Finite Group Databasebeta

← #11923000

Subgroups — #11923000

Order 1152.

608 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 608 conjugacy classes of subgroups — too many to draw. Only its 77 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#10596586C212yesmaximal, complemented
2C2×C3×#131⋊C3C211yesmaximal, characteristic, complemented
3#10593641C211yesmaximal, characteristic, complemented
4#10593640C212yesmaximal, complemented
5#10593637C211yesmaximal, characteristic, complemented
6#85451—33nomaximal
7#83805—31nomaximal
8C2×C3×#41⋊C3C2211yescharacteristic
9#64267C2212yes—
10#64266—22no—
11#64265C2212yes—
12#64264—22no—
13#64113—41nomaximal
14C3×#131⋊C3C2211yescharacteristic
15C3×#131⋊C3C2211yescharacteristic
16#5923—36no—
17C2×C3×#131D311yescharacteristic, complemented
18#5822—32no—
19C2×#904—31no—
20C2×#131⋊C3D313yescomplemented
21#5407—33no—
22#5406—36no—
23#5403—33no—
24#5143—31no—
25#5143—31no—
26#5142—31no—
27#5142—31no—
28#5141—32no—
29#5141—32no—
30#5132—32no—
31C2×#819—31no—
32#3828—42no—
33C3×#41⋊C3D411yescharacteristic
34C3×#41⋊C3C2311yescharacteristic
35C2×C3×SL(2,𝔽3)D411yescharacteristic
36C3⋊GL(2,𝔽3)—22no—
37C3⋊#226—22no—
38C2×C32×C8—41no—
39#3746—41no—
40#3745—41no—
41#3744—42no—
42#3067—91noSylow
43C2×C3×#41C2×D311yescharacteristic
44C2×#235—32no—
45C2×#41⋊C3C2×D313yes—
46#986—36no—
47#985—66no—
48#984—36no—
49#983—66no—
50C3×#131—31no—
51C3×#131—31no—
52C3×#131C2×D311yescharacteristic
53C3×#131C2×D311yescharacteristic
54C2×C3×M16—31no—
55C22×C3×C8—31no—
56#951—32no—
57#951—32no—
58#950—62no—
59#950—62no—
60#949—32no—
61#949—32no—
62#942—62no—
63#932—62no—
64C2×#208—32no—
65C2×Dic12—31no—
66#904—61no—
67#904—61no—
68#904—61no—
69#904—61no—
70#904—123no—
71C2×D24—31no—
72C2×#204—32no—
73#131⋊C3C2×D313yes—
74#131⋊C3C2×D313yes—
75#824—32no—
76#824—32no—
77#823—32no—
78#823—32no—
79#819—32no—
80#819—31no—
81#819—31no—
82C2×C32×C4—41no—
83#638—82no—
84C4⋊(C3⋊D3)—42no—
85C4×C3⋊D3—82no—
86#634—42no—
87C3×SL(2,𝔽3)C2×D411yescharacteristic
88C32×C8—41no—
89C32×C8—41no—
90#575—92no—
91C2×#135—91no—
92C2×#131C3⋊D311yescharacteristic, complemented
93#473—91no—
94#473—91no—
95#472—91no—
96#472—91no—
97#471—92no—
98#471—92no—
99C2×#108—91no—
100#413—92no—
101C3×#41—31no—
102C3×#41—31no—
103C3×#41D1211yescharacteristic
104C3×#41C22×D311yescharacteristic
105C2×C3×Q8D1211yescharacteristic
106C2×C3×D4—31no—
107C22×C3×C4—31no—
108C2×#67—62no—
109#235—62no—
110#235—62no—
111#235—62no—
112#235—62no—
113#235—126no—
114C2×D12—32no—
115C2×C4×D3—62no—
116C2×Dic6—32no—
117#41⋊C3D1213yes—
118#41⋊C3C22×D313yes—
119C2×SL(2,𝔽3)D1213yes—
120GL(2,𝔽3)—66no—
121#226—66no—
122C3×M16—31no—
123C3×M16—31no—
124C3×M16—61no—
125C2×C3×C8—31no—
126C2×C3×C8—31no—
127C2×C3×C8—31no—
128C2×C3×C8—31no—
129C2×C3×C8—31no—
130C2×C3×C8—43no—
131C2×C3×C8S411yescharacteristic
132#216—62no—
133#216—62no—
134#215—62no—
135#215—62no—
136#214—62no—
137#214—62no—
138#213—62no—
139#213—62no—
140#208—32no—
141#208—32no—
142#208—62no—
143C2×C3⋊C8—62no—
144Dic12—61no—
145Dic12—61no—
146Dic12—123no—
147D24—61no—
148D24—61no—
149D24—123no—
150#204—62no—
151#204—62no—
152#204—126no—
153C22×C32—41no—
154C2×C3⋊D3—82no—
155C32×C4—41no—
156C32×C4—41no—
157C3⋊Dic3—82no—
158C2×#41—92no—
159C2×#41C2×C3⋊D311yescharacteristic
160#137—92no—
161#137—92no—
162#137—182no—
163AGL(1,ℤ/8)—92no—
164AGL(1,ℤ/8)—92no—
165AGL(1,ℤ/8)—182no—
166#135—181no—
167#135—181no—
168#135—181no—
169#135—182no—
170#135—182no—
171#135—181no—
172C2×Q16—91no—
173C2×SD16—92no—
174C2×D8—91no—
175#131—31no—
176#131—31no—
177#131C2×C3⋊D311yescharacteristic
178#131C2×C3⋊D311yescharacteristic
179C2×M16—31no—
180C2×M16—92no—
181C22×C8—31no—
182#108—92no—
183#108—91no—
184#108—91no—
185#101—92no—
186#101—92no—
187#100—92no—
188#100—92no—
189C23×C3—31no—
190C22×D3—62no—
191C3×Q8—31no—
192C3×Q8C2×D1211yescharacteristic
193C3×D4—31no—
194C3×D4—31no—
195C3×D4—61no—
196C2×C3×C4—31no—
197C2×C3×C4—31no—
198C2×C3×C4—31no—
199C2×C3×C4—31no—
200C2×C3×C4—31no—
201C2×C3×C4—43no—
202C2×C3×C4C2×S411yescharacteristic
203#67—122no—
204#67—122no—
205#67—246no—
206C2×Dic3—62no—
207D12—62no—
208D12—62no—
209D12—126no—
210C4×D3—122no—
211C4×D3—122no—
212C4×D3—246no—
213Dic6—62no—
214Dic6—62no—
215Dic6—126no—
216SL(2,𝔽3)C2×D1213yes—
217C3×C8—31no—
218C3×C8—31no—
219C3×C8—43no—
220C3×C8—43no—
221C3×C8C2×S411yescharacteristic
222C3×C8C2×S411yescharacteristic
223C3⋊C8—62no—
224C3⋊C8—62no—
225C2×C32—41no—
226C2×C32—81no—
227C3⋊D3—162no—
228#41—31no—
229#41—31no—
230#41—182no—
231#41—182no—
232#41—182no—
233#41—182no—
234#41C4⋊(C3⋊D3)11yescharacteristic
235#41C22×C3⋊D311yescharacteristic
236C2×Q8—92no—
237C2×Q8C4⋊(C3⋊D3)11yescharacteristic
238C2×D4—31no—
239C2×D4—92no—
240C2×D4—182no—
241C22×C4—31no—
242C22×C4—182no—
243Q16—181no—
244Q16—182no—
245Q16—182no—
246Q16—181no—
247SD16—182no—
248SD16—182no—
249SD16—182no—
250SD16—182no—
251SD16—182no—
252SD16—182no—
253D8—181no—
254D8—182no—
255D8—182no—
256D8—181no—
257M16—31no—
258M16—31no—
259M16—61no—
260M16—92no—
261M16—92no—
262M16—182no—
263C2×C8—31no—
264C2×C8—31no—
265C2×C8—31no—
266C2×C8—31no—
267C2×C8—31no—
268C2×C8—182no—
269C2×C8C3⋊S411yescharacteristic
270C22×C3—31no—
271C22×C3—31no—
272C22×C3—43no—
273C22×C3—121no—
274C22×C3C4⋊S411yescharacteristic
275C2×D3—122no—
276C2×D3—246no—
277C2×D3—242no—
278C3×C4—31no—
279C3×C4—31no—
280C3×C4—43no—
281C3×C4—43no—
282C3×C4C4⋊S411yescharacteristic
283C3×C4C22×S411yescharacteristic
284Dic3—122no—
285Dic3—246no—
286C32—41noSylow
287C23—31no—
288C23—182no—
289Q8—31no—
290Q8—182no—
291Q8—182no—
292Q8C2×C4⋊(C3⋊D3)11yescharacteristic
293D4—31no—
294D4—31no—
295D4—61no—
296D4—182no—
297D4—182no—
298D4—362no—
299D4—362no—
300C2×C4—31no—
301C2×C4—31no—
302C2×C4—31no—
303C2×C4—31no—
304C2×C4—31no—
305C2×C4—182no—
306C2×C4—362no—
307C2×C4—362no—
308C2×C4C2×C3⋊S411yescharacteristic
309C8—31no—
310C8—31no—
311C8—182no—
312C8—182no—
313C8C2×C3⋊S411yescharacteristic
314C8C2×C3⋊S411yescharacteristic
315C2×C3—21no—
316C2×C3—43no—
317C2×C3—61no—
318C2×C3—61no—
319C2×C3—83no—
320C2×C3C2×C4⋊S411yescharacteristic
321D3—242no—
322D3—486no—
323C22—31no—
324C22—31no—
325C22—121no—
326C22—362no—
327C22—722no—
328C22C4⋊(C3⋊S4)11yescharacteristic
329C4—31no—
330C4—31no—
331C4—362no—
332C4C4⋊(C3⋊S4)11yescharacteristic
333C4C22×C3⋊S411yescharacteristic
334C3—43no—
335C3#8545111yescharacteristic, complemented
336C2—21no—
337C2—61no—
338C2—61no—
339C2—722no—
340C2C2×C4⋊(C3⋊S4)11yescharacteristic
subgroupG/H
centerC4C4⋊(C3⋊S4)
commutator, cosocleC3×#41⋊C3C23
FrattiniC4C22×C3⋊S4
FittingC2×C3×#131D3
socleC2×C3C2×C4⋊S4
hypercenterC2×C8C3⋊S4
hypocenterC3×SL(2,𝔽3)C2×D4
LCSG ▷ C3×#41⋊C3 ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. #10596586C2semidirect
2. C2×C3×#131⋊C3C2semidirect
3. #10593641C2semidirect
4. #10593640C2semidirect
5. #10593637C2semidirect
17. C2×C3×#131D3semidirect
20. C2×#131⋊C3D3semidirect
92. C2×#131C3⋊D3semidirect
335. C3#85451semidirect

Maximal subgroups

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

subgroupindex
#105965862
C2×C3×#131⋊C32
#105936412
#105936402
#105936372
#854513
#838053
#641134