Finite Group Databasebeta

← #11922923

Subgroups — #11922923

Order 1152.

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

normal subgroups only

This group has 538 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#10593659C211yesmaximal, characteristic, complemented
3#10593658C211yesmaximal, characteristic, complemented
4#10593653C211yesmaximal, characteristic, complemented
5C2×C3⋊#859C211yesmaximal, characteristic, complemented
6#10593623C211yesmaximal, characteristic, complemented
7C4×C3⋊#226C211yesmaximal, characteristic, complemented
8#85375—33nomaximal
9#83534—31nomaximal
10C2×C3×#41⋊C3C2211yescharacteristic
11C22×C3×SL(2,𝔽3)C2211yescharacteristic
12C2×C3⋊#226C2211yescharacteristic
13#64086—41nomaximal
14C3×C4×SL(2,𝔽3)C2211yescharacteristic
15C3⋊#859C2211yescharacteristic
16C3⋊#859C2211yescharacteristic
17C3⋊#859C2211yescharacteristic
18C3×#548D311yescharacteristic
19#5541—31no—
20#548⋊C3D313yes—
21#5425—33no—
22#5424—33no—
23#5419—33no—
24C2×#859—33no—
25#5389—33no—
26C4×#226—33no—
27#5239—31no—
28#5233—31no—
29#5228—31no—
30C2×#835—31no—
31#5056—31no—
32#5054—31no—
33#5051—31no—
34#5045—31no—
35#5041—31no—
36#5033—31no—
37C4×#216—31no—
38#5022—31no—
39D4×C3⋊C8—31no—
40C22×C3⋊Dic3—41no—
41C3×#41⋊C3—21no—
42C2×C3×SL(2,𝔽3)—21no—
43C2×C3×SL(2,𝔽3)C2311yescharacteristic
44C3⋊#226—21no—
45C32×C22⋊C4—41no—
46#3757—41no—
47#3750—41no—
48#3750—41no—
49C4×C3⋊Dic3—41no—
50C4×C3⋊Dic3—41no—
51#3300—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—
64#935—61no—
65C2×#214—31no—
66#926—31no—
67C2×C4⋊Dic3—31no—
68#924—31no—
69#924—31no—
70C22×C3⋊C8—31no—
71C22×C3⋊C8—31no—
72#889—61no—
73#879—61no—
74#877—61no—
75#877—123no—
76#870—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#807—31no—
96#807—31no—
97C4⋊(C3⋊C8)—31no—
98C4×C3⋊C8—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—44no—
107C3×SL(2,𝔽3)#4111yescharacteristic
108#552—91no—
109#548C3⋊D311yescharacteristic
110#488—91no—
111#485—91no—
112#480—91no—
113#479—91no—
114#467—91no—
115#465—91no—
116#464—91no—
117#452—91no—
118#451—91no—
119C4×Q16—91no—
120C8×D4—91no—
121#416—91no—
122C2×#103—91no—
123C3×#41—21no—
124C3×#41—61no—
125C2×C3×Q8—21no—
126C2×C3×Q8—31no—
127C2×C3×Q8—61no—
128C2×C3×Q8C22×D311yescharacteristic
129C2×C3×D4—31no—
130C2×C3×D4—31no—
131C22×C3×C4—31no—
132C22×C3×C4—31no—
133C22×Dic3—61no—
134C22×Dic3—123no—
135C2×Dic6—31no—
136#41⋊C3—23no—
137C2×SL(2,𝔽3)—23no—
138C2×SL(2,𝔽3)C22×D313yes—
139#226—63no—
140C3×C4⋊C4—31no—
141C3×C4⋊C4—31no—
142C3×C22⋊C4—31no—
143C3×C22⋊C4—43no—
144C3×C22⋊C4—61no—
145C3×C22⋊C4S411yescharacteristic
146C3×C42—31no—
147#217—61no—
148#217—61no—
149#217—61no—
150#217—123no—
151#216—61no—
152#216—61no—
153#216—121no—
154#214—121no—
155C4⋊Dic3—31no—
156C4⋊Dic3—31no—
157C4⋊Dic3—31no—
158C4⋊Dic3—61no—
159Dic3⋊C4—61no—
160Dic3⋊C4—61no—
161Dic3⋊C4—61no—
162Dic3⋊C4—61no—
163Dic3⋊C4—61no—
164Dic3⋊C4—123no—
165Dic3⋊C4—123no—
166C4×Dic3—61no—
167C4×Dic3—61no—
168C4×Dic3—123no—
169C4×Dic3—123no—
170C2×C3⋊C8—31no—
171C2×C3⋊C8—31no—
172C2×C3⋊C8—31no—
173C2×C3⋊C8—31no—
174C2×C3⋊C8—121no—
175C2×C3⋊C8—121no—
176C22×C32—42no—
177C22×C32—41no—
178C22×C32—81no—
179C22×C32—81no—
180C32×C4—81no—
181C32×C4—81no—
182C3⋊Dic3—42no—
183C3⋊Dic3—42no—
184C3⋊Dic3—81no—
185C3⋊Dic3—81no—
186C2×#41C2×C3⋊D311yescharacteristic
187C22×Q8C2×C3⋊D311yescharacteristic
188C2×Q16—91no—
189C2×Q16—91no—
190C2×SD16—91no—
191C22×C8—91no—
192C22×C8—91no—
193#126—181no—
194#125—91no—
195#125—181no—
196#124—31no—
197#123—181no—
198C22⋊Q8—31no—
199C22⋊Q8—91no—
200C22⋊Q8—91no—
201#121—31no—
202C4×Q8—91no—
203C4×Q8C2×C3⋊D311yescharacteristic
204C4×D4—31no—
205#117—91no—
206#117—181no—
207C2×C4⋊C4—91no—
208#107—91no—
209#107—91no—
210#106—91no—
211C4⋊C8—91no—
212#103—91no—
213#103—91no—
214#103—91no—
215#103—91no—
216#103—91no—
217#103—91no—
218#103—91no—
219#102—91no—
220#102—91no—
221#102—91no—
222C22⋊C8—91no—
223C22⋊C8—91no—
224C4×C8—91no—
225C23×C3—31no—
226C23×C3—43no—
227C23×C3C2×S411yescharacteristic
228C3×Q8—31no—
229C3×Q8—61no—
230C3×Q8—61no—
231C3×Q8#23711yescharacteristic
232C3×D4—61no—
233C3×D4—61no—
234C3×D4—62no—
235C2×C3×C4—31no—
236C2×C3×C4—31no—
237C2×C3×C4—31no—
238C2×C3×C4—43no—
239C2×C3×C4—43no—
240C2×C3×C4—61no—
241C2×C3×C4—61no—
242C2×C3×C4—61no—
243C2×C3×C4—61no—
244C2×C3×C4C2×S411yescharacteristic
245C2×C3×C4C2×S411yescharacteristic
246C2×Dic3—61no—
247C2×Dic3—61no—
248C2×Dic3—61no—
249C2×Dic3—61no—
250C2×Dic3—123no—
251C2×Dic3—123no—
252C2×Dic3—123no—
253C2×Dic3—123no—
254C2×Dic3—1212no—
255C2×Dic3—122no—
256Dic6—61no—
257Dic6—61no—
258SL(2,𝔽3)#23713yes—
259C3⋊C8—61no—
260C3⋊C8—61no—
261C3⋊C8—61no—
262C3⋊C8—61no—
263C2×C32—41no—
264C2×C32—41no—
265C2×C32—41no—
266C2×C32—82no—
267#41—21no—
268#41—61no—
269C2×Q8—21no—
270C2×Q8—31no—
271C2×Q8—61no—
272C2×Q8—91no—
273C2×Q8C22×C3⋊D311yescharacteristic
274C2×D4—31no—
275C2×D4—31no—
276C22×C4—31no—
277C22×C4—31no—
278C22×C4—181no—
279Q16—181no—
280Q16—181no—
281Q16—361no—
282SD16—361no—
283C2×C8—91no—
284C2×C8—91no—
285C2×C8—91no—
286C2×C8—91no—
287C2×C8—361no—
288C2×C8—361no—
289C4⋊C4—31no—
290C4⋊C4—31no—
291C4⋊C4—91no—
292C4⋊C4—91no—
293C4⋊C4—91no—
294C4⋊C4—181no—
295C4⋊C4—181no—
296C4⋊C4—181no—
297C4⋊C4—181no—
298C4⋊C4—181no—
299C4⋊C4—181no—
300C22⋊C4—31no—
301C22⋊C4—61no—
302C22⋊C4—181no—
303C22⋊C4—181no—
304C22⋊C4—181no—
305C22⋊C4C3⋊S411yescharacteristic
306C42—31no—
307C42—181no—
308C42—181no—
309C22×C3—21no—
310C22×C3—21no—
311C22×C3—46no—
312C22×C3—43no—
313C22×C3—61no—
314C22×C3—61no—
315C22×C3—61no—
316C22×C3—83no—
317C22×C3—83no—
318C22×C3#98512yes—
319C22×C3C22×S411yescharacteristic
320C3×C4—21no—
321C3×C4—21no—
322C3×C4—31no—
323C3×C4—31no—
324C3×C4—61no—
325C3×C4—61no—
326C3×C4—83no—
327C3×C4—83no—
328Dic3—126no—
329Dic3—126no—
330Dic3—121no—
331Dic3—121no—
332Dic3—121no—
333Dic3—121no—
334Dic3—243no—
335Dic3—243no—
336C32—41noSylow
337C23—31no—
338C23C2×C3⋊S411yescharacteristic
339Q8—31no—
340Q8—61no—
341Q8—61no—
342Q8—181no—
343Q8—181no—
344Q8#383011yescharacteristic
345D4—61no—
346D4—61no—
347D4—62no—
348C2×C4—31no—
349C2×C4—31no—
350C2×C4—31no—
351C2×C4—61no—
352C2×C4—61no—
353C2×C4—61no—
354C2×C4—61no—
355C2×C4—181no—
356C2×C4—181no—
357C2×C4—181no—
358C2×C4—181no—
359C2×C4—362no—
360C2×C4C2×C3⋊S411yescharacteristic
361C2×C4C2×C3⋊S411yescharacteristic
362C8—181no—
363C8—181no—
364C8—181no—
365C8—181no—
366C2×C3—22no—
367C2×C3—43no—
368C2×C3—43no—
369C2×C3—43no—
370C2×C3—86no—
371C2×C3—121no—
372C2×C3#591411yescharacteristic
373C2×C3C2×#98511yescharacteristic
374C2×C3#592711yescharacteristic
375C22—21no—
376C22—21no—
377C22—61no—
378C22—61no—
379C22—61no—
380C22#6426612yes—
381C22C22×C3⋊S411yescharacteristic
382C4—21no—
383C4—21no—
384C4—31no—
385C4—31no—
386C4—61no—
387C4—61no—
388C4—361no—
389C4—361no—
390C4—361no—
391C4—361no—
392C3—43no—
393C3#8537511yescharacteristic, complemented
394C2—22no—
395C2—121no—
396C2#1059657711yescharacteristic
397C2C2×#6426611yescharacteristic
398C2#1059659011yescharacteristic
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. #10593659C2semidirect
3. #10593658C2semidirect
4. #10593653C2semidirect
5. C2×C3⋊#859C2semidirect
6. #10593623C2semidirect
7. C4×C3⋊#226C2semidirect
393. C3#85375semidirect

Maximal subgroups

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

subgroupindex
C3×#548⋊C32
#105936592
#105936582
#105936532
C2×C3⋊#8592
#105936232
C4×C3⋊#2262
#853753
#835343
#640864