Finite Group Databasebeta

← #190336

Subgroups — #190336

Order 512.

1013 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34185C211yesmaximal, characteristic
2#20726C211yesmaximal, characteristic
3#17364C211yesmaximal, characteristic
4#8773C218yesmaximal, complemented
5#8070C212yesmaximal, complemented
6#7976C212yesmaximal, complemented
7C22×C4×D4C2211yescharacteristic
8#2920C2212yes—
9#2920C2212yes—
10#2353C414yes—
11#2353C2214yes—
12#2352—22no—
13#2352—22no—
14#2313—21no—
15#2312—21no—
16#2308—21no—
17#2283C2211yescharacteristic
18#2283C2211yescharacteristic
19#2280C412yes—
20C2×#380C412yes—
21#2273—21no—
22#1821—21no—
23#1800—22no—
24#1783—21no—
25C4×#100C2214yes—
26#1466C2214yes—
27#1462—21no—
28#1462—21no—
29#1460—22no—
30#1300C2218yescomplemented
31#1288C2218yescomplemented
32C23×D4D411yescharacteristic
33C24×C4C2×C411yescharacteristic
34C24×C4C2×C411yescharacteristic
35C2×C4×D4—22no—
36C2×C4×D4—22no—
37C2×C4×D4—22no—
38C2×C4×D4—22no—
39C2×C4×D4C2×C412yes—
40C2×C4×D4D412yes—
41C2×C4×D4D412yes—
42C2×C4×D4C2312yes—
43C2×#117—22no—
44C22×C4⋊C4—22no—
45C22×C4⋊C4D411yescharacteristic
46C22×C22⋊C4—22no—
47C22×C22⋊C4D411yescharacteristic
48C22×C22⋊C4D411yescharacteristic
49C22×C42C2311yescharacteristic
50#432—22no—
51#432—22no—
52#432—22no—
53#432C2314yes—
54#431—22no—
55#431—22no—
56#431—22no—
57#431—22no—
58#423—22no—
59#423—22no—
60C2×#100—24no—
61#407—22no—
62#407—22no—
63C4×M16—22no—
64C4×M16C2314yes—
65#388—24no—
66#388—24no—
67#387—24no—
68#387—24no—
69#386—24no—
70#386—24no—
71#383C2×C412yes—
72#383C2312yes—
73#382—21no—
74#382—21no—
75#381—21no—
76#381—21no—
77#381C2×C414yes—
78#380—22no—
79#380—22no—
80#380—22no—
81#380C2×C414yes—
82#380C2×C412yes—
83#380C2×C412yes—
84#379—22no—
85C4×C4⋊C4—21no—
86C4×C4⋊C4—21no—
87C4×C22⋊C4—22no—
88C4×C22⋊C4—21no—
89C4×C22⋊C4—21no—
90C4×C22⋊C4—24no—
91C4×C22⋊C4—24no—
92C4×C22⋊C4C2×C414yes—
93#376C2×C412yes—
94#376C2312yes—
95C2×#95C2×C414yes—
96#343—24no—
97#323—24no—
98#323—24no—
99C25C22⋊C411yescharacteristic
100C25C22⋊C411yescharacteristic
101C22×D4—22no—
102C22×D4—22no—
103C22×D4—44no—
104C22×D4C22⋊C412yes—
105C22×D4C2×D412yes—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—24no—
115C23×C4—82no—
116C23×C4—82no—
117C23×C4C22⋊C412yes—
118C23×C4C22⋊C412yes—
119C23×C4C22⋊C411yescharacteristic
120C23×C4C22⋊C411yescharacteristic
121C23×C4C22×C412yes—
122C23×C4C22×C412yes—
123C23×C4C22×C411yescharacteristic
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C2×M16—24no—
127C2×M16—42no—
128C4×D4—24no—
129C4×D4—24no—
130C4×D4—24no—
131C4×D4—24no—
132C4×D4—24no—
133C4×D4—24no—
134C4×D4—24no—
135C4×D4—24no—
136#117—84no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—84no—
143C2×C4⋊C4C22⋊C412yes—
144C2×C4⋊C4C2×D412yes—
145C2×C22⋊C4—22no—
146C2×C22⋊C4—22no—
147C2×C22⋊C4—22no—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—22no—
150C2×C22⋊C4—22no—
151C2×C22⋊C4—24no—
152C2×C22⋊C4—24no—
153C2×C22⋊C4—24no—
154C2×C22⋊C4—48no—
155C2×C22⋊C4C22⋊C412yes—
156C2×C22⋊C4C22⋊C412yes—
157C2×C22⋊C4C2×D412yes—
158C2×C22⋊C4C2×D412yes—
159C2×C42—24no—
160C2×C42—22no—
161C2×C42—22no—
162C2×C42C22×C411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42#4112yes—
166C2×C42#4112yes—
167C2×C42#4112yes—
168C2×C42#4112yes—
169C2×C42C2411yescharacteristic
170C4⋊C8—24no—
171C4⋊C8—42no—
172C4⋊C8—42no—
173C4⋊C8—42no—
174#100—48no—
175C22⋊C8—24no—
176C22⋊C8—42no—
177C22⋊C8—42no—
178C22⋊C8—42no—
179#97—24no—
180#97—24no—
181C4×C8—24no—
182C4×C8—42no—
183#95—24no—
184#95—22no—
185#95—24no—
186#95—22no—
187#95—22no—
188#95—22no—
189#95—22no—
190#95—22no—
191#95C22×C414yes—
192#95C22×C414yes—
193C24—21no—
194C24—21no—
195C24—24no—
196C24—21no—
197C24—21no—
198C24—42no—
199C24—81no—
200C24—81no—
201C24—81no—
202C24—81no—
203C24C2×C22⋊C412yes—
204C24C2×C22⋊C411yescharacteristic
205C24C2×C22⋊C412yes—
206C2×D4—24no—
207C2×D4—24no—
208C2×D4—24no—
209C2×D4—24no—
210C2×D4—42no—
211C2×D4—42no—
212C2×D4—48no—
213C2×D4—48no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—44no—
231C22×C4—44no—
232C22×C4—48no—
233C22×C4—82no—
234C22×C4—82no—
235C22×C4—82no—
236C22×C4—82no—
237C22×C4—81no—
238C22×C4—82no—
239C22×C4—82no—
240C22×C4—82no—
241C22×C4—82no—
242C22×C4—81no—
243C22×C4—81no—
244C22×C4—82no—
245C22×C4—82no—
246C22×C4—84no—
247C22×C4—84no—
248C22×C4C2×C22⋊C411yescharacteristic
249C22×C4C2×C22⋊C411yescharacteristic
250C22×C4C2×C22⋊C412yes—
251C22×C4C2×C22⋊C412yes—
252C22×C4C2×C22⋊C411yescharacteristic
253C22×C4#11711yescharacteristic
254C22×C4#11711yescharacteristic
255C22×C4#11711yescharacteristic
256C22×C4#11711yescharacteristic
257C22×C4#12312yes—
258C22×C4#12312yes—
259C22×C4#12312yes—
260C22×C4#12312yes—
261C22×C4C23×C411yescharacteristic
262C22×C4C22×D411yescharacteristic
263C22×C4C22×D411yescharacteristic
264M16—48no—
265M16—84no—
266C2×C8—44no—
267C2×C8—44no—
268C4⋊C4—24no—
269C4⋊C4—24no—
270C4⋊C4—44no—
271C4⋊C4—44no—
272C4⋊C4—84no—
273C22⋊C4—24no—
274C22⋊C4—24no—
275C22⋊C4—24no—
276C22⋊C4—24no—
277C22⋊C4—44no—
278C22⋊C4—44no—
279C22⋊C4—48no—
280C22⋊C4—84no—
281C22⋊C4—84no—
282C42—82no—
283C42—82no—
284C42—84no—
285C42#11712yes—
286C42#11712yes—
287C42#12312yes—
288C42#12312yes—
289C42#12312yes—
290C42#12312yes—
291C42C2×#4112yes—
292C42C2×#4112yes—
293C23—24no—
294C23—24no—
295C23—22no—
296C23—22no—
297C23—22no—
298C23—22no—
299C23—22no—
300C23—22no—
301C23—41no—
302C23—41no—
303C23—42no—
304C23—42no—
305C23—42no—
306C23—42no—
307C23—42no—
308C23—42no—
309C23—81no—
310C23—81no—
311C23—81no—
312C23—81no—
313C23—81no—
314C23—81no—
315C23—81no—
316C23—81no—
317C23—88no—
318C23—88no—
319C23#38111yescharacteristic
320C23#38111yescharacteristic
321C23#42114yes—
322C23C22×C22⋊C411yescharacteristic
323D4—48no—
324D4—48no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—168no—
362C2×C4#38112yes—
363C2×C4#38112yes—
364C2×C4#38111yescharacteristic
365C2×C4#38111yescharacteristic
366C2×C4C2×#11711yescharacteristic
367C2×C4C2×#11711yescharacteristic
368C2×C4C2×#12312yes—
369C2×C4C2×#12312yes—
370C8—84no—
371C8—84no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—24no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—44no—
382C22—44no—
383C22—44no—
384C22—42no—
385C22—42no—
386C22—42no—
387C22—42no—
388C22—42no—
389C22—82no—
390C22—82no—
391C22—84no—
392C22—84no—
393C22—84no—
394C22—84no—
395C22C2×#38111yescharacteristic
396C22C2×#42112yes—
397C4—22no—
398C4—22no—
399C4—22no—
400C4—22no—
401C4—21no—
402C4—21no—
403C4—21no—
404C4—42no—
405C4—42no—
406C4—84no—
407C4—84no—
408C4#182912yes—
409C2—22no—
410C2—41no—
411C2—41no—
412C2—44no—
413C2—82no—
414C2#2070511yescharacteristic
415C2#2070811yescharacteristic
416C2C2×#182911yescharacteristic
subgroupG/H
center, socleC22C2×#381
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8773C2semidirect
5. #8070C2semidirect
6. #7976C2semidirect
30. #1300C22semidirect
31. #1288C22semidirect

Maximal subgroups

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

subgroupindex
#341852
#207262
#173642
#87732
#80702
#79762