Finite Group Databasebeta

← #11788824

Subgroups — #11788824

Order 1152.

763 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 763 conjugacy classes of subgroups — too many to draw. Only its 148 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
1#10596287C211yesmaximal, characteristic
2C3×#5812C211yesmaximal, characteristic, complemented
3C2×C3×#836C211yesmaximal, characteristic, complemented
4#10591113C212yesmaximal
5#10589158C212yesmaximal, complemented
6#71663—31nomaximal
7#70105—31nomaximal
8C22×C32×Q8C2211yescharacteristic
9C2×#3828C2211yescharacteristic
10#64156C412yescomplemented
11#64155—22no—
12#64153C412yescomplemented
13#64138—22no—
14#64135C2211yescharacteristic
15C3×Q8×Dic3—24no—
16C3×#926—22no—
17C3×#926C2212yes—
18C2×C3×#208C2212yes—
19C3×#836—22no—
20#63570—24no—
21C3×#519—31no—
22#5816—31no—
23#5816—31no—
24#5816—61no—
25#5812D311yescharacteristic, complemented
26C2×C3×#101—31no—
27C2×#836D311yescharacteristic, complemented
28#5139—32no—
29#5005—32no—
30#4557—32no—
31#4531—32no—
32C2×C32×Q8—22no—
33C2×C32×Q8—22no—
34C2×C32×Q8C2×C412yes—
35C2×C32×Q8D412yes—
36C22×C32×C4D412yes—
37C22×C32×C4C2311yescharacteristic
38C2×#638—21no—
39#3828—22no—
40#3828—22no—
41C2×C4⋊(C3⋊D3)C2×C411yescharacteristic
42C2×C4×C3⋊D3—21no—
43C2×#634C2×C411yescharacteristic
44#3757—21no—
45#3752—24no—
46C4⋊(C3⋊Dic3)—21no—
47#3750—24no—
48C4×C3⋊Dic3C2×C412yes—
49C3×#217—22no—
50C3×C4⋊Dic3—24no—
51C3×C4⋊Dic3—22no—
52C3×C4×Dic3—24no—
53C3×C4×Dic3D414yes—
54C3×#208—22no—
55C3×#208—22no—
56C2×C3×C3⋊C8—22no—
57#1912—91noSylow
58C22×C3×Q8—21no—
59C22×C3×Q8C2×D311yescharacteristic
60C22×C3×Q8C2×D311yescharacteristic
61C2×#235—31no—
62C2×#235—31no—
63C2×#235—61no—
64C2×C3×M16—32no—
65C3×C4×Q8—64no—
66C3×#117—32no—
67C3×#117—62no—
68#947—32no—
69#947—32no—
70#947—62no—
71#946—64no—
72#946—62no—
73#946—62no—
74Q8×Dic3—24no—
75#944—32no—
76#944—32no—
77#944—62no—
78#929—64no—
79#929—62no—
80#929—62no—
81#926—22no—
82#926—31no—
83#926—31no—
84#926—61no—
85#926C2×D312yes—
86C2×#208C2×D312yes—
87C3×#101—62no—
88#836—22no—
89#825—64no—
90#805—64no—
91C23×C32C2×D411yescharacteristic
92C22×C3⋊D3—21no—
93C32×Q8—24no—
94C32×Q8—24no—
95C2×C32×C4—22no—
96C2×C32×C4—22no—
97C2×C32×C4C22⋊C414yes—
98C2×C32×C4C22×C411yescharacteristic
99C2×C32×C4C2×D411yescharacteristic
100C2×C32×C4C2×D412yes—
101C2×C32×C4#4112yes—
102#638—41no—
103#638—41no—
104C2×C3⋊Dic3—22no—
105C2×C3⋊Dic3—21no—
106C4⋊(C3⋊D3)—22no—
107C4×C3⋊D3—42no—
108#634—22no—
109C2×C3×Dic3—24no—
110C3×C3⋊C8—42no—
111#536—91no—
112#519—31no—
113#421—92no—
114C2×#101—31no—
115H(3,ℤ/4)—92no—
116C2×C3×Q8—22no—
117C2×C3×Q8—22no—
118C2×C3×Q8—22no—
119C2×C3×Q8—22no—
120C2×C3×Q8—22no—
121C2×C3×Q8—22no—
122C2×C3×Q8—42no—
123C2×C3×Q8—42no—
124C2×C3×Q8C4×D312yes—
125C2×C3×Q8C4×D312yes—
126C2×C3×Q8D1212yes—
127C2×C3×Q8#6712yes—
128C22×C3×C4—22no—
129C22×C3×C4—21no—
130C22×C3×C4D1212yes—
131C22×C3×C4#6712yes—
132C22×C3×C4C22×D311yescharacteristic
133C22×C3×C4C22×D311yescharacteristic
134C2×#67—62no—
135C2×#67—61no—
136C2×#67—61no—
137#235—62no—
138#235—62no—
139#235—62no—
140#235—62no—
141#235—122no—
142#235—122no—
143C2×D12—31no—
144C2×D12—31no—
145C2×D12—61no—
146C2×C4×D3—62no—
147C2×C4×D3—61no—
148C2×C4×D3—61no—
149C2×Dic6—31no—
150C2×Dic6—31no—
151C2×Dic6—61no—
152C3×M16—62no—
153C3×M16—62no—
154C2×C3×C8—62no—
155C3×C4⋊C4—64no—
156C3×C4⋊C4—62no—
157C3×C22⋊C4—62no—
158C3×C42—34no—
159C3×C42—64no—
160#217—22no—
161#217—62no—
162#217—61no—
163#217—61no—
164(C2×D3)⋊C4—68no—
165(C2×D3)⋊C4—64no—
166(C2×D3)⋊C4—64no—
167C4⋊Dic3—24no—
168C4⋊Dic3—22no—
169C4⋊Dic3—62no—
170C4⋊Dic3—61no—
171C4⋊Dic3—61no—
172Dic3⋊C4—68no—
173Dic3⋊C4—64no—
174Dic3⋊C4—64no—
175C4×Dic3—24no—
176C4×Dic3—32no—
177C4×Dic3—32no—
178C4×Dic3—62no—
179C4×Dic3#6714yes—
180#208—22no—
181#208—22no—
182C2×C3⋊C8—22no—
183C22×C32—41no—
184C22×C32C2×C22⋊C411yescharacteristic
185C22×C32#12011yescharacteristic
186C22×C32#12311yescharacteristic
187C2×C3⋊D3—42no—
188C2×C3⋊D3—41no—
189C32×C4—24no—
190C32×C4C4×D412yes—
191C32×C4#12112yes—
192C3⋊Dic3—42no—
193C3⋊Dic3—41no—
194C3×Dic3—44no—
195C2×#41—91no—
196C22×Q8D3211yescharacteristic
197C2×M16—32no—
198C4⋊Q8—92no—
199#124—92no—
200#123—182no—
201C22⋊Q8—182no—
202C4×Q8—64no—
203#117—32no—
204#117—62no—
205#117—91no—
206#104—184no—
207#101—62no—
208C23×C3—21no—
209C23×C3C2×D1211yescharacteristic
210C23×C3C2×#6711yescharacteristic
211C22×D3—62no—
212C22×D3—61no—
213C22×D3—61no—
214C3×Q8—24no—
215C3×Q8—24no—
216C3×Q8—24no—
217C3×Q8—24no—
218C3×Q8—44no—
219C3×Q8—44no—
220C2×C3×C4—22no—
221C2×C3×C4—22no—
222C2×C3×C4—22no—
223C2×C3×C4—22no—
224C2×C3×C4—24no—
225C2×C3×C4—22no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—22no—
229C2×C3×C4—42no—
230C2×C3×C4—42no—
231C2×C3×C4—64no—
232C2×C3×C4(C2×D3)⋊C414yes—
233C2×C3×C4(C2×D3)⋊C414yes—
234C2×C3×C4C2×C4×D311yescharacteristic
235C2×C3×C4C2×C4×D311yescharacteristic
236C2×C3×C4C2×D1211yescharacteristic
237C2×C3×C4#23512yes—
238C2×C3×C4D3×D412yes—
239C2×C3×C4#23712yes—
240C2×C3×C4C2×#6711yescharacteristic
241C2×C3×C4C2×#6712yes—
242#67—122no—
243#67—122no—
244#67—121no—
245#67—121no—
246#67—121no—
247#67—121no—
248C2×Dic3—24no—
249C2×Dic3—64no—
250C2×Dic3—62no—
251C2×Dic3—62no—
252C2×Dic3—62no—
253C2×Dic3—61no—
254C2×Dic3—61no—
255D12—62no—
256D12—62no—
257D12—122no—
258C4×D3—124no—
259C4×D3—122no—
260C4×D3—122no—
261Dic6—62no—
262Dic6—62no—
263Dic6—122no—
264C3×C8—122no—
265C3⋊C8—42no—
266C2×C32—21no—
267C2×C32—21no—
268C2×C32—21no—
269C2×C32#38611yescharacteristic
270C3⋊D3—81no—
271#41—182no—
272#41—182no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—91no—
276C2×Q8#62412yes—
277C2×Q8#62612yes—
278C2×D4—91no—
279C2×D4—181no—
280C22×C4—181no—
281C22×C4#62612yes—
282C22×C4C2×D3211yescharacteristic
283M16—62no—
284M16—62no—
285C2×C8—62no—
286C4⋊C4—64no—
287C4⋊C4—62no—
288C4⋊C4—184no—
289C4⋊C4—181no—
290C22⋊C4—62no—
291C22⋊C4—181no—
292C22⋊C4—184no—
293C42—34no—
294C42—64no—
295C42—92no—
296C22×C3—21no—
297C22×C3—21no—
298C22×C3—21no—
299C22×C3—41no—
300C22×C3—41no—
301C22×C3—42no—
302C22×C3#88211yescharacteristic
303C22×C3#88611yescharacteristic
304C22×C3C2×(C2×D3)⋊C411yescharacteristic
305C22×C3C2×(C2×D3)⋊C411yescharacteristic
306C22×C3#92911yescharacteristic
307C22×C3#95311yescharacteristic
308C2×D3—122no—
309C2×D3—124no—
310C2×D3—122no—
311C2×D3—122no—
312C2×D3—121no—
313C2×D3—121no—
314C3×C4—24no—
315C3×C4—24no—
316C3×C4—22no—
317C3×C4—22no—
318C3×C4—44no—
319C3×C4—124no—
320C3×C4#67⋊C412yes—
321C3×C4C4×#6712yes—
322C3×C4#93012yes—
323C3×C4#93912yes—
324Dic3—44no—
325Dic3—122no—
326Dic3—124no—
327Dic3—122no—
328Dic3—122no—
329Dic3—121no—
330Dic3—121no—
331C32#191211yescharacteristic, complemented, Sylow
332C23—181no—
333C23C2×#62611yescharacteristic
334Q8—24no—
335Q8—24no—
336Q8—182no—
337D4—182no—
338D4—361no—
339D4—361no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—64no—
343C2×C4—182no—
344C2×C4—181no—
345C2×C4—362no—
346C2×C4#372214yes—
347C2×C4#380412yes—
348C2×C4C2×#62411yescharacteristic
349C2×C4C2×#62611yescharacteristic
350C2×C4D3×#6712yes—
351C8—122no—
352C2×C3—22no—
353C2×C3—21no—
354C2×C3—21no—
355C2×C3—21no—
356C2×C3—21no—
357C2×C3—21no—
358C2×C3—21no—
359C2×C3—21no—
360C2×C3—41no—
361C2×C3—41no—
362C2×C3#495811yescharacteristic
363C2×C3#521311yescharacteristic
364D3—242no—
365D3—241no—
366D3—241no—
367C22—41no—
368C22—362no—
369C22—361no—
370C22C2×#372211yescharacteristic
371C22#6396611yescharacteristic
372C22#6397711yescharacteristic
373C4—24no—
374C4—124no—
375C4—362no—
376C4—361no—
377C4#6395212yes—
378C4#6397812yes—
379C3—21no—
380C3#7010511yescharacteristic, complemented
381C3#7166311yescharacteristic, complemented
382C2—21no—
383C2—21no—
384C2—21no—
385C2—721no—
386C2#1059171111yescharacteristic
subgroupG/H
centerC2#10591711
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C2×D32
FittingC22×C32×Q8C22
socleC2×C32#386
prime radicalC2×#3828C22
cosocleC22×C32×C4C23
hypercenterC22×Q8D32
hypocenterC32#1912
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
2. C3×#5812C2semidirect
3. C2×C3×#836C2semidirect
5. #10589158C2semidirect
10. #64156C4semidirect
12. #64153C4semidirect
25. #5812D3semidirect
27. C2×#836D3semidirect
331. C32#1912semidirect
380. C3#70105semidirect
381. C3#71663semidirect

Maximal subgroups

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

subgroupindex
#105962872
C3×#58122
C2×C3×#8362
#105911132
#105891582
#716633
#701053