Finite Group Databasebeta

← #11922313

Subgroups — #11922313

Order 1152.

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

normal subgroups only

This group has 637 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×C3⋊#994C211yesmaximal, characteristic, complemented
2C2×C3×#985C211yesmaximal, characteristic, complemented
3#10593660C211yesmaximal, characteristic, complemented
4#10593203C212yesmaximal, complemented
5#10593194C212yesmaximal, complemented
6#85529—31nomaximal
7#83912—31nomaximal
8C2×C3×#41⋊C3C2211yescharacteristic
9C3⋊#994—22no—
10C2×C3⋊(#41⋊C3)C2212yes—
11C3×#985—22no—
12C2×C3×GL(2,𝔽3)C2211yescharacteristic, complemented
13C2×C3×#226C2211yescharacteristic, complemented
14#63856—41nomaximal
15C3⋊#859C2212yescomplemented
16C2×#1009D311yescharacteristic, complemented
17C2×#994—31no—
18C2×#985D311yescharacteristic, complemented
19C2×C3×#135—31no—
20#5833—31no—
21C2×#949—31no—
22#5427—32no—
23#5426—31no—
24#5426—61no—
25#5425—32no—
26#5232—31no—
27#5187—32no—
28#5172—32no—
29#5169—32no—
30#5156—32no—
31C2×#821—31no—
32#5110—31no—
33C2×C3×C4×D3—41no—
34C2×C3×Dic6—41no—
35C3×#41⋊C3D412yescomplemented
36C2×C3×SL(2,𝔽3)C2311yescharacteristic
37C3⋊(#41⋊C3)—22no—
38C3×GL(2,𝔽3)—21no—
39C3×#226—21no—
40C4⋊(C3⋊Dic3)—41no—
41#3724—42no—
42(C2×D3)⋊Dic3—42no—
43#3012—91noSylow
44C2×C3×#41—31no—
45C2×C3×#41C2×D311yescharacteristic
46#1009—22no—
47#1009—62no—
48C2×#239C2×D312yes—
49C2×D3×D4—32no—
50C2×#235—31no—
51C22×D12—31no—
52#994—62no—
53C2×#41⋊C3—21no—
54C2×#41⋊C3—32no—
55C2×#41⋊C3C2×D311yescharacteristic
56#985—22no—
57C2×GL(2,𝔽3)C2×D311yescharacteristic, complemented
58C2×#226C2×D311yescharacteristic, complemented
59C3×#135—62no—
60C3×#135—62no—
61C2×C3×Q16—31no—
62C2×C3×SD16—31no—
63C2×C3×SD16—31no—
64C2×C3×D8—31no—
65C22×C3×C8—31no—
66#949—64no—
67#946—32no—
68#946—121no—
69C2×#215—32no—
70#939—62no—
71#938—32no—
72C2×#213—32no—
73#930—61no—
74C2×C4⋊Dic3—31no—
75#924—61no—
76C2×#208—31no—
77#859—32no—
78#859—62no—
79#835—32no—
80#832—32no—
81#821—32no—
82#821—32no—
83(C2×D3)⋊C8—32no—
84(C2×D3)⋊C8—32no—
85C22×C3×D3—41no—
86C2×C32×C4—41no—
87C2×C3⋊Dic3—42no—
88C2×C3×Dic3—41no—
89C2×C3×Dic3—42no—
90C3×C4×D3—82no—
91C3×Dic6—82no—
92C3×SL(2,𝔽3)C2×D411yescharacteristic
93C2×H(5,𝔽2)—31no—
94C2×AGL(1,ℤ/8)—91no—
95C2×#135—31no—
96#537—91no—
97#452—92no—
98#450—92no—
99#449—92no—
100#447—92no—
101#417—91no—
102C2×#102—91no—
103#408—91no—
104C23×D3—61no—
105C3×#41—32no—
106C3×#41—62no—
107C3×#41—62no—
108C3×#41#6712yes—
109C2×C3×Q8—31no—
110C2×C3×Q8—121no—
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—32no—
118C2×#67—62no—
119C22×Dic3—61no—
120#239—22no—
121#239—62no—
122D3×D4—64no—
123D3×D4—64no—
124#235—62no—
125#235—62no—
126C2×D12—32no—
127C2×D12—34no—
128C2×D12—31no—
129C2×D12—62no—
130C2×C4×D3—32no—
131C2×C4×D3—41no—
132C2×Dic6S411yescharacteristic
133#41⋊C3—22no—
134#41⋊C3—62no—
135#41⋊C3#6712yescomplemented
136C2×SL(2,𝔽3)—21no—
137C2×SL(2,𝔽3)C22×D311yescharacteristic
138GL(2,𝔽3)—21no—
139#226—21no—
140C3×Q16—61no—
141C3×Q16—61no—
142C3×SD16—62no—
143C3×SD16—61no—
144C3×SD16—61no—
145C3×D8—62no—
146C2×C3×C8—31no—
147C2×C3×C8—31no—
148C2×C3×C8—62no—
149#217—62no—
150#215—64no—
151#213—64no—
152(C2×D3)⋊C4—62no—
153(C2×D3)⋊C4—122no—
154C4⋊Dic3—32no—
155C4⋊Dic3—61no—
156C4⋊Dic3—121no—
157C4⋊Dic3—241no—
158Dic3⋊C4—62no—
159Dic3⋊C4—122no—
160#208—62no—
161C2×C3⋊C8—32no—
162C22×C32—41no—
163C2×C3×D3—42no—
164C2×C3×D3—81no—
165C2×C3×D3—81no—
166C32×C4—42no—
167C3⋊Dic3—82no—
168C3×Dic3—82no—
169C3×Dic3—81no—
170H(5,𝔽2)—62no—
171H(5,𝔽2)—182no—
172C2×#41—31no—
173C2×#41—32no—
174C2×#41—91no—
175C2×#41D3211yescharacteristic
176C22×D4—91no—
177C22×D4—92no—
178AGL(1,ℤ/8)—184no—
179#135—62no—
180#135—62no—
181C2×Q16—31no—
182C2×SD16—31no—
183C2×SD16—31no—
184C2×SD16—92no—
185C2×D8—31no—
186C2×D8—92no—
187C2×M16—91no—
188C22×C8—31no—
189C22⋊Q8—92no—
190C22⋊Q8—181no—
191#121—92no—
192#121—181no—
193#121—182no—
194C2×C4⋊C4—91no—
195#103—92no—
196#102—92no—
197#102—92no—
198#102—92no—
199C22⋊C8—92no—
200C22⋊C8—92no—
201C23×C3—31no—
202C23×C3—61no—
203C22×D3—32no—
204C22×D3—41no—
205C22×D3—64no—
206C22×D3—241no—
207C22×D3—241no—
208C3×Q8—31no—
209C3×Q8—61no—
210C3×Q8—61no—
211C3×Q8—242no—
212C3×Q8C2×#6711yescharacteristic
213C3×D4—32no—
214C3×D4—61no—
215C3×D4—61no—
216C3×D4—61no—
217C3×D4—122no—
218C2×C3×C4—31no—
219C2×C3×C4—32no—
220C2×C3×C4—32no—
221C2×C3×C4—41no—
222C2×C3×C4—61no—
223C2×C3×C4—81no—
224C2×C3×C4—122no—
225C2×C3×C4—122no—
226C2×C3×C4C2×S411yescharacteristic
227#67—64no—
228#67—242no—
229C2×Dic3—41no—
230C2×Dic3—62no—
231C2×Dic3—122no—
232C2×Dic3—242no—
233C2×Dic3—241no—
234C2×Dic3C2×S412yes—
235D12—34no—
236D12—62no—
237D12—62no—
238C4×D3—62no—
239C4×D3—62no—
240C4×D3—82no—
241Dic6—22no—
242SL(2,𝔽3)—21no—
243SL(2,𝔽3)C2×#6711yescharacteristic
244C3×C8—61no—
245C3×C8—61no—
246C3⋊C8—62no—
247C2×C32—41no—
248C2×C32—41no—
249C2×C32—41no—
250C3×D3—82no—
251C24—181no—
252#41—32no—
253#41—62no—
254#41—62no—
255#41—62no—
256#41—182no—
257#41—182no—
258#41—182no—
259#41#62512yes—
260C2×Q8—31no—
261C2×Q8—31no—
262C2×Q8C2×D3211yescharacteristic
263C2×D4—31no—
264C2×D4—31no—
265C2×D4—61no—
266C2×D4—92no—
267C2×D4—94no—
268C2×D4—92no—
269C2×D4—91no—
270C2×D4—182no—
271C2×D4—184no—
272C2×D4—184no—
273C2×D4—182no—
274C22×C4—31no—
275C22×C4—61no—
276C22×C4—92no—
277C22×C4—181no—
278Q16—61no—
279Q16—61no—
280SD16—62no—
281SD16—61no—
282SD16—61no—
283SD16—184no—
284D8—62no—
285D8—184no—
286M16—182no—
287C2×C8—31no—
288C2×C8—31no—
289C2×C8—62no—
290C2×C8—92no—
291C4⋊C4—92no—
292C4⋊C4—182no—
293C4⋊C4—181no—
294C22⋊C4—182no—
295C22⋊C4—182no—
296C22×C3—32no—
297C22×C3—41no—
298C22×C3—61no—
299C22×C3—61no—
300C22×C3—81no—
301C22×C3—121no—
302C22×C3—121no—
303C22×C3—121no—
304C22×C3C22×S411yescharacteristic
305C2×D3—42no—
306C2×D3—62no—
307C2×D3—62no—
308C2×D3—62no—
309C2×D3—81no—
310C2×D3—81no—
311C2×D3—244no—
312C3×C4—31no—
313C3×C4—31no—
314C3×C4—42no—
315C3×C4—82no—
316C3×C4—121no—
317C3×C4—242no—
318C3×C4GL(2,ℤ/4)12yes—
319Dic3—22no—
320Dic3—81no—
321Dic3—122no—
322Dic3—242no—
323Dic3—482no—
324C32—41noSylow
325C23—31no—
326C23—61no—
327C23—92no—
328C23—184no—
329C23—721no—
330C23—721no—
331Q8—31no—
332Q8—61no—
333Q8—61no—
334Q8—62no—
335Q8C2×#62511yescharacteristic
336D4—32no—
337D4—61no—
338D4—61no—
339D4—61no—
340D4—94no—
341D4—122no—
342D4—182no—
343D4—184no—
344D4—182no—
345D4—722no—
346C2×C4—31no—
347C2×C4—32no—
348C2×C4—32no—
349C2×C4—32no—
350C2×C4—61no—
351C2×C4—122no—
352C2×C4—182no—
353C2×C4—182no—
354C2×C4—182no—
355C2×C4—721no—
356C2×C4D3×S411yescharacteristic
357C8—61no—
358C8—61no—
359C8—182no—
360C2×C3—41no—
361C2×C3—41no—
362C2×C3—41no—
363C2×C3—62no—
364C2×C3—81no—
365C2×C3—81no—
366C2×C3—81no—
367C2×C3—241no—
368C2×C3#592411yescharacteristic
369C2×C3#592511yescharacteristic
370C2×C3C2×GL(2,ℤ/4)11yescharacteristic
371D3—82no—
372D3—122no—
373C22—32no—
374C22—61no—
375C22—61no—
376C22—121no—
377C22—121no—
378C22—121no—
379C22—182no—
380C22—182no—
381C22—182no—
382C22—724no—
383C22C2×D3×S411yescharacteristic
384C4—31no—
385C4—31no—
386C4—62no—
387C4—121no—
388C4—362no—
389C4#6420912yes—
390C3—41no—
391C3—81no—
392C3#8552911yescharacteristic, complemented
393C2—62no—
394C2—241no—
395C2—362no—
396C2#1059644711yescharacteristic
397C2#1059644911yescharacteristic
398C2C2×#6420911yescharacteristic
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×C3⋊#994C2semidirect
2. C2×C3×#985C2semidirect
3. #10593660C2semidirect
4. #10593203C2semidirect
5. #10593194C2semidirect
12. C2×C3×GL(2,𝔽3)C22semidirect
13. C2×C3×#226C22semidirect
15. C3⋊#859C22semidirect
16. C2×#1009D3semidirect
18. C2×#985D3semidirect
35. C3×#41⋊C3D4semidirect
57. C2×GL(2,𝔽3)C2×D3semidirect
58. C2×#226C2×D3semidirect
135. #41⋊C3#67semidirect
392. C3#85529semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊#9942
C2×C3×#9852
#105936602
#105932032
#105931942
#855293
#839123
#638564