91no—
194C22⋊Q8—181no—
195#121—91no—
196C4×Q8—91no—
197C4×Q8C2×C3⋊D311yescharacteristic
198C4×Q8C2×C3⋊D311yescharacteristic
199C4×D4—31no—
200C4×D4—31no—
201C4×D4—91no—
202#117—31no—
203C2×C4⋊C4—31no—
204#107—91no—
205#107—91no—
206#107—91no—
207#107—91no—
208C4⋊C8—91no—
209C4⋊C8—91no—
210#103—91no—
211#103—91no—
212#103—91no—
213#103—91no—
214#102—91no—
215#102—91no—
216#102—91no—
217#102—91no—
218C22⋊C8—91no—
219C22⋊C8—91no—
220#97—91no—
221C4×C8—91no—
222C23×C3—31no—
223C22×D3—61no—
224C22×D3—123no—
225C3×Q8—31no—
226C3×Q8#23511yescharacteristic
227C3×D4—32no—
228C3×D4—61no—
229C2×C3×C4—31no—
230C2×C3×C4—31no—
231C2×C3×C4—31no—
232C2×C3×C4—43no—
233C2×C3×C4—43no—
234C2×C3×C4—43no—
235C2×C3×C4—61no—
236C2×C3×C4—61no—
237C2×C3×C4—241no—
238C2×C3×C4C2×S411yescharacteristic
239C2×C3×C4C2×S411yescharacteristic
240C2×C3×C4C2×S411yescharacteristic
241#67—241no—
242C2×Dic3—61no—
243C2×Dic3—61no—
244C2×Dic3—61no—
245C2×Dic3—123no—
246C2×Dic3—123no—
247C2×Dic3—123no—
248D12—61no—
249D12—61no—
250C4×D3—241no—
251Dic6—61no—
252Dic6—61no—
253Dic6—241no—
254SL(2,𝔽3)#23513yes—
255C3⋊C8—61no—
256C3⋊C8—61no—
257C3⋊C8—61no—
258C3⋊C8—61no—
259C2×C32—41no—
260C2×C32—41no—
261C2×C32—41no—
262C3⋊D3—161no—
263#41—21no—
264#41—61no—
265C2×Q8—91no—
266C2×Q8—181no—
267C2×Q8C22×C3⋊D311yescharacteristic
268C2×D4—31no—
269C2×D4—91no—
270C2×D4—181no—
271C22×C4—31no—
272C22×C4—61no—
273C22×C4—181no—
274Q16—181no—
275Q16—181no—
276SD16—182no—
277SD16—181no—
278SD16—181no—
279D8—182no—
280C2×C8—91no—
281C2×C8—91no—
282C2×C8—91no—
283C2×C8—91no—
284C4⋊C4—31no—
285C4⋊C4—31no—
286C4⋊C4—31no—
287C4⋊C4—91no—
288C4⋊C4—91no—
289C4⋊C4—181no—
290C4⋊C4—181no—
291C4⋊C4—181no—
292C4⋊C4C3⋊S411yescharacteristic
293C22⋊C4—31no—
294C22⋊C4—31no—
295C22⋊C4—181no—
296C22⋊C4—181no—
297C22⋊C4—181no—
298C22⋊C4—181no—
299C22⋊C4—181no—
300C42—31no—
301C42—31no—
302C42—181no—
303C22×C3—43no—
304C22×C3—61no—
305C22×C3—61no—
306C22×C3—61no—
307C22×C3C22×S411yescharacteristic
308C2×D3—121no—
309C2×D3—121no—
310C2×D3—121no—
311C2×D3—243no—
312C2×D3—243no—
313C2×D3—243no—
314C3×C4—21no—
315C3×C4—21no—
316C3×C4—21no—
317C3×C4—31no—
318C3×C4—31no—
319C3×C4—61no—
320C3×C4—61no—
321C3×C4—83no—
322C3×C4—83no—
323C3×C4—83no—
324Dic3—121no—
325Dic3—121no—
326Dic3—121no—
327Dic3—243no—
328Dic3—243no—
329Dic3—243no—
330C32—41noSylow
331C23—31no—
332C23—181no—
333Q8—31no—
334Q8—181no—
335Q8—181no—
336Q8—721no—
337Q8#382811yescharacteristic
338D4—32no—
339D4—61no—
340D4—181no—
341D4—181no—
342D4—721no—
343C2×C4—31no—
344C2×C4—31no—
345C2×C4—31no—
346C2×C4—61no—
347C2×C4—61no—
348C2×C4—181no—
349C2×C4—181no—
350C2×C4—181no—
351C2×C4—241no—
352C2×C4—721no—
353C2×C4C2×C3⋊S411yescharacteristic
354C2×C4C2×C3⋊S411yescharacteristic
355C2×C4C2×C3⋊S411yescharacteristic
356C8—181no—
357C8—181no—
358C8—181no—
359C8—181no—
360C2×C3—43no—
361C2×C3—43no—
362C2×C3—43no—
363C2×C3—121no—
364C2×C3#591211yescharacteristic
365C2×C3#592511yescharacteristic
366C2×C3#592711yescharacteristic
367D3—241no—
368D3—483no—
369C22—61no—
370C22—61no—
371C22—61no—
372C22—361no—
373C22—361no—
374C22—361no—
375C22C22×C3⋊S411yescharacteristic
376C4—21no—
377C4—21no—
378C4—21no—
379C4—31no—
380C4—31no—
381C4—61no—
382C4—61no—
383C4—361no—
384C4—361no—
385C4—361no—
386C3—43no—
387C3#8537111yescharacteristic, complemented
388C2—121no—
389C2—721no—
390C2#1059657511yescharacteristic
391C2#1059658811yescharacteristic
392C2#1059659011yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC3×#520D3
socleC22×C3C22×S4
prime radical#10593661C2
hypercenterC4⋊C4C3⋊S4
hypocenterC3×SL(2,𝔽3)#41
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C3×#520⋊C3C2semidirect
3. #10593659C2semidirect
4. #10593629C2semidirect
5. #10593628C2semidirect
6. #10593625C2semidirect
7. C4×C3⋊#226C2semidirect
18. C3×#520D3semidirect
20. #520⋊C3D3semidirect
108. #520C3⋊D3semidirect
387. C3#85371semidirect

Maximal subgroups

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

subgroupindex
C3×#520⋊C32
#105936612
#105936592
#105936292
#105936282
#105936252
C4×C3⋊#2262
#853713
#825693
#641074