Finite Group Databasebeta

← #11922892

Subgroups — #11922892

Order 1152.

540 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 69 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×C4×#41⋊C3C211yesmaximal, characteristic, complemented
2#10593661C211yesmaximal, characteristic
3#10593659C211yesmaximal, characteristic, complemented
4#10593629C211yesmaximal, characteristic, complemented
5C4×C3⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
6#10593623C211yesmaximal, characteristic, complemented
7C4×C3⋊#226C211yesmaximal, characteristic, complemented
8#85343—33nomaximal
9#82528—31nomaximal
10C2×C3×#41⋊C3C2211yescharacteristic
11C2×C3⋊GL(2,𝔽3)C2211yescharacteristic
12C2×C3⋊#226C2211yescharacteristic
13#64081—41nomaximal
14C3×C4×SL(2,𝔽3)C2211yescharacteristic
15C3×C4×SL(2,𝔽3)C2211yescharacteristic
16C3⋊#859C2211yescharacteristic
17C3⋊#859C2211yescharacteristic
18C3×C4×#41D311yescharacteristic, complemented
19#5479—31no—
20C4×#41⋊C3D313yescomplemented
21#5427—33no—
22#5425—33no—
23#5395—33no—
24C4×GL(2,𝔽3)—33no—
25#5389—33no—
26C4×#226—33no—
27#5239—31no—
28#5238—31no—
29C4×#216—31no—
30#5028—31no—
31C4×#215—31no—
32#5022—31no—
33C4×#214—31no—
34#5016—31no—
35C4×#213—31no—
36#5009—31no—
37#5003—31no—
38#4996—31no—
39#4925—31no—
40C2×C4×C3⋊D3—41no—
41C3×#41⋊C3—21no—
42C2×C3×SL(2,𝔽3)C2311yescharacteristic
43C3⋊GL(2,𝔽3)—21no—
44C3⋊#226—21no—
45C32×C42—41no—
46#3752—41no—
47#3752—41no—
48#3750—41no—
49#3750—41no—
50C4×C3⋊Dic3—41no—
51#3102—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×C42—31no—
62C2×#216—31no—
63C2×#215—31no—
64C2×#214—31no—
65C2×#213—31no—
66#930—31no—
67#929—61no—
68C4×#67—61no—
69#924—31no—
70#876—61no—
71#875—31no—
72C4×D12—31no—
73#872—61no—
74#872—123no—
75#870—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#808—31no—
92#808—31no—
93#807—31no—
94#807—31no—
95C4⋊(C3⋊C8)—31no—
96C4⋊(C3⋊C8)—31no—
97C4×C3⋊C8—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—
106C4×C3⋊D3—44no—
107C3×SL(2,𝔽3)#4111yescharacteristic
108#529—91no—
109C4×#41C3⋊D311yescharacteristic, complemented
110#485—91no—
111#482—91no—
112#479—91no—
113#478—91no—
114#464—91no—
115#463—91no—
116#452—91no—
117#449—91no—
118C4×Q16—91no—
119C4×SD16—91no—
120C4×SD16—91no—
121C4×D8—91no—
122#431—91no—
123C3×#41—21no—
124C3×#41—61no—
125C2×C3×Q8C22×D311yescharacteristic
126C2×C3×D4—31no—
127C22×C3×C4—31no—
128C22×C3×C4—31no—
129C22×C3×C4—31no—
130C2×#67—61no—
131C2×D12—31no—
132C2×C4×D3—61no—
133C2×C4×D3—123no—
134C2×Dic6—31no—
135#41⋊C3—23no—
136C2×SL(2,𝔽3)C22×D313yes—
137GL(2,𝔽3)—63no—
138#226—63no—
139C3×C4⋊C4—31no—
140C3×C4⋊C4—31no—
141C3×C22⋊C4—61no—
142C3×C42—31no—
143C3×C42—31no—
144C3×C42—31no—
145C3×C42—43no—
146C3×C42S411yescharacteristic
147#217—61no—
148#216—61no—
149#216—61no—
150#215—61no—
151#215—61no—
152#214—62no—
153#213—62no—
154(C2×D3)⋊C4—61no—
155(C2×D3)⋊C4—61no—
156(C2×D3)⋊C4—61no—
157(C2×D3)⋊C4—61no—
158(C2×D3)⋊C4—123no—
159(C2×D3)⋊C4—123no—
160C4⋊Dic3—31no—
161C4⋊Dic3—31no—
162Dic3⋊C4—61no—
163Dic3⋊C4—61no—
164Dic3⋊C4—61no—
165Dic3⋊C4—61no—
166Dic3⋊C4—123no—
167Dic3⋊C4—123no—
168C4×Dic3—61no—
169C4×Dic3—123no—
170C2×C3⋊C8—31no—
171C2×C3⋊C8—31no—
172C2×C3⋊C8—31no—
173C2×C3⋊C8—31no—
174C22×C32—41no—
175C2×C3⋊D3—42no—
176C2×C3⋊D3—81no—
177C2×C3⋊D3—81no—
178C32×C4—42no—
179C32×C4—81no—
180C32×C4—81no—
181C3⋊Dic3—42no—
182C3⋊Dic3—81no—
183C3⋊Dic3—81no—
184C2×#41C2×C3⋊D311yescharacteristic
185C2×Q16—91no—
186C2×SD16—91no—
187C2×SD16—91no—
188C2×D8—91no—
189#126—181no—
190#125—91no—
191#124—91no—
192#123—181no—
193C22⋊Q8—91no—
194#121—91no—
195C4×Q8—91no—
196C4×Q8C2×C3⋊D311yescharacteristic
197C4×Q8C2×C3⋊D311yescharacteristic
198C4×D4—31no—
199C4×D4—31no—
200C4×D4—91no—
201C4×D4—181no—
202#117—31no—
203#117—181no—
204C2×C42—31no—
205#107—91no—
206#107—91no—
207#106—91no—
208#106—91no—
209C4⋊C8—91no—
210C4⋊C8—91no—
211#103—91no—
212#103—91no—
213#103—91no—
214#103—91no—
215#102—91no—
216#102—91no—
217#102—91no—
218#102—91no—
219C22⋊C8—91no—
220C22⋊C8—91no—
221C4×C8—91no—
222C4×C8—91no—
223C23×C3—31no—
224C22×D3—61no—
225C22×D3—123no—
226C3×Q8—31no—
227C3×Q8#23511yescharacteristic
228C3×D4—32no—
229C3×D4—61no—
230C2×C3×C4—31no—
231C2×C3×C4—31no—
232C2×C3×C4—31no—
233C2×C3×C4—43no—
234C2×C3×C4—43no—
235C2×C3×C4—43no—
236C2×C3×C4—61no—
237C2×C3×C4—61no—
238C2×C3×C4—62no—
239C2×C3×C4—121no—
240C2×C3×C4C2×S411yescharacteristic
241C2×C3×C4C2×S411yescharacteristic
242C2×C3×C4C2×S411yescharacteristic
243#67—122no—
244C2×Dic3—61no—
245C2×Dic3—61no—
246C2×Dic3—61no—
247C2×Dic3—123no—
248C2×Dic3—123no—
249C2×Dic3—123no—
250D12—61no—
251D12—61no—
252C4×D3—1212no—
253C4×D3—122no—
254Dic6—61no—
255Dic6—61no—
256SL(2,𝔽3)#23513yes—
257C3⋊C8—61no—
258C3⋊C8—61no—
259C3⋊C8—61no—
260C3⋊C8—61no—
261C2×C32—41no—
262C2×C32—41no—
263C2×C32—41no—
264C3⋊D3—82no—
265#41—21no—
266#41—61no—
267C2×Q8—91no—
268C2×Q8C22×C3⋊D311yescharacteristic
269C2×D4—31no—
270C2×D4—91no—
271C2×D4—181no—
272C22×C4—31no—
273C22×C4—31no—
274C22×C4—31no—
275C22×C4—181no—
276Q16—181no—
277Q16—181no—
278SD16—182no—
279SD16—181no—
280SD16—181no—
281D8—182no—
282C2×C8—91no—
283C2×C8—91no—
284C2×C8—91no—
285C2×C8—91no—
286C4⋊C4—31no—
287C4⋊C4—31no—
288C4⋊C4—91no—
289C4⋊C4—91no—
290C4⋊C4—181no—
291C4⋊C4—181no—
292C4⋊C4—181no—
293C4⋊C4—181no—
294C22⋊C4—61no—
295C22⋊C4—181no—
296C22⋊C4—181no—
297C22⋊C4—181no—
298C22⋊C4—181no—
299C22⋊C4—181no—
300C42—31no—
301C42—31no—
302C42—31no—
303C42—181no—
304C42C3⋊S411yescharacteristic
305C22×C3—43no—
306C22×C3—61no—
307C22×C3—61no—
308C22×C3—61no—
309C22×C3C22×S411yescharacteristic
310C2×D3—126no—
311C2×D3—121no—
312C2×D3—121no—
313C2×D3—121no—
314C2×D3—243no—
315C2×D3—243no—
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—31no—
319C3×C4—31no—
320C3×C4—46no—
321C3×C4—61no—
322C3×C4—61no—
323C3×C4—83no—
324C3×C4—83no—
325C3×C4#98512yes—
326Dic3—126no—
327Dic3—121no—
328Dic3—121no—
329Dic3—121no—
330Dic3—243no—
331Dic3—243no—
332C32—41noSylow
333C23—31no—
334C23—181no—
335Q8—31no—
336Q8—181no—
337Q8—181no—
338Q8#382811yescharacteristic
339D4—32no—
340D4—61no—
341D4—181no—
342D4—181no—
343D4—362no—
344C2×C4—31no—
345C2×C4—31no—
346C2×C4—31no—
347C2×C4—61no—
348C2×C4—61no—
349C2×C4—62no—
350C2×C4—121no—
351C2×C4—181no—
352C2×C4—181no—
353C2×C4—181no—
354C2×C4—362no—
355C2×C4C2×C3⋊S411yescharacteristic
356C2×C4C2×C3⋊S411yescharacteristic
357C2×C4C2×C3⋊S411yescharacteristic
358C8—181no—
359C8—181no—
360C8—181no—
361C8—181no—
362C2×C3—43no—
363C2×C3—43no—
364C2×C3—43no—
365C2×C3—121no—
366C2×C3#591211yescharacteristic
367C2×C3C2×#98511yescharacteristic
368C2×C3#592311yescharacteristic
369D3—246no—
370D3—241no—
371C22—61no—
372C22—61no—
373C22—61no—
374C22—361no—
375C22—361no—
376C22—361no—
377C22C22×C3⋊S411yescharacteristic
378C4—21no—
379C4—21no—
380C4—31no—
381C4—31no—
382C4—61no—
383C4—61no—
384C4—361no—
385C4—361no—
386C4—361no—
387C4#6426612yes—
388C3—43no—
389C3#8534311yescharacteristic, complemented
390C2—121no—
391C2—721no—
392C2#1059657511yescharacteristic
393C2C2×#6426611yescharacteristic
394C2#1059658611yescharacteristic
subgroupG/H
centerC2×C4C2×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FrattiniC22C22×C3⋊S4
FittingC3×C4×#41D3
socleC22×C3C22×S4
prime radical#10593661C2
hypercenterC42C3⋊S4
hypocenterC3×SL(2,𝔽3)#41
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C3×C4×#41⋊C3C2semidirect
3. #10593659C2semidirect
4. #10593629C2semidirect
5. C4×C3⋊GL(2,𝔽3)C2semidirect
6. #10593623C2semidirect
7. C4×C3⋊#226C2semidirect
18. C3×C4×#41D3semidirect
20. C4×#41⋊C3D3semidirect
109. C4×#41C3⋊D3semidirect
389. C3#85343semidirect

Maximal subgroups

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

subgroupindex
C3×C4×#41⋊C32
#105936612
#105936592
#105936292
C4×C3⋊GL(2,𝔽3)2
#105936232
C4×C3⋊#2262
#853433
#825283
#640814