Finite Group Databasebeta

← #11923066

Subgroups — #11923066

Order 1152.

742 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 742 conjugacy classes of subgroups — too many to draw. Only its 85 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
1C22×C3×#41⋊C3C211yesmaximal, characteristic
2C2×#64265C211yesmaximal, characteristic, complemented
3#10593660C211yesmaximal, characteristic, complemented
4#10593659C212yesmaximal, complemented
5#10593653C212yesmaximal, complemented
6#85517—33nomaximal
7#84084—31nomaximal
8C2×C3×#41⋊C3—22no—
9C2×C3×#41⋊C3C2211yescharacteristic
10C2×C3×#41⋊C3C2211yescharacteristic
11C22×C3×SL(2,𝔽3)C2211yescharacteristic
12#64265—22no—
13C2×C3⋊#226C2212yescomplemented
14#64133—41nomaximal
15C3⋊#859C2212yescomplemented
16C22×C3×#41D311yescharacteristic
17C22×#41⋊C3D313yes—
18C2×#984—33no—
19C2×#951—31no—
20C2×#951—31no—
21C2×#924—31no—
22#5426—33no—
23#5425—36no—
24#5419—36no—
25#5239—32no—
26#5239—32no—
27#5232—31no—
28#5232—31no—
29#5228—32no—
30#5218—32no—
31#5207—31no—
32C22×C32×C4—41no—
33C2×#634—41no—
34C3×#41⋊C3—21no—
35C3×#41⋊C3D412yes—
36C2×C3×SL(2,𝔽3)—21no—
37C2×C3×SL(2,𝔽3)C2311yescharacteristic
38C3⋊#226—22no—
39#3757—42no—
40C4⋊(C3⋊Dic3)—41no—
41#3750—42no—
42#3007—91noSylow
43C2×C3×#41—22no—
44C2×C3×#41—31no—
45C2×C3×#41—31no—
46C2×C3×#41—62no—
47C2×C3×#41C2×D311yescharacteristic
48C2×C3×#41C2×D311yescharacteristic
49C22×C3×Q8C2×D311yescharacteristic
50C22×C3×D4—31no—
51C23×C3×C4—31no—
52C22×Dic6—31no—
53C2×#41⋊C3—26no—
54C2×#41⋊C3C2×D313yes—
55C2×#41⋊C3C2×D313yes—
56C22×SL(2,𝔽3)C2×D313yes—
57#984—66no—
58C2×#226—36no—
59C2×#217—61no—
60#951—62no—
61#951—64no—
62#951—62no—
63C2×#216—32no—
64C2×#216—32no—
65C2×#214—32no—
66C2×#214—32no—
67C2×C4⋊Dic3—31no—
68#924—32no—
69#924—32no—
70#924—61no—
71#924—61no—
72#924—123no—
73C2×Dic3⋊C4—61no—
74C2×#208—31no—
75C2×#208—31no—
76#859—36no—
77#835—32no—
78#835—32no—
79#832—32no—
80#832—32no—
81#830—32no—
82#830—32no—
83C23×C32—41no—
84C2×C32×C4—41no—
85C2×C32×C4—41no—
86C2×C32×C4—82no—
87C2×C3⋊Dic3—42no—
88C2×C3⋊Dic3—42no—
89#634—82no—
90C3×SL(2,𝔽3)C2×D411yescharacteristic
91C22×#41C3⋊D311yescharacteristic
92C2×#137—91no—
93C2×#137—91no—
94C2×C22⋊Q8—91no—
95#452—92no—
96#452—92no—
97#451—92no—
98#450—92no—
99#417—91no—
100#417—91no—
101#407—91no—
102C24×C3—31no—
103C3×#41—21no—
104C3×#41—32no—
105C3×#41—34no—
106C3×#41—62no—
107C3×#41—62no—
108C3×#41—61no—
109C3×#41—62no—
110C3×#41#6712yes—
111C2×C3×Q8—21no—
112C2×C3×Q8—31no—
113C2×C3×Q8—61no—
114C2×C3×Q8C22×D311yescharacteristic
115C2×C3×D4—31no—
116C2×C3×D4—31no—
117C2×C3×D4—61no—
118C2×C3×D4—62no—
119C2×C3×D4—62no—
120C22×C3×C4—31no—
121C22×C3×C4—31no—
122C22×C3×C4—31no—
123C22×C3×C4—31no—
124C22×C3×C4—31no—
125C22×C3×C4—43no—
126C22×C3×C4—62no—
127C22×C3×C4—62no—
128C22×C3×C4S411yescharacteristic
129C22×Dic3—61no—
130C22×Dic3—61no—
131C2×Dic6—32no—
132C2×Dic6—62no—
133C2×Dic6—62no—
134C2×Dic6—61no—
135C2×Dic6—123no—
136#41⋊C3—23no—
137#41⋊C3#6716yes—
138C2×SL(2,𝔽3)—23no—
139C2×SL(2,𝔽3)C22×D313yes—
140#226—66no—
141#217—62no—
142#217—62no—
143#217—126no—
144#216—62no—
145#216—64no—
146#216—62no—
147#214—64no—
148#214—64no—
149C4⋊Dic3—32no—
150C4⋊Dic3—61no—
151C4⋊Dic3—123no—
152Dic3⋊C4—62no—
153Dic3⋊C4—62no—
154Dic3⋊C4—126no—
155#208—62no—
156#208—62no—
157C2×C3⋊C8—32no—
158C2×C3⋊C8—32no—
159C22×C32—42no—
160C22×C32—41no—
161C22×C32—81no—
162C22×C32—81no—
163C32×C4—42no—
164C32×C4—81no—
165C3⋊Dic3—82no—
166C3⋊Dic3—82no—
167C2×#41—22no—
168C2×#41—31no—
169C2×#41—31no—
170C2×#41—62no—
171C2×#41C2×C3⋊D311yescharacteristic
172C2×#41C2×C3⋊D311yescharacteristic
173C22×Q8—91no—
174C22×Q8C2×C3⋊D311yescharacteristic
175C22×D4—31no—
176C23×C4—31no—
177#137—182no—
178#137—184no—
179#137—182no—
180C2×Q16—92no—
181C2×Q16—92no—
182C2×SD16—92no—
183C2×SD16—92no—
184C2×M16—91no—
185C2×M16—91no—
186C22⋊Q8—92no—
187C22⋊Q8—92no—
188C22⋊Q8—181no—
189C22⋊Q8—181no—
190C2×C4⋊C4—91no—
191C2×C4⋊C4—181no—
192C2×C22⋊C4—181no—
193#103—92no—
194#103—92no—
195#102—92no—
196#102—92no—
197C22⋊C8—92no—
198C22⋊C8—92no—
199C23×C3—31no—
200C23×C3—31no—
201C23×C3—43no—
202C23×C3—62no—
203C23×C3—62no—
204C23×C3—121no—
205C23×C3C2×S411yescharacteristic
206C3×Q8—31no—
207C3×Q8—32no—
208C3×Q8—61no—
209C3×Q8C2×#6711yescharacteristic
210C3×D4—32no—
211C3×D4—32no—
212C3×D4—61no—
213C3×D4—61no—
214C3×D4—64no—
215C2×C3×C4—22no—
216C2×C3×C4—31no—
217C2×C3×C4—32no—
218C2×C3×C4—32no—
219C2×C3×C4—31no—
220C2×C3×C4—43no—
221C2×C3×C4—43no—
222C2×C3×C4—61no—
223C2×C3×C4—61no—
224C2×C3×C4—62no—
225C2×C3×C4—61no—
226C2×C3×C4—61no—
227C2×C3×C4—62no—
228C2×C3×C4—86no—
229C2×C3×C4C2×S411yescharacteristic
230C2×C3×C4C2×S411yescharacteristic
231C2×Dic3—62no—
232C2×Dic3—62no—
233C2×Dic3—126no—
234C2×Dic3—126no—
235C2×Dic3—122no—
236C2×Dic3—122no—
237Dic6—62no—
238Dic6—62no—
239Dic6—122no—
240Dic6—246no—
241SL(2,𝔽3)C2×#6713yes—
242C3⋊C8—62no—
243C3⋊C8—62no—
244C2×C32—41no—
245C2×C32—41no—
246C2×C32—41no—
247C2×C32—82no—
248C24—31no—
249#41—21no—
250#41—32no—
251#41—34no—
252#41—62no—
253#41—62no—
254#41—61no—
255#41—62no—
256#41#63812yes—
257C2×Q8—21no—
258C2×Q8—31no—
259C2×Q8—61no—
260C2×Q8—92no—
261C2×Q8—182no—
262C2×Q8—182no—
263C2×Q8—181no—
264C2×Q8C22×C3⋊D311yescharacteristic
265C2×D4—31no—
266C2×D4—31no—
267C2×D4—61no—
268C2×D4—62no—
269C2×D4—62no—
270C22×C4—31no—
271C22×C4—31no—
272C22×C4—31no—
273C22×C4—31no—
274C22×C4—31no—
275C22×C4—62no—
276C22×C4—62no—
277C22×C4—181no—
278C22×C4—181no—
279C22×C4C3⋊S411yescharacteristic
280Q16—182no—
281Q16—184no—
282Q16—182no—
283SD16—184no—
284SD16—184no—
285M16—182no—
286M16—182no—
287C2×C8—92no—
288C2×C8—92no—
289C4⋊C4—92no—
290C4⋊C4—182no—
291C4⋊C4—181no—
292C4⋊C4—182no—
293C22⋊C4—182no—
294C22⋊C4—182no—
295C22×C3—21no—
296C22×C3—21no—
297C22×C3—32no—
298C22×C3—46no—
299C22×C3—43no—
300C22×C3—61no—
301C22×C3—61no—
302C22×C3—61no—
303C22×C3—61no—
304C22×C3—61no—
305C22×C3—83no—
306C22×C3—83no—
307C22×C3—124no—
308C22×C3#98412yescomplemented
309C22×C3C22×S411yescharacteristic
310C3×C4—21no—
311C3×C4—31no—
312C3×C4—31no—
313C3×C4—46no—
314C3×C4—61no—
315C3×C4—83no—
316C3×C4GL(2,ℤ/4)12yes—
317Dic3—122no—
318Dic3—122no—
319Dic3—246no—
320Dic3—246no—
321C32—41noSylow
322C23—31no—
323C23—31no—
324C23—62no—
325C23—62no—
326C23—121no—
327C23C2×C3⋊S411yescharacteristic
328Q8—31no—
329Q8—32no—
330Q8—61no—
331Q8—182no—
332Q8—182no—
333Q8—362no—
334Q8C2×#63811yescharacteristic
335D4—32no—
336D4—32no—
337D4—61no—
338D4—61no—
339D4—64no—
340C2×C4—22no—
341C2×C4—31no—
342C2×C4—32no—
343C2×C4—32no—
344C2×C4—31no—
345C2×C4—61no—
346C2×C4—61no—
347C2×C4—62no—
348C2×C4—61no—
349C2×C4—61no—
350C2×C4—62no—
351C2×C4—182no—
352C2×C4—182no—
353C2×C4—362no—
354C2×C4—362no—
355C2×C4C2×C3⋊S411yescharacteristic
356C2×C4C2×C3⋊S411yescharacteristic
357C8—182no—
358C8—182no—
359C2×C3—22no—
360C2×C3—43no—
361C2×C3—43no—
362C2×C3—43no—
363C2×C3—62no—
364C2×C3—86no—
365C2×C3—121no—
366C2×C3C2×#98411yescharacteristic
367C2×C3#592311yescharacteristic
368C2×C3C2×GL(2,ℤ/4)11yescharacteristic
369C22—21no—
370C22—21no—
371C22—32no—
372C22—61no—
373C22—61no—
374C22—61no—
375C22—61no—
376C22—61no—
377C22—124no—
378C22#6426512yescomplemented
379C22C22×C3⋊S411yescharacteristic
380C4—21no—
381C4—31no—
382C4—31no—
383C4—61no—
384C4—362no—
385C4—362no—
386C4#6426912yes—
387C3—43no—
388C3#8551711yescharacteristic, complemented
389C2—22no—
390C2—62no—
391C2—121no—
392C2C2×#6426511yescharacteristic
393C2#1059658611yescharacteristic
394C2C2×#6426911yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC22×C3×#41D3
socleC22×C3C22×S4
prime radicalC22×C3×#41⋊C3C2
hypercenterC22×C4C3⋊S4
hypocenterC3×SL(2,𝔽3)C2×D4
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. C2×#64265C2semidirect
3. #10593660C2semidirect
4. #10593659C2semidirect
5. #10593653C2semidirect
13. C2×C3⋊#226C22semidirect
15. C3⋊#859C22semidirect
308. C22×C3#984semidirect
378. C22#64265semidirect
388. C3#85517semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#41⋊C32
C2×#642652
#105936602
#105936592
#105936532
#855173
#840843
#641334