Finite Group Databasebeta

← #10814894

Subgroups — #10814894

Order 768.

710 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79040C212yesmaximal, complemented
2C4×#904C211yesmaximal, characteristic
3C3×#1848C211yesmaximal, characteristic, complemented
4#70703C212yesmaximal, complemented
5#70699C214yesmaximal, complemented
6#70697C212yesmaximal
7#70529C212yesmaximal, complemented
8#69757C211yesmaximal, characteristic, complemented
9#22803—31nomaximal, Sylow
10C2×#904C2211yescharacteristic
11C4×#235C2212yes—
12C2×C3×#108C2212yes—
13C2×C3×#108C2212yes—
14C3×#423C2212yes—
15C2×C3×C4×C8C2211yescharacteristic
16C2×#825C2214yes—
17#5129C2214yes—
18#5122C2214yes—
19#5113C2212yes—
20#5106C2211yescharacteristic
21C2×#815C2212yes—
22C4×C4⋊Dic3C2211yescharacteristic
23C2×C4⋊(C3⋊C8)C2211yescharacteristic
24#4840—24no—
25#4835—22no—
26#4833—22no—
27C4×Dic12C411yescharacteristic, complemented
28C4×Dic12C411yescharacteristic, complemented
29C4×D24C411yescharacteristic, complemented
30C4×D24C411yescharacteristic, complemented
31C4×#204C412yescomplemented
32C4×#204C412yescomplemented
33#4555C2214yes—
34#4550C2212yes—
35C4×#135—31no—
36#2925—32no—
37#1975—32no—
38#1929—34no—
39#1848D311yescharacteristic, complemented
40#1806—32no—
41#1773—32no—
42#1770—31no—
43C2×#235C2312yes—
44C2×C3×M16C2314yes—
45C22×C3×C8C2311yescharacteristic
46C22×C3×C8C2311yescharacteristic
47C2×C3×C42C2311yescharacteristic
48C4×#67—22no—
49C4×#67—22no—
50#926C2312yes—
51C2×C4⋊Dic3—21no—
52C2×C4×Dic3C2312yes—
53C22×C3⋊C8C2312yes—
54C2×Dic12C2×C411yescharacteristic
55#904—22no—
56#904—22no—
57C2×D24C2×C411yescharacteristic
58C2×#204C2×C412yes—
59C4×D12C2×C412yes—
60C4×D12C2×C412yes—
61#872—22no—
62C42×D3—22no—
63C4×Dic6C2×C412yes—
64C4×Dic6C2×C412yes—
65C3×#108—24no—
66C3×#108—42no—
67C3×C4⋊C8—22no—
68C3×C4⋊C8—22no—
69C3×C4×C8C2×C411yescharacteristic
70C3×C4×C8C2×C411yescharacteristic
71C3×C4×C8C2×C411yescharacteristic
72C3×C4×C8C2×C411yescharacteristic
73C3⋊#95—21no—
74#825—44no—
75#821C2×C412yes—
76#821C2×C412yes—
77(C2×D3)⋊C8—24no—
78(C2×D3)⋊C8—24no—
79#818C2×C411yescharacteristic
80#818C2×C411yescharacteristic
81#817C2×C412yes—
82#816C2×C412yes—
83#816C2×C412yes—
84#815—24no—
85#815—24no—
86C8×Dic3—24no—
87C4⋊(C3⋊C8)D412yes—
88C4⋊(C3⋊C8)D412yes—
89C2×#135—31no—
90C4×#41—32no—
91C4×Q16—31no—
92C4×Q16—31no—
93C4×SD16—32no—
94C4×SD16—32no—
95C4×D8—31no—
96C4×D8—31no—
97#433—64no—
98#432—62no—
99#431—62no—
100C2×#108C2×D312yes—
101C2×#108C2×D312yes—
102#427—31no—
103#423C2×D312yes—
104C2×C4⋊C8—31no—
105C2×#104—34no—
106#416—32no—
107#408—34no—
108#405—34no—
109C2×#97—32no—
110C2×C4×C8C2×D311yescharacteristic
111C4×C4⋊C4—31no—
112#339—34no—
113#336—32no—
114C22×C3×C4#4111yescharacteristic
115C22×C3×C4#4111yescharacteristic
116C22×C3×C4C2411yescharacteristic
117C2×#67—22no—
118C22×Dic3—22no—
119#235—22no—
120#235—22no—
121#235—22no—
122#235—22no—
123C2×D12C22×C412yes—
124C2×C4×D3—22no—
125C2×C4×D3—22no—
126C2×C4×D3—22no—
127C2×Dic6C22×C412yes—
128C3×M16—24no—
129C3×M16—24no—
130C2×C3×C8—21no—
131C2×C3×C8—21no—
132C2×C3×C8—24no—
133C2×C3×C8C22×C411yescharacteristic
134C2×C3×C8C22×C411yescharacteristic
135C2×C3×C8C22×C411yescharacteristic
136C2×C3×C8C22×C411yescharacteristic
137C2×C3×C8#4112yes—
138C2×C3×C8#4112yes—
139C3×C42C22×C411yescharacteristic
140C3×C42C22×C411yescharacteristic
141C3×C42C2×D411yescharacteristic
142C3×C42C2×D411yescharacteristic
143#217—22no—
144(C2×D3)⋊C4—22no—
145(C2×D3)⋊C4—22no—
146C4⋊Dic3—42no—
147C4⋊Dic3C22×C412yes—
148C4⋊Dic3C22×C412yes—
149Dic3⋊C4—22no—
150Dic3⋊C4—22no—
151C4×Dic3—22no—
152C4×Dic3—22no—
153C4×Dic3—22no—
154C4×Dic3—22no—
155C4×Dic3—22no—
156C2×C3⋊C8—22no—
157C2×C3⋊C8—22no—
158C2×C3⋊C8C2×D414yes—
159Dic12—22no—
160D24—22no—
161#204—24no—
162C2×#41—32no—
163#135—62no—
164#135—62no—
165C2×Q16—31no—
166C2×SD16—32no—
167C2×D8—31no—
168C2×M16C22×D314yes—
169C22×C8—32no—
170C22×C8C22×D311yescharacteristic
171C22×C8C22×D311yescharacteristic
172C4×Q8—32no—
173C4×Q8—32no—
174C4×D4—32no—
175C4×D4—32no—
176C4×D4—62no—
177C4×D4—62no—
178#117—32no—
179#117—62no—
180C2×C4⋊C4—61no—
181C2×C42—32no—
182C2×C42—62no—
183C2×C42C22×D311yescharacteristic
184#108—24no—
185#108—42no—
186#107—31no—
187#107—31no—
188#106—32no—
189C4⋊C8—22no—
190C4⋊C8—22no—
191C4⋊C8—32no—
192C4⋊C8—32no—
193#104—124no—
194#103—32no—
195#103—32no—
196#102—32no—
197#102—32no—
198C22⋊C8—64no—
199C22⋊C8—64no—
200#97—64no—
201#97—64no—
202C4×C8—64no—
203C4×C8C4×D311yescharacteristic
204C4×C8C4×D311yescharacteristic
205C4×C8C4×D311yescharacteristic
206C4×C8C4×D311yescharacteristic
207#95—61no—
208C23×C3C2×#4111yescharacteristic
209C22×D3—22no—
210C2×C3×C4—22no—
211C2×C3×C4—41no—
212C2×C3×C4C4×D411yescharacteristic
213C2×C3×C4C4×D411yescharacteristic
214C2×C3×C4C4×D411yescharacteristic
215C2×C3×C4C4×D411yescharacteristic
216C2×C3×C4C23×C411yescharacteristic
217C2×C3×C4C22×D411yescharacteristic
218C2×C3×C4C2×#4111yescharacteristic
219C2×C3×C4C2×#4111yescharacteristic
220C2×C3×C4H(5,𝔽2)11yescharacteristic
221C2×C3×C42−1+411yescharacteristic
222#67—42no—
223#67—42no—
224C2×Dic3—22no—
225C2×Dic3—22no—
226C2×Dic3—22no—
227C2×Dic3—42no—
228C2×Dic3—42no—
229C2×Dic3—42no—
230D12—22no—
231D12—22no—
232C4×D3—42no—
233C4×D3—42no—
234C4×D3—44no—
235C4×D3—82no—
236Dic6—22no—
237Dic6—22no—
238C3×C8—21no—
239C3×C8—21no—
240C3×C8—44no—
241C3×C8#11712yes—
242C3×C8#11712yes—
243C3⋊C8—24no—
244#41—62no—
245#41—62no—
246#41—62no—
247#41—62no—
248C2×Q8—32no—
249C2×D4—32no—
250C2×D4—62no—
251C22×C4—62no—
252C22×C4—62no—
253C22×C4—62no—
254C22×C4—62no—
255C22×C4#23911yescharacteristic
256C22×C4#23911yescharacteristic
257C22×C4C23×D311yescharacteristic
258Q16—62no—
259SD16—64no—
260D8—62no—
261M16—24no—
262M16—24no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—24no—
266C2×C8—34no—
267C2×C8—62no—
268C2×C8—62no—
269C2×C8C2×C4×D311yescharacteristic
270C2×C8C2×C4×D311yescharacteristic
271C2×C8C2×C4×D311yescharacteristic
272C2×C8C2×C4×D311yescharacteristic
273C2×C8#23712yes—
274C2×C8#23912yes—
275C4⋊C4—32no—
276C4⋊C4—32no—
277C4⋊C4—62no—
278C4⋊C4—62no—
279C4⋊C4—122no—
280C22⋊C4—62no—
281C22⋊C4—62no—
282C22⋊C4—62no—
283C42—62no—
284C42—62no—
285C42—62no—
286C42—62no—
287C42—62no—
288C42C2×C4×D311yescharacteristic
289C42C2×C4×D311yescharacteristic
290C42D3×D411yescharacteristic
291C42D3×D411yescharacteristic
292C22×C3—21no—
293C22×C3—21no—
294C22×C3C2×C4×D411yescharacteristic
295C22×C3#55511yescharacteristic
296C22×C3#55611yescharacteristic
297C2×D3—42no—
298C2×D3—42no—
299C2×D3—42no—
300C3×C4—21no—
301C3×C4—21no—
302C3×C4—21no—
303C3×C4#44412yes—
304C3×C4C2×#11711yescharacteristic
305C3×C4#52011yescharacteristic
306C3×C4#54711yescharacteristic
307C3×C4#55011yescharacteristic
308Dic3—42no—
309Dic3—42no—
310Dic3—42no—
311Dic3—82no—
312C23—62no—
313C23C2×#23911yescharacteristic
314Q8—62no—
315Q8—62no—
316D4—62no—
317D4—62no—
318D4—122no—
319D4—122no—
320C2×C4—22no—
321C2×C4—41no—
322C2×C4—62no—
323C2×C4—62no—
324C2×C4—62no—
325C2×C4—124no—
326C2×C4—122no—
327C2×C4—122no—
328C2×C4—122no—
329C2×C4—122no—
330C2×C4—122no—
331C2×C4—242no—
332C2×C4#89311yescharacteristic
333C2×C4#89311yescharacteristic
334C2×C4#89311yescharacteristic
335C2×C4#89311yescharacteristic
336C2×C4C22×C4×D311yescharacteristic
337C2×C4C2×D3×D411yescharacteristic
338C2×C4C2×#23711yescharacteristic
339C2×C4#100411yescharacteristic
340C2×C4C2×#23911yescharacteristic
341C2×C4#100711yescharacteristic
342C8—21no—
343C8—21no—
344C8—44no—
345C8—64no—
346C8#89212yes—
347C8#89212yes—
348C2×C3—21no—
349C2×C3—21no—
350C2×C3#235411yescharacteristic
351C2×C3C2×#44411yescharacteristic
352C2×C3#295711yescharacteristic
353D3—82no—
354C22—21no—
355C22—21no—
356C22—122no—
357C22—122no—
358C22—122no—
359C22C2×#89311yescharacteristic
360C22#573211yescharacteristic
361C22#573411yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—122no—
366C4—122no—
367C4—122no—
368C4—242no—
369C4#489412yes—
370C4C2×#89211yescharacteristic
371C4#550411yescharacteristic
372C4#571811yescharacteristic
373C4#572611yescharacteristic
374C3#2280311yescharacteristic, complemented, Sylow
375C2—21no—
376C2—21no—
377C2—242no—
378C2#7432411yescharacteristic
379C2C2×#489411yescharacteristic
380C2#7924911yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1848C2
socleC22×C3C2×C4×D4
prime radicalC2×#904C22
cosocleC22×C3×C4C24
hypercenter#1848D3
hypocenterC3#22803
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1848

Complements

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

NH ≅ G/Ntype
1. #79040C2semidirect
3. C3×#1848C2semidirect
4. #70703C2semidirect
5. #70699C2semidirect
7. #70529C2semidirect
8. #69757C2semidirect
27. C4×Dic12C4semidirect
28. C4×Dic12C4semidirect
29. C4×D24C4semidirect
30. C4×D24C4semidirect
31. C4×#204C4semidirect
32. C4×#204C4semidirect
39. #1848D3semidirect
374. C3#22803semidirect

Maximal subgroups

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

subgroupindex
#790402
C4×#9042
C3×#18482
#707032
#706992
#706972
#705292
#697572
#228033