Finite Group Databasebeta

← #11922944

Subgroups — #11922944

Order 1152.

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

normal subgroups only

This group has 542 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×D4×SL(2,𝔽3)C211yesmaximal, characteristic
2#10593654C211yesmaximal, characteristic, complemented
3#10593653C211yesmaximal, characteristic, complemented
4#10593653C211yesmaximal, characteristic, complemented
5C2×C3⋊#859C211yesmaximal, characteristic, complemented
6#10593623C211yesmaximal, characteristic, complemented
7#10593622C211yesmaximal, characteristic, complemented
8#85394—33nomaximal
9#83552—31nomaximal
10C22×C3×SL(2,𝔽3)C2211yescharacteristic
11C22×C3×SL(2,𝔽3)C2211yescharacteristic
12C2×C3⋊#226C2211yescharacteristic
13#64144—41nomaximal
14C3×C4×SL(2,𝔽3)C2211yescharacteristic
15C3⋊#859C2211yescharacteristic
16C3⋊#859C2211yescharacteristic
17C3⋊#859C2211yescharacteristic
18C3×D4×Q8D311yescharacteristic
19#5541—31no—
20D4×SL(2,𝔽3)D313yes—
21#5420—33no—
22#5419—33no—
23#5419—33no—
24C2×#859—33no—
25#5389—33no—
26#5388—33no—
27#5228—31no—
28#5228—31no—
29#5225—31no—
30C2×#835—31no—
31#5090—31no—
32#5082—31no—
33#5052—31no—
34#5051—31no—
35#5045—31no—
36#5043—31no—
37#5030—31no—
38#5022—31no—
39#5011—31no—
40C2×C32×D4—41no—
41C22×C3⋊Dic3—41no—
42C2×C3×SL(2,𝔽3)—21no—
43C2×C3×SL(2,𝔽3)—21no—
44C2×C3×SL(2,𝔽3)C2311yescharacteristic
45C3⋊#226—21no—
46#3757—41no—
47#3757—41no—
48#3757—41no—
49#3750—41no—
50#3750—41no—
51#3312—91noSylow
52C22×C3×Q8C2×D311yescharacteristic
53C22×C3×Q8C2×D311yescharacteristic
54C22×SL(2,𝔽3)C2×D313yes—
55C22×SL(2,𝔽3)C2×D313yes—
56C2×#226—33no—
57C3×C4⋊Q8—31no—
58C3×C22⋊Q8—31no—
59C3×C22⋊Q8—31no—
60C3×C4×Q8C2×D311yescharacteristic
61C3×C4×D4—31no—
62C2×#216—31no—
63C2×#216—31no—
64C2×#216—31no—
65#935—61no—
66#935—123no—
67#926—31no—
68C2×C4⋊Dic3—31no—
69#924—31no—
70#924—31no—
71C2×#208—31no—
72C22×C3⋊C8—31no—
73#889—61no—
74#879—61no—
75#877—61no—
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#835—31no—
89#835—31no—
90#835—31no—
91#830—31no—
92#830—31no—
93#810—31no—
94#808—31no—
95#808—31no—
96#807—31no—
97C4⋊(C3⋊C8)—31no—
98#803—31no—
99C23×C32—41no—
100C23×C32—41no—
101C32×D4—82no—
102C2×C32×C4—41no—
103C2×C3⋊Dic3—41no—
104C2×C3⋊Dic3—41no—
105C2×C3⋊Dic3—41no—
106C2×C3⋊Dic3—41no—
107C2×C3⋊Dic3—82no—
108C3×SL(2,𝔽3)#4111yescharacteristic
109#552—91no—
110D4×Q8C3⋊D311yescharacteristic
111#491—91no—
112#485—91no—
113#483—91no—
114#479—91no—
115#470—91no—
116#467—91no—
117#462—91no—
118#451—91no—
119#451—91no—
120#441—91no—
121#435—91no—
122#419—91no—
123C2×#103—91no—
124C2×C3×Q8—21no—
125C2×C3×Q8—21no—
126C2×C3×Q8—31no—
127C2×C3×Q8—31no—
128C2×C3×Q8—61no—
129C2×C3×Q8—61no—
130C2×C3×Q8C22×D311yescharacteristic
131C2×C3×D4—43no—
132C2×C3×D4S411yescharacteristic
133C22×C3×C4—31no—
134C22×C3×C4—31no—
135C22×Dic3—61no—
136C22×Dic3—123no—
137C2×Dic6—31no—
138C2×SL(2,𝔽3)—23no—
139C2×SL(2,𝔽3)—23no—
140C2×SL(2,𝔽3)C22×D313yes—
141#226—63no—
142C3×C4⋊C4—31no—
143C3×C4⋊C4—31no—
144C3×C4⋊C4—61no—
145C3×C22⋊C4—31no—
146C3×C22⋊C4—31no—
147C3×C42—31no—
148#217—61no—
149#217—61no—
150#217—61no—
151#217—123no—
152#217—123no—
153#217—123no—
154#216—61no—
155#216—61no—
156#216—121no—
157#216—121no—
158C4⋊Dic3—31no—
159C4⋊Dic3—31no—
160C4⋊Dic3—31no—
161C4⋊Dic3—61no—
162Dic3⋊C4—61no—
163Dic3⋊C4—61no—
164Dic3⋊C4—61no—
165Dic3⋊C4—61no—
166Dic3⋊C4—61no—
167Dic3⋊C4—123no—
168Dic3⋊C4—123no—
169C4×Dic3—61no—
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—
181C22×C32—81no—
182C22×C32—81no—
183C22×C32—81no—
184C32×C4—81no—
185C3⋊Dic3—81no—
186C3⋊Dic3—81no—
187C3⋊Dic3—81no—
188C3⋊Dic3—81no—
189C22×Q8C2×C3⋊D311yescharacteristic
190C22×Q8C2×C3⋊D311yescharacteristic
191C2×Q16—91no—
192C2×Q16—91no—
193C2×Q16—91no—
194C2×M16—91no—
195C22×C8—91no—
196C4⋊Q8—31no—
197#126—181no—
198#125—91no—
199#125—181no—
200#123—181no—
201C22⋊Q8—31no—
202C22⋊Q8—31no—
203C22⋊Q8—91no—
204C22⋊Q8—91no—
205C4×Q8—91no—
206C4×Q8C2×C3⋊D311yescharacteristic
207C4×D4—31no—
208#117—91no—
209#117—181no—
210C2×C4⋊C4—91no—
211#107—91no—
212#106—91no—
213#106—91no—
214C4⋊C8—91no—
215#103—91no—
216#103—91no—
217#103—91no—
218#103—91no—
219#103—91no—
220#103—91no—
221#103—91no—
222#103—91no—
223#103—91no—
224#103—91no—
225C22⋊C8—91no—
226C22⋊C8—91no—
227#97—91no—
228C23×C3—43no—
229C23×C3—43no—
230C23×C3C2×S411yescharacteristic
231C23×C3C2×S411yescharacteristic
232C3×Q8—31no—
233C3×Q8—61no—
234C3×Q8—61no—
235C3×Q8—61no—
236C3×Q8—61no—
237C3×Q8#23711yescharacteristic
238C3×D4—22no—
239C3×D4—86no—
240C2×C3×C4—31no—
241C2×C3×C4—31no—
242C2×C3×C4—31no—
243C2×C3×C4—31no—
244C2×C3×C4—43no—
245C2×C3×C4—61no—
246C2×C3×C4—61no—
247C2×C3×C4—61no—
248C2×C3×C4—61no—
249C2×C3×C4C2×S411yescharacteristic
250C2×Dic3—61no—
251C2×Dic3—61no—
252C2×Dic3—61no—
253C2×Dic3—61no—
254C2×Dic3—123no—
255C2×Dic3—123no—
256C2×Dic3—123no—
257C2×Dic3—123no—
258C2×Dic3—122no—
259C2×Dic3—246no—
260Dic6—61no—
261Dic6—61no—
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—
271C2×C32—161no—
272C2×Q8—21no—
273C2×Q8—21no—
274C2×Q8—31no—
275C2×Q8—31no—
276C2×Q8—61no—
277C2×Q8—61no—
278C2×Q8—91no—
279C2×Q8C22×C3⋊D311yescharacteristic
280C2×D4C3⋊S411yescharacteristic
281C22×C4—31no—
282C22×C4—31no—
283C22×C4—181no—
284Q16—181no—
285Q16—181no—
286Q16—361no—
287Q16—361no—
288M16—361no—
289C2×C8—91no—
290C2×C8—91no—
291C2×C8—91no—
292C2×C8—91no—
293C2×C8—361no—
294C4⋊C4—31no—
295C4⋊C4—31no—
296C4⋊C4—61no—
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—31no—
308C22⋊C4—181no—
309C22⋊C4—181no—
310C22⋊C4—181no—
311C42—31no—
312C42—181no—
313C42—181no—
314C22×C3—21no—
315C22×C3—21no—
316C22×C3—21no—
317C22×C3—21no—
318C22×C3—21no—
319C22×C3—46no—
320C22×C3—43no—
321C22×C3—83no—
322C22×C3—83no—
323C22×C3—83no—
324C22×C3—83no—
325C22×C3—83no—
326C22×C3#98312yes—
327C22×C3C22×S411yescharacteristic
328C3×C4—21no—
329C3×C4—31no—
330C3×C4—31no—
331C3×C4—61no—
332C3×C4—61no—
333C3×C4—61no—
334C3×C4—83no—
335Dic3—121no—
336Dic3—121no—
337Dic3—121no—
338Dic3—121no—
339Dic3—243no—
340Dic3—243no—
341Dic3—243no—
342Dic3—243no—
343C32—41noSylow
344C23C2×C3⋊S411yescharacteristic
345C23C2×C3⋊S411yescharacteristic
346Q8—31no—
347Q8—61no—
348Q8—61no—
349Q8—61no—
350Q8—61no—
351Q8—181no—
352Q8—181no—
353Q8#383011yescharacteristic
354D4—22no—
355C2×C4—31no—
356C2×C4—31no—
357C2×C4—31no—
358C2×C4—31no—
359C2×C4—61no—
360C2×C4—61no—
361C2×C4—61no—
362C2×C4—61no—
363C2×C4—181no—
364C2×C4—181no—
365C2×C4—181no—
366C2×C4—181no—
367C2×C4—362no—
368C2×C4C2×C3⋊S411yescharacteristic
369C8—181no—
370C8—181no—
371C8—181no—
372C8—181no—
373C2×C3—22no—
374C2×C3—41no—
375C2×C3—43no—
376C2×C3—43no—
377C2×C3—43no—
378C2×C3—86no—
379C2×C3—163no—
380C2×C3#591411yescharacteristic
381C2×C3C2×#98311yescharacteristic
382C2×C3#592711yescharacteristic
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22#6426412yes—
389C22C22×C3⋊S411yescharacteristic
390C4—21no—
391C4—31no—
392C4—31no—
393C4—61no—
394C4—61no—
395C4—61no—
396C4—361no—
397C4—361no—
398C4—361no—
399C4—361no—
400C3—43no—
401C3#8539411yescharacteristic, complemented
402C2—22no—
403C2—41no—
404C2#1059657711yescharacteristic
405C2C2×#6426411yescharacteristic
406C2#1059659011yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC3×D4×Q8D3
socleC22×C3C22×S4
prime radicalC3×D4×SL(2,𝔽3)C2
hypercenterC2×D4C3⋊S4
hypocenterC3×SL(2,𝔽3)#41
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #10593654C2semidirect
3. #10593653C2semidirect
4. #10593653C2semidirect
5. C2×C3⋊#859C2semidirect
6. #10593623C2semidirect
7. #10593622C2semidirect
401. C3#85394semidirect

Maximal subgroups

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

subgroupindex
C3×D4×SL(2,𝔽3)2
#105936542
#105936532
C2×C3⋊#8592
#105936232
#105936222
#853943
#835523
#641444