Finite Group Databasebeta

← #109738

Subgroups — #109738

Order 512.

494 conjugacy classes of subgroups.

The subgroup lattice has 371 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20551C211yesmaximal, characteristic
2#17822C211yesmaximal, characteristic
3#10856C211yesmaximal, characteristic
4#8119C211yesmaximal, characteristic
5#8093C211yesmaximal, characteristic
6#8071C211yesmaximal, characteristic
7#7941C211yesmaximal, characteristic
8#2920C2211yescharacteristic
9#2920C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C2×#423C2211yescharacteristic
13#2283C2211yescharacteristic
14C4×#117C2211yescharacteristic
15#1771—21no—
16#1771—21no—
17#1768—21no—
18#1768—21no—
19#1753—21no—
20#1753—21no—
21C2×#344—21no—
22C2×#344—21no—
23#1573C414yescomplemented
24#1571—21no—
25#1570—21no—
26#1569—22no—
27#1307C414yescomplemented
28#1303—21no—
29#1303—21no—
30#1300—22no—
31#1284C414yes—
32C22×M16—21no—
33C22×M16D411yescharacteristic
34C22×M16D411yescharacteristic
35C2×#117—21no—
36C22×C42C2311yescharacteristic
37#432—21no—
38#432—21no—
39#432—21no—
40#432—21no—
41#432C2×C412yes—
42#432C2×C412yes—
43#432C2×C412yes—
44#432D412yes—
45#431—21no—
46#431—21no—
47#431—21no—
48#431C2×C412yes—
49#423—21no—
50#423—21no—
51#423—21no—
52#423—21no—
53#423—21no—
54#423—21no—
55#423—21no—
56#423—21no—
57#423C2×C412yes—
58#423Q812yes—
59C2×C4⋊C8—21no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8C2×C412yes—
63C2×C4⋊C8D411yescharacteristic
64C2×C4⋊C8D411yescharacteristic
65#407—21no—
66#407C2×C412yes—
67C2×C22⋊C8—21no—
68C4×M16—21no—
69C4×M16—21no—
70C2×#97—21no—
71C2×#97—21no—
72C2×#97—21no—
73C2×C4×C8—21no—
74C2×C4×C8—21no—
75#383Q812yes—
76#382—21no—
77#381C2×C412yes—
78#380—21no—
79C4×C4⋊C4—21no—
80C4×C4⋊C4—21no—
81C4×C4⋊C4D411yescharacteristic
82C4×C4⋊C4D411yescharacteristic
83C4×C22⋊C4—21no—
84C4×C22⋊C4—21no—
85C4×C22⋊C4C2×C412yes—
86#376—21no—
87#376—21no—
88C43—21no—
89#344—42no—
90#344—42no—
91#336—21no—
92#336—21no—
93#336—21no—
94#336—21no—
95#335—24no—
96#334—24no—
97C23×C4C22×C411yescharacteristic
98C23×C4C2×D411yescharacteristic
99C23×C4#4111yescharacteristic
100C2×M16—22no—
101C2×M16—21no—
102C2×M16—21no—
103C2×M16—41no—
104C2×M16—41no—
105C2×M16—41no—
106C2×M16—41no—
107C2×M16—41no—
108C2×M16—41no—
109C2×M16—41no—
110C2×M16—41no—
111C2×M16—41no—
112C2×M16—41no—
113C2×M16—41no—
114C2×M16C4⋊C412yes—
115C2×M16C4⋊C412yes—
116C22×C8—21no—
117C22×C8—21no—
118C22×C8—21no—
119C22×C8—21no—
120C22×C8—22no—
121#117—22no—
122#117—22no—
123#117—41no—
124#117—41no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—41no—
128C2×C22⋊C4—22no—
129C2×C22⋊C4—22no—
130C2×C42—21no—
131C2×C42—21no—
132C2×C42—22no—
133C2×C42—41no—
134C2×C42—41no—
135C2×C42C22×C411yescharacteristic
136C2×C42C22×C411yescharacteristic
137C2×C42C2×D411yescharacteristic
138C2×C42C2×D411yescharacteristic
139C2×C42C2×D411yescharacteristic
140C2×C42C2×Q811yescharacteristic
141C2×C42C2×Q811yescharacteristic
142C2×C42#4111yescharacteristic
143C2×C42#4111yescharacteristic
144C2×C42#4111yescharacteristic
145C2×C42#4111yescharacteristic
146C2×C42#4111yescharacteristic
147C4⋊C8—22no—
148C4⋊C8—22no—
149C4⋊C8—22no—
150C4⋊C8—22no—
151C4⋊C8—22no—
152C4⋊C8—22no—
153C4⋊C8—22no—
154C4⋊C8—22no—
155C4⋊C8—22no—
156C4⋊C8—21no—
157C4⋊C8—21no—
158C4⋊C8—41no—
159C4⋊C8—41no—
160C4⋊C8—41no—
161C4⋊C8C22⋊C412yes—
162C4⋊C8C22⋊C412yes—
163C22⋊C8—22no—
164C22⋊C8—22no—
165C22⋊C8—22no—
166C22⋊C8—22no—
167C22⋊C8—41no—
168C22⋊C8C4214yes—
169#97—22no—
170#97—22no—
171#97—22no—
172#97—22no—
173#97—81no—
174#97—81no—
175C4×C8—22no—
176C4×C8—41no—
177C4×C8—81no—
178C4×C8—81no—
179#95—22no—
180#95—22no—
181#95—22no—
182#95—22no—
183#95—41no—
184#95C4⋊C412yes—
185#95C4⋊C412yes—
186C24C4×D411yescharacteristic
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—42no—
194C22×C4—81no—
195C22×C4—81no—
196C22×C4C2×C4211yescharacteristic
197C22×C4C2×C4⋊C411yescharacteristic
198C22×C4C2×C4⋊C411yescharacteristic
199C22×C4#11711yescharacteristic
200C22×C4#11711yescharacteristic
201C22×C4C4×D411yescharacteristic
202C22×C4C4×D411yescharacteristic
203C22×C4C4×D411yescharacteristic
204C22×C4C4×Q811yescharacteristic
205C22×C4C4×Q811yescharacteristic
206C22×C4#12011yescharacteristic
207C22×C4#12111yescharacteristic
208C22×C4C22⋊Q811yescharacteristic
209C22×C4C22⋊Q811yescharacteristic
210C22×C4#12311yescharacteristic
211C22×C4#12311yescharacteristic
212C22×C4#12411yescharacteristic
213C22×C4#12611yescharacteristic
214M16—42no—
215M16—42no—
216M16—81no—
217M16—81no—
218M16—82no—
219M16—81no—
220M16—81no—
221M16—82no—
222M16—81no—
223M16—81no—
224C2×C8—24no—
225C2×C8—22no—
226C2×C8—22no—
227C2×C8—22no—
228C2×C8—22no—
229C2×C8—41no—
230C2×C8—41no—
231C2×C8—41no—
232C2×C8—41no—
233C2×C8—41no—
234C2×C8—41no—
235C2×C8—41no—
236C2×C8—41no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—42no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—81no—
244C4⋊C4—81no—
245C22⋊C4—24no—
246C22⋊C4—42no—
247C22⋊C4—82no—
248C42—42no—
249C42—42no—
250C42—41no—
251C42—41no—
252C42—41no—
253C42—41no—
254C42—82no—
255C42—82no—
256C42C2×C22⋊C411yescharacteristic
257C42#11711yescharacteristic
258C42C4×D411yescharacteristic
259C42C4×D411yescharacteristic
260C42C4×D411yescharacteristic
261C42C4×D411yescharacteristic
262C42C4×Q811yescharacteristic
263C42C4×Q811yescharacteristic
264C42#12111yescharacteristic
265C42C22⋊Q811yescharacteristic
266C42C22⋊Q811yescharacteristic
267C42#12311yescharacteristic
268C42#12511yescharacteristic
269C42#12511yescharacteristic
270C42#12611yescharacteristic
271C42C4⋊Q811yescharacteristic
272C23—21no—
273C23—21no—
274C23—41no—
275C23—41no—
276C23C4×C4⋊C411yescharacteristic
277C23#38511yescharacteristic
278C23#38811yescharacteristic
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—44no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4C4×C22⋊C411yescharacteristic
311C2×C4C4×C4⋊C411yescharacteristic
312C2×C4#38511yescharacteristic
313C2×C4#38611yescharacteristic
314C2×C4#38711yescharacteristic
315C2×C4#38711yescharacteristic
316C2×C4#38711yescharacteristic
317C2×C4#38711yescharacteristic
318C2×C4#38811yescharacteristic
319C2×C4#38911yescharacteristic
320C2×C4#38911yescharacteristic
321C2×C4#39111yescharacteristic
322C2×C4#45311yescharacteristic
323C2×C4#45411yescharacteristic
324C2×C4#45511yescharacteristic
325C2×C4#45611yescharacteristic
326C8—44no—
327C8—42no—
328C8—42no—
329C8—42no—
330C8—42no—
331C8—82no—
332C8—82no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—81no—
342C22—81no—
343C22#143611yescharacteristic
344C22#185911yescharacteristic
345C22#186011yescharacteristic
346C4—21no—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—82no—
357C4—82no—
358C4—84no—
359C4#185611yescharacteristic
360C4#185711yescharacteristic
361C4#188211yescharacteristic
362C4#188311yescharacteristic
363C2—21no—
364C2—21no—
365C2—41no—
366C2—41no—
367C2#1089911yescharacteristic
368C2#1090511yescharacteristic
369C2#1091711yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
socleC22#1436
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
23. #1573C4semidirect
27. #1307C4semidirect

Maximal subgroups

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

subgroupindex
#205512
#178222
#108562
#81192
#80932
#80712
#79412