Finite Group Databasebeta

← #10804039

Subgroups — #10804039

Order 768.

654 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79170C211yesmaximal, characteristic
2#78543C212yesmaximal
3C3×#1772C211yesmaximal, characteristic
4#70704C211yesmaximal, characteristic
5#70693C211yesmaximal, characteristic
6#70662C211yesmaximal, characteristic
7#69890C212yesmaximal
8#69779C211yesmaximal, characteristic
9#69749C211yesmaximal, characteristic
10#68748C212yesmaximal
11#68732C212yesmaximal
12#21733—31nomaximal, Sylow
13#5517C2212yes—
14C2×C3×#108C2211yescharacteristic
15C3×#427C2211yescharacteristic
16C3×#405C2212yes—
17C2×C3×C4×C8C2211yescharacteristic
18C2×#822C2212yes—
19#5119C2212yes—
20#5118C2212yes—
21C2×#819C2211yescharacteristic
22#5106C2211yescharacteristic
23#5101C2212yes—
24C2×#815C2211yescharacteristic
25C2×#815C2211yescharacteristic
26C2×C8×Dic3C2211yescharacteristic
27C2×C8×Dic3C2211yescharacteristic
28#4999C2212yes—
29#4998C2212yes—
30#4932C2211yescharacteristic
31C2×#803C2211yescharacteristic
32#4878C412yescomplemented
33#4874C412yescomplemented
34#4858C412yescomplemented
35#4855C412yescomplemented
36#4701—24no—
37#4679—24no—
38C3×#339C2212yes—
39#4555C2212yes—
40#4532C2212yes—
41#4523C2212yes—
42#3160—31no—
43#2961—32no—
44#1995—32no—
45#1946—32no—
46#1945—31no—
47#1945—31no—
48#1848—31no—
49#1773—32no—
50#1772—31no—
51#1772D311yescharacteristic
52#1726—31no—
53C2×C3×M16C2312yes—
54C22×C3×C8C2311yescharacteristic
55C22×C3×C8C2311yescharacteristic
56C3×#117C2312yes—
57C2×C3×C42C2311yescharacteristic
58#926C2312yes—
59C2×C4×Dic3C2311yescharacteristic
60C2×C4×Dic3C2311yescharacteristic
61C2×#208C2312yes—
62C22×C3⋊C8C2311yescharacteristic
63C22×C3⋊C8C2311yescharacteristic
64#890C2×C414yes—
65#888C2×C414yes—
66#878—24no—
67C4×Dic6—24no—
68C3×#108—22no—
69C3×#107C2×C412yes—
70C3×#106C2×C412yes—
71C3×#97—22no—
72C3×C4×C8—22no—
73C3×C4×C8Q814yescomplemented
74C3⋊#95—22no—
75#822—24no—
76#819—22no—
77#818C2×C412yes—
78#817C2×C412yes—
79#815—22no—
80#815D412yes—
81#815D412yes—
82#815Q814yescomplemented
83Dic3⋊C8—24no—
84Dic3⋊C8—24no—
85C8×Dic3—22no—
86C8×Dic3—22no—
87C8×Dic3C2×C411yescharacteristic
88C8×Dic3C2×C411yescharacteristic
89C8×Dic3C2×C411yescharacteristic
90C8×Dic3C2×C411yescharacteristic
91#808C2×C414yes—
92#807C2×C414yes—
93C4⋊(C3⋊C8)—24no—
94#803—22no—
95#533—32no—
96#500—32no—
97#499—32no—
98#499—32no—
99#498—32no—
100#446—64no—
101#446—64no—
102C2×#108—32no—
103C2×#108—31no—
104C2×#108C2×D311yescharacteristic
105#427—31no—
106#427—32no—
107#427C2×D311yescharacteristic
108#424—32no—
109#424—32no—
110#423—32no—
111#405—32no—
112#405C2×D312yes—
113C2×#97—31no—
114C2×#97—31no—
115C2×#97—31no—
116C2×C4×C8—31no—
117C2×C4×C8—31no—
118C2×C4×C8C2×D311yescharacteristic
119#376—31no—
120#339—32no—
121#339—32no—
122#339—32no—
123#339C2×D312yes—
124C22×C3×C4—21no—
125C22×C3×C4C2411yescharacteristic
126C22×Dic3—21no—
127C22×Dic3#4112yes—
128C2×Dic6—24no—
129C3×M16—22no—
130C3×M16—22no—
131C2×C3×C8—22no—
132C2×C3×C8C22×C411yescharacteristic
133C2×C3×C8C22×C411yescharacteristic
134C2×C3×C8C2×D411yescharacteristic
135C2×C3×C8C2×D411yescharacteristic
136C2×C3×C8C2×Q812yes—
137C2×C3×C8C2×Q812yes—
138C2×C3×C8C2×Q812yes—
139C2×C3×C8#4112yes—
140C3×C4⋊C4C22×C414yes—
141C3×C22⋊C4—22no—
142C3×C42—22no—
143C3×C42—21no—
144C3×C42C2×Q812yes—
145#217—22no—
146C4⋊Dic3—24no—
147C4⋊Dic3C22×C414yes—
148Dic3⋊C4—24no—
149Dic3⋊C4—24no—
150C4×Dic3—22no—
151C4×Dic3—21no—
152C4×Dic3C22×C411yescharacteristic
153C4×Dic3C22×C411yescharacteristic
154C4×Dic3C2×D412yes—
155C4×Dic3C2×Q812yes—
156#208—22no—
157#208—22no—
158C2×C3⋊C8—21no—
159C2×C3⋊C8—21no—
160C2×C3⋊C8—22no—
161C2×C3⋊C8C22×C411yescharacteristic
162C2×C3⋊C8C22×C411yescharacteristic
163C2×C3⋊C8C2×D412yes—
164C2×C3⋊C8C2×Q812yes—
165C2×C3⋊C8#4112yes—
166C2×M16—32no—
167C2×M16C22×D312yes—
168C22×C8—31no—
169C22×C8—31no—
170C22×C8C22×D311yescharacteristic
171C22×C8C22×D311yescharacteristic
172C4⋊Q8—34no—
173#125—34no—
174C22⋊Q8—64no—
175C4×Q8—64no—
176#117—32no—
177#117C22×D312yes—
178C2×C42—31no—
179C2×C42—31no—
180C2×C42C22×D311yescharacteristic
181#108—22no—
182#108—64no—
183#108—62no—
184#107—32no—
185#107—34no—
186#107C4×D312yes—
187#106—32no—
188#106—34no—
189#106C4×D312yes—
190C4⋊C8—64no—
191C4⋊C8—64no—
192C4⋊C8—64no—
193#97—22no—
194#97—32no—
195#97—32no—
196#97—34no—
197#97—62no—
198#97—62no—
199C4×C8—22no—
200C4×C8—31no—
201C4×C8—31no—
202C4×C8—31no—
203C4×C8—31no—
204C4×C8—62no—
205C4×C8—62no—
206C4×C8Dic614yescomplemented
207#95—62no—
208C23×C3C2×#4111yescharacteristic
209C2×C3×C4—24no—
210C2×C3×C4—22no—
211C2×C3×C4—42no—
212C2×C3×C4C23×C411yescharacteristic
213C2×C3×C4C22×D411yescharacteristic
214C2×C3×C4C22×Q811yescharacteristic
215C2×C3×C4C22×Q811yescharacteristic
216C2×C3×C4C2×#4111yescharacteristic
217C2×C3×C4C2×#4111yescharacteristic
218C2×Dic3—24no—
219C2×Dic3—22no—
220C2×Dic3—42no—
221C2×Dic3—42no—
222C2×Dic3C4×D412yes—
223C2×Dic3C4×D412yes—
224Dic6—84no—
225C3×C8—42no—
226C3×C8C4×Q812yes—
227C3×C8C4×Q812yes—
228C3×C8C4⋊Q812yes—
229C3×C8C4⋊Q812yes—
230C3⋊C8—22no—
231C3⋊C8—42no—
232C3⋊C8C4×Q812yes—
233C3⋊C8C4×Q812yes—
234C2×Q8—64no—
235C22×C4—21no—
236C22×C4—32no—
237C22×C4—61no—
238C22×C4C23×D311yescharacteristic
239M16—22no—
240M16—22no—
241M16—62no—
242M16—62no—
243C2×C8—22no—
244C2×C8—31no—
245C2×C8—31no—
246C2×C8—32no—
247C2×C8—32no—
248C2×C8—32no—
249C2×C8—61no—
250C2×C8—61no—
251C2×C8—62no—
252C2×C8C2×Dic612yes—
253C2×C8C2×Dic612yes—
254C2×C8C2×C4×D311yescharacteristic
255C2×C8C2×C4×D311yescharacteristic
256C2×C8#23512yes—
257C2×C8D3×D411yescharacteristic
258C2×C8D3×D411yescharacteristic
259C2×C8D3×Q812yes—
260C4⋊C4—34no—
261C4⋊C4—64no—
262C4⋊C4—64no—
263C4⋊C4—64no—
264C4⋊C4C2×C4×D314yes—
265C22⋊C4—22no—
266C22⋊C4—62no—
267C42—22no—
268C42—21no—
269C42—31no—
270C42—31no—
271C42—32no—
272C42—32no—
273C42—61no—
274C42—62no—
275C42C2×Dic612yes—
276C22×C3—22no—
277C22×C3C2×C4×D411yescharacteristic
278C22×C3#53311yescharacteristic
279C22×C3#53311yescharacteristic
280C3×C4—42no—
281C3×C4—44no—
282C3×C4C2×C4×Q811yescharacteristic
283C3×C4C2×C4×Q811yescharacteristic
284C3×C4C2×C4⋊Q811yescharacteristic
285C3×C4#53211yescharacteristic
286Dic3—22no—
287Dic3—22no—
288Dic3—42no—
289Dic3—44no—
290C23D3×#4111yescharacteristic
291Q8—244no—
292C2×C4—22no—
293C2×C4—24no—
294C2×C4—32no—
295C2×C4—32no—
296C2×C4—42no—
297C2×C4—64no—
298C2×C4—62no—
299C2×C4—122no—
300C2×C4—122no—
301C2×C4C22×Dic611yescharacteristic
302C2×C4C22×C4×D311yescharacteristic
303C2×C4C2×#23511yescharacteristic
304C2×C4C2×D3×D411yescharacteristic
305C2×C4C2×D3×Q811yescharacteristic
306C2×C4D3×#4111yescharacteristic
307C8—32no—
308C8—32no—
309C8—42no—
310C8—62no—
311C8—122no—
312C8C4×Dic612yes—
313C8C4×Dic612yes—
314C8#88812yes—
315C8#88812yes—
316C2×C3—21no—
317C2×C3—21no—
318C2×C3C4×C4⋊Q811yescharacteristic
319C2×C3#295712yes—
320C22—22no—
321C22#551711yescharacteristic
322C22C4×D3×D411yescharacteristic
323C22#557811yescharacteristic
324C4—42no—
325C4—44no—
326C4—62no—
327C4—62no—
328C4—122no—
329C4—124no—
330C4C2×C4×Dic611yescharacteristic
331C4C2×#88811yescharacteristic
332C4#554811yescharacteristic
333C4C4×D3×Q811yescharacteristic
334C3#2173311yescharacteristic, complemented, Sylow
335C2—21no—
336C2—21no—
337C2C4×#88811yescharacteristic
338C2#7989012yes—
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1772C2
socleC22×C3C2×C4×D4
prime radicalC23×C3C2×#41
cosocleC22×C3×C4C24
hypercenter#1772D3
hypocenterC3#21733
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1772

Complements

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

NH ≅ G/Ntype
32. #4878C4semidirect
33. #4874C4semidirect
34. #4858C4semidirect
35. #4855C4semidirect
73. C3×C4×C8Q8semidirect
82. #815Q8semidirect
206. C4×C8Dic6semidirect
334. C3#21733semidirect

Maximal subgroups

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

subgroupindex
#791702
#785432
C3×#17722
#707042
#706932
#706622
#698902
#697792
#697492
#687482
#687322
#217333