Finite Group Databasebeta

← #11922927

Subgroups — #11922927

Order 1152.

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

normal subgroups only

This group has 536 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×#548⋊C3C211yesmaximal, characteristic
2#10593660C211yesmaximal, characteristic, complemented
3#10593659C211yesmaximal, characteristic, complemented
4#10593653C211yesmaximal, characteristic, complemented
5C2×C3⋊#859C211yesmaximal, characteristic, complemented
6#10593622C211yesmaximal, characteristic, complemented
7#10593620C211yesmaximal, characteristic, complemented
8#85377—33nomaximal
9#83538—31nomaximal
10C2×C3×#41⋊C3C2211yescharacteristic
11C22×C3×SL(2,𝔽3)C2211yescharacteristic
12C2×C3⋊#226C2211yescharacteristic
13#64087—41nomaximal
14C3×C4×SL(2,𝔽3)C2211yescharacteristic
15C3⋊#859C2211yescharacteristic
16C3⋊#859C2211yescharacteristic
17C3⋊#859C2211yescharacteristic
18C3×#548D311yescharacteristic
19#5543—31no—
20#548⋊C3D313yes—
21#5426—33no—
22#5425—33no—
23#5419—33no—
24C2×#859—33no—
25#5388—33no—
26#5386—33no—
27#5239—31no—
28#5232—31no—
29#5228—31no—
30C2×#835—31no—
31#5055—31no—
32#5054—31no—
33#5052—31no—
34#5043—31no—
35#5041—31no—
36#5034—31no—
37#5030—31no—
38#5020—31no—
39#5011—31no—
40C22×C3⋊Dic3—41no—
41C2×#634—41no—
42C3×#41⋊C3—21no—
43C2×C3×SL(2,𝔽3)—21no—
44C2×C3×SL(2,𝔽3)C2311yescharacteristic
45C3⋊#226—21no—
46C32×C22⋊C4—41no—
47#3757—41no—
48C4⋊(C3⋊Dic3)—41no—
49#3750—41no—
50#3750—41no—
51#3316—91noSylow
52C2×C3×#41C2×D311yescharacteristic
53C22×C3×Q8C2×D311yescharacteristic
54C2×#41⋊C3C2×D313yes—
55C22×SL(2,𝔽3)C2×D313yes—
56C2×#226—33no—
57C3×#124—31no—
58C3×C22⋊Q8—31no—
59C3×#121—31no—
60C3×C4×Q8C2×D311yescharacteristic
61C3×C4×D4—31no—
62C2×#216—31no—
63C2×#216—31no—
64D4×Dic3—61no—
65C2×#214—31no—
66C2×C4⋊Dic3—31no—
67C2×C4⋊Dic3—31no—
68#924—31no—
69#924—31no—
70C2×#208—31no—
71C22×C3⋊C8—31no—
72#890—61no—
73#878—61no—
74#878—61no—
75#878—123no—
76#869—31no—
77C4×Dic6—31no—
78C4×SL(2,𝔽3)C2×D313yes—
79#859—33no—
80#859—33no—
81#859—33no—
82#835—31no—
83#835—31no—
84#835—31no—
85#835—31no—
86#835—31no—
87#835—31no—
88#832—31no—
89#832—31no—
90#832—31no—
91#830—31no—
92#830—31no—
93#810—31no—
94#808—31no—
95#808—31no—
96#808—31no—
97C4⋊(C3⋊C8)—31no—
98#803—31no—
99C23×C32—41no—
100C2×C32×C4—41no—
101C2×C32×C4—41no—
102C2×C3⋊Dic3—41no—
103C2×C3⋊Dic3—41no—
104C2×C3⋊Dic3—41no—
105C2×C3⋊Dic3—41no—
106C2×C3⋊Dic3—82no—
107#634—82no—
108C3×SL(2,𝔽3)#4111yescharacteristic
109#554—91no—
110#548C3⋊D311yescharacteristic
111#489—91no—
112#483—91no—
113#481—91no—
114#475—91no—
115#470—91no—
116#465—91no—
117#464—91no—
118#452—91no—
119#451—91no—
120#441—91no—
121#435—91no—
122#417—91no—
123C2×#103—91no—
124C3×#41—21no—
125C3×#41—61no—
126C2×C3×Q8—21no—
127C2×C3×Q8—31no—
128C2×C3×Q8—61no—
129C2×C3×Q8C22×D311yescharacteristic
130C2×C3×D4—31no—
131C2×C3×D4—31no—
132C22×C3×C4—31no—
133C22×C3×C4—31no—
134C22×Dic3—61no—
135C22×Dic3—61no—
136C22×Dic3—123no—
137C2×Dic6—31no—
138C2×Dic6—61no—
139C2×Dic6—123no—
140#41⋊C3—23no—
141C2×SL(2,𝔽3)—23no—
142C2×SL(2,𝔽3)C22×D313yes—
143#226—63no—
144C3×C4⋊C4—31no—
145C3×C4⋊C4—31no—
146C3×C22⋊C4—31no—
147C3×C22⋊C4—43no—
148C3×C22⋊C4—61no—
149C3×C22⋊C4S411yescharacteristic
150C3×C42—31no—
151#217—61no—
152#217—61no—
153#217—123no—
154#216—61no—
155#216—61no—
156#216—121no—
157#214—121no—
158C4⋊Dic3—31no—
159C4⋊Dic3—31no—
160C4⋊Dic3—31no—
161C4⋊Dic3—61no—
162C4⋊Dic3—61no—
163C4⋊Dic3—61no—
164C4⋊Dic3—123no—
165Dic3⋊C4—61no—
166Dic3⋊C4—61no—
167Dic3⋊C4—61no—
168Dic3⋊C4—123no—
169Dic3⋊C4—123no—
170C4×Dic3—61no—
171#208—121no—
172C2×C3⋊C8—31no—
173C2×C3⋊C8—31no—
174C2×C3⋊C8—31no—
175C2×C3⋊C8—31no—
176C2×C3⋊C8—121no—
177C22×C32—42no—
178C22×C32—41no—
179C22×C32—81no—
180C22×C32—81no—
181C32×C4—81no—
182C32×C4—81no—
183C3⋊Dic3—42no—
184C3⋊Dic3—81no—
185C3⋊Dic3—81no—
186C3⋊Dic3—81no—
187C2×#41C2×C3⋊D311yescharacteristic
188C22×Q8C2×C3⋊D311yescharacteristic
189C2×Q16—91no—
190C2×Q16—91no—
191C2×SD16—91no—
192C2×M16—91no—
193C22×C8—91no—
194C4⋊Q8—91no—
195#125—181no—
196#124—31no—
197C22⋊Q8—31no—
198C22⋊Q8—91no—
199C22⋊Q8—91no—
200C22⋊Q8—181no—
201C22⋊Q8—181no—
202#121—31no—
203C4×Q8—91no—
204C4×Q8C2×C3⋊D311yescharacteristic
205C4×D4—31no—
206C4×D4—181no—
207C2×C4⋊C4—91no—
208C2×C4⋊C4—91no—
209#106—91no—
210#106—91no—
211#106—91no—
212C4⋊C8—91no—
213#103—91no—
214#103—91no—
215#103—91no—
216#103—91no—
217#103—91no—
218#103—91no—
219#103—91no—
220#102—91no—
221#102—91no—
222#102—91no—
223C22⋊C8—91no—
224C22⋊C8—91no—
225#97—91no—
226C23×C3—31no—
227C23×C3—43no—
228C23×C3C2×S411yescharacteristic
229C3×Q8—31no—
230C3×Q8—61no—
231C3×Q8—61no—
232C3×Q8#23711yescharacteristic
233C3×D4—61no—
234C3×D4—61no—
235C3×D4—62no—
236C2×C3×C4—31no—
237C2×C3×C4—31no—
238C2×C3×C4—31no—
239C2×C3×C4—43no—
240C2×C3×C4—43no—
241C2×C3×C4—61no—
242C2×C3×C4—61no—
243C2×C3×C4—61no—
244C2×C3×C4—61no—
245C2×C3×C4C2×S411yescharacteristic
246C2×C3×C4C2×S411yescharacteristic
247C2×Dic3—61no—
248C2×Dic3—61no—
249C2×Dic3—61no—
250C2×Dic3—61no—
251C2×Dic3—123no—
252C2×Dic3—123no—
253C2×Dic3—123no—
254C2×Dic3—123no—
255C2×Dic3—241no—
256C2×Dic3—241no—
257C2×Dic3—246no—
258Dic6—61no—
259Dic6—61no—
260Dic6—241no—
261Dic6—246no—
262SL(2,𝔽3)#23713yes—
263C3⋊C8—61no—
264C3⋊C8—61no—
265C3⋊C8—61no—
266C3⋊C8—61no—
267C2×C32—41no—
268C2×C32—41no—
269C2×C32—41no—
270C2×C32—82no—
271#41—21no—
272#41—61no—
273C2×Q8—21no—
274C2×Q8—31no—
275C2×Q8—61no—
276C2×Q8—91no—
277C2×Q8—181no—
278C2×Q8C22×C3⋊D311yescharacteristic
279C2×D4—31no—
280C2×D4—31no—
281C22×C4—31no—
282C22×C4—31no—
283C22×C4—181no—
284C22×C4—181no—
285Q16—181no—
286Q16—181no—
287Q16—361no—
288SD16—361no—
289M16—361no—
290C2×C8—91no—
291C2×C8—91no—
292C2×C8—91no—
293C2×C8—91no—
294C2×C8—361no—
295C4⋊C4—31no—
296C4⋊C4—31no—
297C4⋊C4—91no—
298C4⋊C4—91no—
299C4⋊C4—91no—
300C4⋊C4—181no—
301C4⋊C4—181no—
302C4⋊C4—181no—
303C4⋊C4—181no—
304C4⋊C4—181no—
305C4⋊C4—181no—
306C22⋊C4—31no—
307C22⋊C4—61no—
308C22⋊C4—181no—
309C22⋊C4—181no—
310C22⋊C4C3⋊S411yescharacteristic
311C42—31no—
312C42—181no—
313C22×C3—21no—
314C22×C3—21no—
315C22×C3—46no—
316C22×C3—43no—
317C22×C3—61no—
318C22×C3—61no—
319C22×C3—61no—
320C22×C3—83no—
321C22×C3—83no—
322C22×C3#98412yes—
323C22×C3C22×S411yescharacteristic
324C3×C4—21no—
325C3×C4—21no—
326C3×C4—31no—
327C3×C4—31no—
328C3×C4—61no—
329C3×C4—61no—
330C3×C4—83no—
331C3×C4—83no—
332Dic3—126no—
333Dic3—121no—
334Dic3—121no—
335Dic3—121no—
336Dic3—121no—
337Dic3—243no—
338Dic3—243no—
339Dic3—243no—
340C32—41noSylow
341C23—31no—
342C23C2×C3⋊S411yescharacteristic
343Q8—31no—
344Q8—61no—
345Q8—61no—
346Q8—181no—
347Q8—181no—
348Q8—721no—
349Q8#383011yescharacteristic
350D4—61no—
351D4—61no—
352D4—62no—
353C2×C4—31no—
354C2×C4—31no—
355C2×C4—31no—
356C2×C4—61no—
357C2×C4—61no—
358C2×C4—61no—
359C2×C4—61no—
360C2×C4—181no—
361C2×C4—181no—
362C2×C4—181no—
363C2×C4—181no—
364C2×C4—721no—
365C2×C4—721no—
366C2×C4C2×C3⋊S411yescharacteristic
367C2×C4C2×C3⋊S411yescharacteristic
368C8—181no—
369C8—181no—
370C8—181no—
371C8—181no—
372C2×C3—22no—
373C2×C3—43no—
374C2×C3—43no—
375C2×C3—43no—
376C2×C3—86no—
377C2×C3—121no—
378C2×C3#591411yescharacteristic
379C2×C3C2×#98411yescharacteristic
380C2×C3#592611yescharacteristic
381C22—21no—
382C22—21no—
383C22—61no—
384C22—61no—
385C22—61no—
386C22#6426512yes—
387C22C22×C3⋊S411yescharacteristic
388C4—21no—
389C4—21no—
390C4—31no—
391C4—31no—
392C4—61no—
393C4—61no—
394C4—361no—
395C4—361no—
396C4—361no—
397C4—361no—
398C3—43no—
399C3#8537711yescharacteristic, complemented
400C2—22no—
401C2—121no—
402C2#1059657711yescharacteristic
403C2C2×#6426511yescharacteristic
404C2#1059658911yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC3×#548D3
socleC22×C3C22×S4
prime radicalC3×#548⋊C3C2
hypercenterC22⋊C4C3⋊S4
hypocenterC3×SL(2,𝔽3)#41
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
2. #10593660C2semidirect
3. #10593659C2semidirect
4. #10593653C2semidirect
5. C2×C3⋊#859C2semidirect
6. #10593622C2semidirect
7. #10593620C2semidirect
399. C3#85377semidirect

Maximal subgroups

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

subgroupindex
C3×#548⋊C32
#105936602
#105936592
#105936532
C2×C3⋊#8592
#105936222
#105936202
#853773
#835383
#640874