Finite Group Databasebeta

← #11922901

Subgroups — #11922901

Order 1152.

514 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 81 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×#519⋊C3C211yesmaximal, characteristic
2#10593654C211yesmaximal, characteristic
3C2×C3⋊#859C211yesmaximal, characteristic
4#10593625C211yesmaximal, characteristic, complemented
5#10593623C211yesmaximal, characteristic, complemented
6#10593623C211yesmaximal, characteristic, complemented
7#10593620C211yesmaximal, characteristic, complemented
8#85351—33nomaximal
9#82550—31nomaximal
10C22×C3×SL(2,𝔽3)C2211yescharacteristic
11#64087—41nomaximal
12C3×C4×SL(2,𝔽3)C2211yescharacteristic
13C3×C4×SL(2,𝔽3)C2211yescharacteristic
14C3⋊#859C2211yescharacteristic
15C3⋊#859C2211yescharacteristic
16C3⋊#859C2211yescharacteristic
17C3⋊#859C2211yescharacteristic
18C3×#519D311yescharacteristic
19#5519—31no—
20#519⋊C3D313yes—
21#5420—33no—
22C2×#859—33no—
23#5391—33no—
24#5389—33no—
25#5389—33no—
26#5386—33no—
27#5225—31no—
28C2×#835—31no—
29#5022—31no—
30#5022—31no—
31#5022—31no—
32#5022—31no—
33#5021—31no—
34#5021—31no—
35#5020—31no—
36#5020—31no—
37#4999—31no—
38#4998—31no—
39#4995—31no—
40C22×C3⋊Dic3—41no—
41C2×#634—41no—
42C2×C3×SL(2,𝔽3)Q812yescomplemented
43C2×C3×SL(2,𝔽3)C2311yescharacteristic
44C32×C22⋊C4—41no—
45#3757—41no—
46C4⋊(C3⋊Dic3)—41no—
47#3750—41no—
48#3750—41no—
49#3082—91noSylow
50C22×C3×Q8C2×D311yescharacteristic
51C22×SL(2,𝔽3)C2×D313yes—
52C3×C4×Q8—31no—
53C3×C4×Q8—31no—
54C3×C4×Q8C2×D311yescharacteristic
55C3×C4×Q8C2×D311yescharacteristic
56C3×#117—31no—
57C3×#117—31no—
58#926—31no—
59C2×C4⋊Dic3—31no—
60C2×#208—31no—
61C22×C3⋊C8—31no—
62#890—61no—
63#888—61no—
64#878—61no—
65#878—61no—
66#878—123no—
67#870—31no—
68#870—31no—
69#869—31no—
70#869—31no—
71C4×SL(2,𝔽3)C2×D313yes—
72C4×SL(2,𝔽3)C2×D313yes—
73#859—33no—
74#859—33no—
75#859—33no—
76#859—33no—
77#835—31no—
78#835—31no—
79#835—31no—
80#835—31no—
81#835—31no—
82#835—31no—
83#835—31no—
84#835—31no—
85#808—31no—
86#808—31no—
87#808—31no—
88#808—31no—
89#807—31no—
90#807—31no—
91#807—31no—
92#807—31no—
93C4⋊(C3⋊C8)—31no—
94C4⋊(C3⋊C8)—31no—
95C4⋊(C3⋊C8)—31no—
96C4⋊(C3⋊C8)—31no—
97C23×C32—41no—
98C2×C32×C4—41no—
99C2×C32×C4—41no—
100C2×C3⋊Dic3—41no—
101C2×C3⋊Dic3—41no—
102C2×C3⋊Dic3—41no—
103C2×C3⋊Dic3—41no—
104C2×C3⋊Dic3—82no—
105#634—82no—
106C3×SL(2,𝔽3)C2×Q811yescharacteristic
107#544—91no—
108#519C3⋊D311yescharacteristic
109#479—91no—
110#479—91no—
111#479—91no—
112#479—91no—
113#477—91no—
114#477—91no—
115#475—91no—
116#475—91no—
117#428—91no—
118#427—91no—
119#424—91no—
120#419—91no—
121C2×#103—91no—
122C2×C3×Q8—31no—
123C2×C3×Q8—32no—
124C2×C3×Q8Dic612yescomplemented
125C2×C3×Q8C22×D311yescharacteristic
126C22×C3×C4—31no—
127C22×Dic3—61no—
128C22×Dic3—123no—
129C2×Dic6—61no—
130C2×Dic6—61no—
131C2×Dic6—123no—
132C2×SL(2,𝔽3)Dic616yescomplemented
133C2×SL(2,𝔽3)C22×D313yes—
134C3×C4⋊C4—31no—
135C3×C4⋊C4—31no—
136C3×C4⋊C4—31no—
137C3×C4⋊C4—31no—
138C3×C22⋊C4—31no—
139C3×C22⋊C4—43no—
140C3×C22⋊C4S411yescharacteristic
141C3×C42—31no—
142C3×C42—31no—
143C3×C42—61no—
144#217—61no—
145#217—123no—
146C4⋊Dic3—31no—
147C4⋊Dic3—31no—
148C4⋊Dic3—31no—
149C4⋊Dic3—31no—
150C4⋊Dic3—61no—
151C4⋊Dic3—61no—
152C4⋊Dic3—61no—
153C4⋊Dic3—123no—
154Dic3⋊C4—61no—
155Dic3⋊C4—61no—
156Dic3⋊C4—61no—
157Dic3⋊C4—61no—
158Dic3⋊C4—123no—
159Dic3⋊C4—123no—
160C4×Dic3—61no—
161#208—62no—
162C2×C3⋊C8—31no—
163C2×C3⋊C8—31no—
164C2×C3⋊C8—31no—
165C2×C3⋊C8—31no—
166C2×C3⋊C8—62no—
167C22×C32—42no—
168C22×C32—41no—
169C22×C32—81no—
170C22×C32—81no—
171C32×C4—81no—
172C32×C4—81no—
173C3⋊Dic3—42no—
174C3⋊Dic3—81no—
175C3⋊Dic3—81no—
176C3⋊Dic3—81no—
177C22×Q8C2×C3⋊D311yescharacteristic
178C2×M16—91no—
179C22×C8—91no—
180C4⋊Q8—91no—
181C4⋊Q8—91no—
182C4⋊Q8—181no—
183#125—91no—
184#125—91no—
185#125—181no—
186C22⋊Q8—181no—
187C22⋊Q8—181no—
188C4×Q8—31no—
189C4×Q8—31no—
190C4×Q8C2×C3⋊D311yescharacteristic
191C4×Q8C2×C3⋊D311yescharacteristic
192#117—31no—
193#117—31no—
194#117—91no—
195C2×C4⋊C4—91no—
196#107—91no—
197#107—91no—
198#107—91no—
199#107—91no—
200#106—91no—
201#106—91no—
202#106—91no—
203#106—91no—
204C4⋊C8—91no—
205C4⋊C8—91no—
206C4⋊C8—91no—
207C4⋊C8—91no—
208#103—91no—
209#103—91no—
210#103—91no—
211#103—91no—
212#103—91no—
213#103—91no—
214#103—91no—
215#103—91no—
216C23×C3—43no—
217C23×C3C2×S411yescharacteristic
218C3×Q8—32no—
219C3×Q8—31no—
220C3×Q8—61no—
221C3×Q8C2×Dic611yescharacteristic
222C2×C3×C4—32no—
223C2×C3×C4—32no—
224C2×C3×C4—31no—
225C2×C3×C4—31no—
226C2×C3×C4—31no—
227C2×C3×C4—31no—
228C2×C3×C4—43no—
229C2×C3×C4—43no—
230C2×C3×C4C2×S411yescharacteristic
231C2×C3×C4C2×S411yescharacteristic
232C2×Dic3—61no—
233C2×Dic3—61no—
234C2×Dic3—61no—
235C2×Dic3—61no—
236C2×Dic3—123no—
237C2×Dic3—123no—
238C2×Dic3—123no—
239C2×Dic3—123no—
240C2×Dic3—241no—
241C2×Dic3—246no—
242Dic6—241no—
243Dic6—241no—
244Dic6—246no—
245SL(2,𝔽3)C2×Dic613yes—
246C3⋊C8—61no—
247C3⋊C8—61no—
248C3⋊C8—61no—
249C3⋊C8—61no—
250C2×C32—41no—
251C2×C32—41no—
252C2×C32—41no—
253C2×C32—82no—
254C2×Q8—31no—
255C2×Q8—32no—
256C2×Q8—181no—
257C2×Q8—181no—
258C2×Q8#63412yescomplemented
259C2×Q8C22×C3⋊D311yescharacteristic
260C22×C4—31no—
261C22×C4—181no—
262M16—182no—
263C2×C8—91no—
264C2×C8—91no—
265C2×C8—91no—
266C2×C8—91no—
267C2×C8—182no—
268C4⋊C4—31no—
269C4⋊C4—31no—
270C4⋊C4—31no—
271C4⋊C4—31no—
272C4⋊C4—91no—
273C4⋊C4—91no—
274C4⋊C4—91no—
275C4⋊C4—91no—
276C4⋊C4—181no—
277C4⋊C4—181no—
278C4⋊C4—181no—
279C4⋊C4—181no—
280C4⋊C4—181no—
281C4⋊C4—181no—
282C4⋊C4—181no—
283C22⋊C4—31no—
284C22⋊C4—181no—
285C22⋊C4C3⋊S411yescharacteristic
286C42—31no—
287C42—31no—
288C42—61no—
289C42—181no—
290C22×C3—21no—
291C22×C3—21no—
292C22×C3—46no—
293C22×C3—43no—
294C22×C3—83no—
295C22×C3—83no—
296C22×C3A4⋊Q812yes—
297C22×C3C22×S411yescharacteristic
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—31no—
301C3×C4—31no—
302C3×C4—32no—
303C3×C4—61no—
304C3×C4—61no—
305C3×C4—83no—
306C3×C4—83no—
307Dic3—126no—
308Dic3—121no—
309Dic3—121no—
310Dic3—121no—
311Dic3—121no—
312Dic3—243no—
313Dic3—243no—
314Dic3—243no—
315C32—41noSylow
316C23C2×C3⋊S411yescharacteristic
317Q8—32no—
318Q8—31no—
319Q8—61no—
320Q8—721no—
321Q8—721no—
322Q8C2×#63411yescharacteristic
323C2×C4—32no—
324C2×C4—32no—
325C2×C4—31no—
326C2×C4—31no—
327C2×C4—31no—
328C2×C4—31no—
329C2×C4—181no—
330C2×C4—181no—
331C2×C4—181no—
332C2×C4—181no—
333C2×C4—721no—
334C2×C4C2×C3⋊S411yescharacteristic
335C2×C4C2×C3⋊S411yescharacteristic
336C8—181no—
337C8—181no—
338C8—181no—
339C8—181no—
340C2×C3—22no—
341C2×C3—43no—
342C2×C3—43no—
343C2×C3—43no—
344C2×C3—86no—
345C2×C3C2×A4⋊Q811yescharacteristic
346C2×C3#592611yescharacteristic
347C2×C3#592711yescharacteristic
348C22—21no—
349C22—21no—
350C22#6425912yes—
351C22C22×C3⋊S411yescharacteristic
352C4—21no—
353C4—21no—
354C4—31no—
355C4—31no—
356C4—32no—
357C4—61no—
358C4—61no—
359C4—361no—
360C4—361no—
361C4—361no—
362C4—361no—
363C3—43no—
364C3#8535111yescharacteristic, complemented
365C2—22no—
366C2C2×#6425911yescharacteristic
367C2#1059658911yescharacteristic
368C2#1059659011yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC3×#519D3
socleC22×C3C22×S4
prime radicalC22×C3×SL(2,𝔽3)C22
hypercenterC22⋊C4C3⋊S4
hypocenterC3×SL(2,𝔽3)C2×Q8
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
4. #10593625C2semidirect
5. #10593623C2semidirect
6. #10593623C2semidirect
7. #10593620C2semidirect
42. C2×C3×SL(2,𝔽3)Q8semidirect
124. C2×C3×Q8Dic6semidirect
132. C2×SL(2,𝔽3)Dic6semidirect
258. C2×Q8#634semidirect
364. C3#85351semidirect

Maximal subgroups

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

subgroupindex
C3×#519⋊C32
#105936542
C2×C3⋊#8592
#105936252
#105936232
#105936202
#853513
#825503
#640874