Finite Group Databasebeta

← #11922398

Subgroups — #11922398

Order 1152.

611 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 611 conjugacy classes of subgroups — too many to draw. Only its 59 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
1C2×#64267C211yesmaximal, characteristic, complemented
2C2×C3×#985C211yesmaximal, characteristic, complemented
3#5827⋊C3C211yesmaximal, characteristic, complemented
4#10593215C212yesmaximal, complemented
5#10593208C212yesmaximal, complemented
6#85368—31nomaximal
7#83901—31nomaximal
8C2×C3×#41⋊C3C2211yescharacteristic
9#64267—22no—
10C2×C3⋊GL(2,𝔽3)C2212yes—
11C3×#985—22no—
12C2×C3×GL(2,𝔽3)C2211yescharacteristic, complemented
13C2×C3×#226C2211yescharacteristic, complemented
14#63909—41nomaximal
15Dic3×SL(2,𝔽3)C2212yescomplemented
16C2×#986—31no—
17C2×#986—61no—
18C2×#985D311yescharacteristic, complemented
19C2×C3×#135—31no—
20#5830—31no—
21#5827D311yescharacteristic, complemented
22C2×#949—31no—
23#520⋊C3—31no—
24#5395—32no—
25C4⋊GL(2,𝔽3)—32no—
26#5232—31no—
27#5185—32no—
28#5165—32no—
29#5163—32no—
30#5150—32no—
31C2×#821—31no—
32#5102—31no—
33C2×C4⋊(C3⋊D3)—41no—
34C2×C3×C4×D3—41no—
35C3×#41⋊C3D412yescomplemented
36C2×C3×SL(2,𝔽3)C2311yescharacteristic
37C2×#626—42no—
38C3⋊GL(2,𝔽3)—22no—
39C3×GL(2,𝔽3)—21no—
40C3×#226—21no—
41C3×C4⋊Dic3—41no—
42#3722—42no—
43#3045—91noSylow
44C2×C3×#41—31no—
45C2×C3×#41C2×D311yescharacteristic
46C22×D12—31no—
47C2×#41⋊C3—21no—
48C2×#41⋊C3C2×D311yescharacteristic
49#986—64no—
50#986—62no—
51#985—22no—
52C2×GL(2,𝔽3)—32no—
53C2×GL(2,𝔽3)—62no—
54C2×GL(2,𝔽3)C2×D311yescharacteristic, complemented
55C2×#226C2×D311yescharacteristic, complemented
56C3×#135—62no—
57C3×#135—62no—
58C2×C3×Q16—31no—
59C2×C3×SD16—31no—
60C2×C3×SD16—31no—
61C2×C3×D8—31no—
62C22×C3×C8—31no—
63#949—62no—
64#949—62no—
65#947—32no—
66Q8×Dic3C2×D312yes—
67C2×#215—32no—
68#940—32no—
69#937—62no—
70D4×Dic3—32no—
71C2×#213—32no—
72#930—61no—
73#930—61no—
74#926—31no—
75C2×C4⋊Dic3—31no—
76C2×#208—31no—
77#895—121no—
78C4×SL(2,𝔽3)—32no—
79#835—32no—
80#832—32no—
81#821—32no—
82#821—32no—
83Dic3⋊C8—32no—
84Dic3⋊C8—32no—
85C22×C3⋊D3—42no—
86C22×C3×D3—41no—
87C2×C32×C4—41no—
88C4⋊(C3⋊D3)—82no—
89C2×C3×Dic3—41no—
90C2×C3×Dic3—42no—
91C3×C4×D3—82no—
92C3×SL(2,𝔽3)C2×D411yescharacteristic
93#626—84no—
94C2×AGL(1,ℤ/8)—91no—
95C2×#135—31no—
96#535—91no—
97#520—31no—
98#464—92no—
99#463—92no—
100#460—92no—
101#459—92no—
102#424—91no—
103#417—91no—
104C2×#102—91no—
105C23×D3—61no—
106C3×#41—32no—
107C3×#41—62no—
108C3×#41—62no—
109C3×#41D1212yescomplemented
110C2×C3×Q8—31no—
111C2×C3×Q8C22×D311yescharacteristic
112C2×C3×D4—31no—
113C2×C3×D4—31no—
114C2×C3×D4—61no—
115C22×C3×C4—31no—
116C22×C3×C4—61no—
117C2×#67—62no—
118C2×#67—62no—
119C22×Dic3—61no—
120C2×D12—32no—
121C2×D12—62no—
122C2×D12—62no—
123C2×D12—61no—
124C2×D12—121no—
125C2×D12—122no—
126C2×D12—241no—
127C2×C4×D3—41no—
128#41⋊C3—22no—
129#41⋊C3#6712yescomplemented
130C2×SL(2,𝔽3)—21no—
131C2×SL(2,𝔽3)C22×D311yescharacteristic
132GL(2,𝔽3)—21no—
133GL(2,𝔽3)—64no—
134GL(2,𝔽3)—62no—
135#226—21no—
136C3×Q16—61no—
137C3×Q16—61no—
138C3×SD16—62no—
139C3×SD16—61no—
140C3×SD16—61no—
141C3×D8—62no—
142C2×C3×C8—31no—
143C2×C3×C8—31no—
144C2×C3×C8—62no—
145C3×C4⋊C4—121no—
146#217—32no—
147#215—62no—
148#215—62no—
149#213—64no—
150(C2×D3)⋊C4—62no—
151(C2×D3)⋊C4—62no—
152(C2×D3)⋊C4—122no—
153C4⋊Dic3—31no—
154C4⋊Dic3—32no—
155C4⋊Dic3S411yescharacteristic
156C4×Dic3—32no—
157#208—62no—
158C2×C3⋊C8—32no—
159C22×C32—41no—
160C2×C3⋊D3—82no—
161C2×C3⋊D3—82no—
162C2×C3⋊D3—82no—
163C2×C3×D3—42no—
164C2×C3×D3—81no—
165C2×C3×D3—81no—
166C32×C4—42no—
167C3×Dic3—82no—
168C3×Dic3—81no—
169C2×#41—31no—
170C2×#41D3211yescharacteristic
171C22×D4—91no—
172AGL(1,ℤ/8)—182no—
173AGL(1,ℤ/8)—182no—
174#135—62no—
175#135—62no—
176C2×Q16—31no—
177C2×SD16—31no—
178C2×SD16—31no—
179C2×SD16—92no—
180C2×D8—31no—
181C2×D8—92no—
182C2×M16—91no—
183C22×C8—31no—
184#127—92no—
185#124—92no—
186#121—181no—
187#121—181no—
188#120—182no—
189C4×Q8—32no—
190C4×D4—92no—
191#117—91no—
192C2×C4⋊C4—91no—
193C4⋊C8—92no—
194C4⋊C8—92no—
195#103—92no—
196#102—92no—
197#102—92no—
198#102—92no—
199C23×C3—31no—
200C23×C3—61no—
201C22×D3—41no—
202C22×D3—62no—
203C22×D3—122no—
204C22×D3—122no—
205C22×D3—242no—
206C22×D3—241no—
207C22×D3—241no—
208C3×Q8—31no—
209C3×Q8—61no—
210C3×Q8—61no—
211C3×Q8C2×D1211yescharacteristic
212C3×D4—32no—
213C3×D4—61no—
214C3×D4—61no—
215C3×D4—61no—
216C3×D4—122no—
217C2×C3×C4—31no—
218C2×C3×C4—32no—
219C2×C3×C4—32no—
220C2×C3×C4—41no—
221C2×C3×C4—61no—
222C2×C3×C4—81no—
223C2×C3×C4—122no—
224C2×C3×C4—122no—
225C2×C3×C4C2×S411yescharacteristic
226#67—242no—
227#67—242no—
228C2×Dic3—32no—
229C2×Dic3—41no—
230C2×Dic3—241no—
231C2×Dic3C2×S412yes—
232D12—62no—
233D12—62no—
234D12—122no—
235D12—242no—
236D12—244no—
237D12—244no—
238C4×D3—82no—
239SL(2,𝔽3)—21no—
240SL(2,𝔽3)C2×#6711yescharacteristic
241C3×C8—61no—
242C3×C8—61no—
243C3⋊C8—62no—
244C2×C32—41no—
245C2×C32—41no—
246C2×C32—41no—
247C3⋊D3—162no—
248C3×D3—82no—
249C24—181no—
250#41—32no—
251#41—62no—
252#41—62no—
253#41#62612yescomplemented
254C2×Q8—31no—
255C2×Q8C2×D3211yescharacteristic
256C2×D4—31no—
257C2×D4—31no—
258C2×D4—61no—
259C2×D4—92no—
260C2×D4—182no—
261C2×D4—182no—
262C2×D4—182no—
263C2×D4—182no—
264C2×D4—181no—
265C22×C4—31no—
266C22×C4—61no—
267C22×C4—181no—
268Q16—61no—
269Q16—61no—
270SD16—62no—
271SD16—61no—
272SD16—61no—
273SD16—182no—
274SD16—182no—
275D8—62no—
276D8—184no—
277M16—182no—
278C2×C8—31no—
279C2×C8—31no—
280C2×C8—62no—
281C2×C8—92no—
282C4⋊C4—31no—
283C4⋊C4—92no—
284C4⋊C4—91no—
285C22⋊C4—92no—
286C22⋊C4—182no—
287C22⋊C4—182no—
288C42—92no—
289C22×C3—32no—
290C22×C3—41no—
291C22×C3—61no—
292C22×C3—61no—
293C22×C3—81no—
294C22×C3—121no—
295C22×C3—121no—
296C22×C3—121no—
297C22×C3C22×S411yescharacteristic
298C2×D3—42no—
299C2×D3—81no—
300C2×D3—81no—
301C2×D3—122no—
302C2×D3—122no—
303C2×D3—122no—
304C2×D3—242no—
305C2×D3—242no—
306C2×D3—242no—
307C2×D3—244no—
308C2×D3—244no—
309C2×D3—482no—
310C2×D3—482no—
311C3×C4—31no—
312C3×C4—31no—
313C3×C4—42no—
314C3×C4—82no—
315C3×C4—121no—
316C3×C4—242no—
317C3×C4C4⋊S412yes—
318Dic3—22no—
319Dic3—62no—
320Dic3—81no—
321C32—41noSylow
322C23—31no—
323C23—61no—
324C23—182no—
325C23—362no—
326C23—721no—
327C23—721no—
328Q8—31no—
329Q8—61no—
330Q8—61no—
331Q8C2×#62611yescharacteristic
332D4—32no—
333D4—61no—
334D4—61no—
335D4—61no—
336D4—122no—
337D4—182no—
338D4—182no—
339D4—362no—
340D4—722no—
341D4—722no—
342C2×C4—31no—
343C2×C4—32no—
344C2×C4—32no—
345C2×C4—32no—
346C2×C4—61no—
347C2×C4—92no—
348C2×C4—122no—
349C2×C4—721no—
350C2×C4D3×S411yescharacteristic
351C8—61no—
352C8—61no—
353C8—182no—
354C2×C3—41no—
355C2×C3—41no—
356C2×C3—41no—
357C2×C3—62no—
358C2×C3—81no—
359C2×C3—81no—
360C2×C3—81no—
361C2×C3—241no—
362C2×C3C2×C4⋊S411yescharacteristic
363C2×C3#592511yescharacteristic
364C2×C3#592611yescharacteristic
365D3—82no—
366D3—242no—
367D3—482no—
368D3—484no—
369C22—32no—
370C22—61no—
371C22—61no—
372C22—121no—
373C22—121no—
374C22—121no—
375C22—362no—
376C22—362no—
377C22—362no—
378C22—724no—
379C22C2×D3×S411yescharacteristic
380C4—31no—
381C4—31no—
382C4—62no—
383C4—121no—
384C4—182no—
385C4#6420712yes—
386C3—41no—
387C3—81no—
388C3#8536811yescharacteristic, complemented
389C2—62no—
390C2—241no—
391C2—722no—
392C2#1059644911yescharacteristic
393C2#1059645911yescharacteristic
394C2C2×#6420711yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC2×C3×#41C2×D3
socleC22×C3C22×S4
hypercenterC2×C4D3×S4
hypocenterC3×SL(2,𝔽3)C2×D4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#64267C2semidirect
2. C2×C3×#985C2semidirect
3. #5827⋊C3C2semidirect
4. #10593215C2semidirect
5. #10593208C2semidirect
12. C2×C3×GL(2,𝔽3)C22semidirect
13. C2×C3×#226C22semidirect
15. Dic3×SL(2,𝔽3)C22semidirect
18. C2×#985D3semidirect
21. #5827D3semidirect
35. C3×#41⋊C3D4semidirect
54. C2×GL(2,𝔽3)C2×D3semidirect
55. C2×#226C2×D3semidirect
109. C3×#41D12semidirect
129. #41⋊C3#67semidirect
253. #41#626semidirect
388. C3#85368semidirect

Maximal subgroups

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

subgroupindex
C2×#642672
C2×C3×#9852
#5827⋊C32
#105932152
#105932082
#853683
#839013
#639094