Finite Group Databasebeta

← #10713595

Subgroups — #10713595

Order 768.

418 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5194C211yesmaximal, characteristic
2C3×#1974C211yesmaximal, characteristic, complemented
3C2×#4567C211yesmaximal, characteristic, complemented
4C2×#4497C211yesmaximal, characteristic, complemented
5C2×#4493C211yesmaximal, characteristic, complemented
6#69795C211yesmaximal, characteristic, complemented
7C2×#4461C211yesmaximal, characteristic, complemented
8#13242—31nomaximal, Sylow
9C2×C3×#135C2211yescharacteristic
10C2×#904C2211yescharacteristic
11C2×C3×#108C2211yescharacteristic
12C2×C3×#104—21no—
13C3×#408—21no—
14C2×C3×C4×C8C2211yescharacteristic
15#5194—22no—
16#5194—22no—
17C2×#829C411yescharacteristic, complemented
18C2×#828C411yescharacteristic, complemented
19C2×#827C411yescharacteristic, complemented
20C2×#826C411yescharacteristic, complemented
21#5110—21no—
22C2×#819C2211yescharacteristic
23C22×C3⋊C16C2211yescharacteristic
24C22×C3⋊C16C2211yescharacteristic
25C2×#805—21no—
26#4567—22no—
27#4497—22no—
28#4493—22no—
29#4461—22no—
30C2×#508—31no—
31C2×#368—31no—
32C2×#359—31no—
33C2×#359—31no—
34C2×#346—31no—
35#1974—31no—
36#1974D311yescharacteristic, complemented
37C2×C3×#41—21no—
38C2×#235—21no—
39C3×#135—21no—
40C3×#135—21no—
41C3×#135—21no—
42C3×#135—21no—
43C2×C3×Q16C2×C411yescharacteristic
44C2×C3×SD16—21no—
45C2×C3×D8C2×C411yescharacteristic
46C2×C3×M16—21no—
47C22×C3×C8D411yescharacteristic
48C22×C3×C8C2311yescharacteristic
49C2×C3×C42D411yescharacteristic
50C2×#208—21no—
51C2×Dic12C2×C411yescharacteristic
52#904—21no—
53#904—21no—
54#904—21no—
55#904—21no—
56C2×D24C2×C411yescharacteristic
57C2×#204—21no—
58C3×#108—21no—
59C3×#108—21no—
60C3×#104—42no—
61C3×C22⋊C8—42no—
62C3×C4×C8—21no—
63C3×C4×C8—21no—
64#829—22no—
65#828—22no—
66#827—22no—
67#826—22no—
68(C2×D3)⋊C8—42no—
69#819—21no—
70#819—21no—
71C2×C3⋊C16—21no—
72C2×C3⋊C16—21no—
73C2×C3⋊C16C2×C411yescharacteristic
74C2×C3⋊C16C2×C411yescharacteristic
75C2×C3⋊C16D412yes—
76C2×C3⋊C16D412yes—
77C2×C3⋊C16D412yes—
78#805—42no—
79C2×#135—31no—
80C2×#135C2×D311yescharacteristic
81#508—62no—
82#508—62no—
83C2×Q32—31no—
84C2×SD32—31no—
85C2×SD32—31no—
86C2×D16—31no—
87C22×C16—31no—
88C22×C16—31no—
89C2×#108—31no—
90C2×#108C2×D311yescharacteristic
91C2×#104—21no—
92C2×#104—61no—
93#408—21no—
94#408—61no—
95C2×C4×C8C2×D311yescharacteristic
96#368—62no—
97#359—62no—
98#359—62no—
99#346—62no—
100C3×#41—41no—
101C3×#41—41no—
102C3×#41—41no—
103C3×#41—41no—
104C2×C3×Q8—21no—
105C2×C3×D4—21no—
106C2×C3×D4—41no—
107C22×C3×C4—21no—
108C22×C3×C4—41no—
109C22×C3×C4C2×D411yescharacteristic
110C2×#67—41no—
111#235—41no—
112#235—41no—
113#235—41no—
114#235—41no—
115C2×D12—21no—
116C2×C4×D3—41no—
117C2×Dic6—21no—
118C3×Q16—21no—
119C3×Q16—21no—
120C3×SD16—41no—
121C3×SD16—41no—
122C3×D8—21no—
123C3×D8—21no—
124C3×M16—41no—
125C3×M16—41no—
126C2×C3×C8—21no—
127C2×C3×C8—21no—
128C2×C3×C8—41no—
129C2×C3×C8D812yes—
130C2×C3×C8C22×C411yescharacteristic
131C2×C3×C8C2×D411yescharacteristic
132C2×C3×C8C2×D411yescharacteristic
133C2×C3×C8C2×D411yescharacteristic
134C2×C3×C8#4111yescharacteristic
135C2×C3×C8#4111yescharacteristic
136C3×C42—21no—
137C3×C42D812yes—
138#208—41no—
139#208—41no—
140C2×C3⋊C8—41no—
141Dic12—21no—
142Dic12—21no—
143D24—21no—
144D24—21no—
145#204—41no—
146#204—41no—
147C3⋊C16—22no—
148C3⋊C16C22⋊C412yes—
149C3⋊C16C22⋊C412yes—
150C2×#41—21no—
151C2×#41—61no—
152#135—21no—
153#135—21no—
154#135—21no—
155#135—21no—
156#135—61no—
157#135—61no—
158#135—61no—
159#135—61no—
160C2×Q16—31no—
161C2×Q16C4×D311yescharacteristic
162C2×SD16—21no—
163C2×SD16—61no—
164C2×D8—31no—
165C2×D8C4×D311yescharacteristic
166C2×M16—21no—
167C2×M16—61no—
168C22×C8#6711yescharacteristic
169C22×C8C22×D311yescharacteristic
170C2×C42#6711yescharacteristic
171Q32—62no—
172SD32—62no—
173SD32—62no—
174D16—62no—
175C2×C16—31no—
176C2×C16—31no—
177C2×C16—32no—
178C2×C16—32no—
179C2×C16—32no—
180C2×C16—61no—
181C2×C16—61no—
182#108—21no—
183#108—21no—
184#108—61no—
185#108—61no—
186#104—42no—
187#104—122no—
188C22⋊C8—42no—
189C22⋊C8—122no—
190C4×C8—21no—
191C4×C8—21no—
192C23×C3—41no—
193C23×C3#13511yescharacteristic
194C22×D3—41no—
195C3×Q8—41no—
196C3×Q8—41no—
197C3×D4—41no—
198C3×D4—41no—
199C3×D4—81no—
200C3×D4—81no—
201C2×C3×C4—22no—
202C2×C3×C4—42no—
203C2×C3×C4—41no—
204C2×C3×C4—81no—
205C2×C3×C4—81no—
206C2×C3×C4C4×D411yescharacteristic
207C2×C3×C4#12111yescharacteristic
208C2×C3×C4#12711yescharacteristic
209C2×C3×C4C2×D811yescharacteristic
210C2×C3×C4C2×D811yescharacteristic
211C2×C3×C4#13511yescharacteristic
212#67—81no—
213#67—81no—
214C2×Dic3—41no—
215D12—41no—
216D12—41no—
217C4×D3—81no—
218C4×D3—81no—
219Dic6—41no—
220Dic6—41no—
221C3×C8—22no—
222C3×C8—81no—
223C3×C8C2×C22⋊C411yescharacteristic
224C3×C8C4×D411yescharacteristic
225C3×C8#12111yescharacteristic
226C3×C8#12411yescharacteristic
227C3⋊C8—81no—
228#41—41no—
229#41—41no—
230#41—41no—
231#41—41no—
232#41—121no—
233#41—121no—
234#41—121no—
235#41—121no—
236C2×Q8—21no—
237C2×Q8—61no—
238C2×D4—21no—
239C2×D4—41no—
240C2×D4—61no—
241C2×D4—121no—
242C22×C4—21no—
243C22×C4—41no—
244C22×C4—121no—
245C22×C4C2×#6711yescharacteristic
246Q16—21no—
247Q16—21no—
248Q16—61no—
249Q16—61no—
250SD16—41no—
251SD16—41no—
252SD16—121no—
253SD16—121no—
254D8—21no—
255D8—21no—
256D8—61no—
257D8—61no—
258M16—41no—
259M16—41no—
260M16—121no—
261M16—121no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—41no—
265C2×C8—121no—
266C2×C8#21312yes—
267C2×C8C2×C4×D311yescharacteristic
268C2×C8#23511yescharacteristic
269C2×C8D3×D411yescharacteristic
270C2×C8D3×D411yescharacteristic
271C2×C8D3×D411yescharacteristic
272C2×C8#23711yescharacteristic
273C42—21no—
274C42#21312yes—
275C16—32no—
276C16—32no—
277C16—62no—
278C22×C3—22no—
279C22×C3—82no—
280C22×C3—81no—
281C22×C3C4×D811yescharacteristic
282C22×C3#46611yescharacteristic
283C22×C3#49511yescharacteristic
284C2×D3—82no—
285C2×D3—81no—
286C3×C4—42no—
287C3×C4—81no—
288C3×C4#39011yescharacteristic
289C3×C4C4×D811yescharacteristic
290C3×C4#46611yescharacteristic
291C3×C4#49311yescharacteristic
292Dic3—81no—
293C23—41no—
294C23—121no—
295C23#95011yescharacteristic
296Q8—41no—
297Q8—41no—
298Q8—121no—
299Q8—121no—
300D4—41no—
301D4—41no—
302D4—81no—
303D4—81no—
304D4—121no—
305D4—121no—
306D4—241no—
307D4—241no—
308C2×C4—22no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—121no—
314C2×C4—241no—
315C2×C4—241no—
316C2×C4C4×#6711yescharacteristic
317C2×C4C2×#21311yescharacteristic
318C2×C4C2×#21311yescharacteristic
319C2×C4#93911yescharacteristic
320C2×C4#94011yescharacteristic
321C2×C4#95011yescharacteristic
322C8—22no—
323C8—81no—
324C8—241no—
325C8D3×C22⋊C411yescharacteristic
326C8#67⋊C411yescharacteristic
327C8#88411yescharacteristic
328C8#88511yescharacteristic
329C2×C3—21no—
330C2×C3—21no—
331C2×C3—161no—
332C2×C3#197011yescharacteristic
333C2×C3#217912yes—
334D3—161no—
335C22—22no—
336C22—82no—
337C22—81no—
338C22—242no—
339C22—241no—
340C22C4×#21311yescharacteristic
341C22#503811yescharacteristic
342C22#505911yescharacteristic
343C4—42no—
344C4—81no—
345C4—241no—
346C4#495711yescharacteristic
347C4C4×#21311yescharacteristic
348C4#503811yescharacteristic
349C4#507311yescharacteristic
350C3#1324211yescharacteristic, complemented, Sylow
351C2—21no—
352C2—21no—
353C2—161no—
354C2—481no—
355C2#6964711yescharacteristic
356C2#7092212yes—
subgroupG/H
centerC2×C4C2×#213
commutatorC2×C3×C8C22×C4
FrattiniC22×C8C22×D3
FittingC3×#1974C2
socleC22×C3C4×D8
prime radicalC2×#5194C2
cosocleC22×C3×C8C23
hypercenter#1974D3
hypocenterC3#13242
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C4×C8 ◁ #1974

Complements

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

NH ≅ G/Ntype
2. C3×#1974C2semidirect
3. C2×#4567C2semidirect
4. C2×#4497C2semidirect
5. C2×#4493C2semidirect
6. #69795C2semidirect
7. C2×#4461C2semidirect
17. C2×#829C4semidirect
18. C2×#828C4semidirect
19. C2×#827C4semidirect
20. C2×#826C4semidirect
36. #1974D3semidirect
350. C3#13242semidirect

Maximal subgroups

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

subgroupindex
C2×#51942
C3×#19742
C2×#45672
C2×#44972
C2×#44932
#697952
C2×#44612
#132423