Finite Group Databasebeta

← #117614

Subgroups — #117614

Order 512.

598 conjugacy classes of subgroups.

The subgroup lattice has 387 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20735C211yesmaximal, characteristic
2#17815C211yesmaximal, characteristic
3#17449C211yesmaximal, characteristic
4#17365C211yesmaximal, characteristic
5#8116C211yesmaximal, characteristic
6#7976C211yesmaximal, characteristic
7#7936C211yesmaximal, characteristic
8C22×C4×Q8C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13C2×#431C2211yescharacteristic
14#2283C2211yescharacteristic
15#1823C412yes—
16#1805—22no—
17#1787C412yes—
18C4×#103—21no—
19C4×#103—21no—
20#1572—22no—
21#1572—22no—
22#1567—21no—
23#1566—21no—
24#1491—22no—
25#1487—22no—
26#1479—22no—
27#1467—22no—
28#1463—22no—
29#1461—22no—
30#1306—22no—
31#1288—22no—
32#1283—22no—
33C23×Q8C2×C411yescharacteristic
34C2×C4×Q8—21no—
35C2×C4×Q8—21no—
36C2×C4×Q8—21no—
37C2×C4×Q8—21no—
38C2×C4×Q8—21no—
39C2×C4×Q8—21no—
40C2×C4×Q8—21no—
41C2×C4×Q8—21no—
42C2×C4×Q8—21no—
43C2×C4×Q8—21no—
44C2×C4×Q8D412yes—
45C2×C4×Q8Q812yes—
46C22×C4⋊C4—21no—
47C22×C4⋊C4C2×C411yescharacteristic
48C22×C42—21no—
49C22×C42C2311yescharacteristic
50#432—21no—
51#432—21no—
52#432—21no—
53#432—21no—
54#432—21no—
55#432—21no—
56#432—21no—
57#432D412yes—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431—21no—
65#431—21no—
66#431—21no—
67#431—21no—
68#431D412yes—
69#431D412yes—
70#423—21no—
71#423—21no—
72#423—21no—
73#423—21no—
74C2×C4⋊C8D412yes—
75C2×#103—22no—
76C2×#103—22no—
77#407—21no—
78#407—21no—
79#407—21no—
80#407D412yes—
81C2×C22⋊C8C2×C412yes—
82C4×M16—21no—
83C4×M16—21no—
84C4×M16—21no—
85C4×M16—21no—
86C2×C4×C8D411yescharacteristic
87C2×C4×C8D411yescharacteristic
88#383—21no—
89#382—21no—
90#381—21no—
91#380C2×C412yes—
92C4×C4⋊C4—21no—
93C4×C22⋊C4—21no—
94#376—21no—
95#340—22no—
96#340—22no—
97#335—24no—
98#334—24no—
99Q8⋊C8—28no—
100#324—28no—
101C22×Q8—21no—
102C22×Q8—21no—
103C22×Q8—21no—
104C22×Q8—21no—
105C22×Q8—21no—
106C22×Q8—44no—
107C22×Q8C4⋊C412yes—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4C22×C411yescharacteristic
111C23×C4C2×D411yescharacteristic
112C23×C4#4111yescharacteristic
113C2×M16—22no—
114C2×M16—41no—
115C2×M16—41no—
116C2×M16—41no—
117C22×C8C22⋊C412yes—
118C22×C8C22⋊C412yes—
119C4×Q8—22no—
120C4×Q8—22no—
121C4×Q8—22no—
122C4×Q8—22no—
123C4×Q8—22no—
124C4×Q8—22no—
125C4×Q8—22no—
126C4×Q8—22no—
127C4×Q8—41no—
128C4×Q8—41no—
129C4×Q8—41no—
130C4×Q8—41no—
131C4×Q8—41no—
132C4×Q8—41no—
133C4×Q8—41no—
134C4×Q8—41no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4C4⋊C412yes—
148C2×C22⋊C4—22no—
149C2×C42—22no—
150C2×C42—22no—
151C2×C42—41no—
152C2×C42—41no—
153C2×C42—41no—
154C2×C42—41no—
155C2×C42—41no—
156C2×C42C2×D411yescharacteristic
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42C2×Q811yescharacteristic
163C2×C42#4111yescharacteristic
164C2×C42#4111yescharacteristic
165C2×C42#4111yescharacteristic
166C2×C42#4111yescharacteristic
167C2×C42#4111yescharacteristic
168C4⋊C8—22no—
169C4⋊C8—22no—
170C4⋊C8—22no—
171C4⋊C8—22no—
172C4⋊C8—22no—
173C4⋊C8—41no—
174C4⋊C8—41no—
175C4⋊C8—41no—
176C4⋊C8—41no—
177C4⋊C8—41no—
178#103—44no—
179#103—44no—
180C22⋊C8—22no—
181C22⋊C8—22no—
182C22⋊C8—22no—
183C22⋊C8—22no—
184C22⋊C8—22no—
185C22⋊C8—22no—
186C22⋊C8—22no—
187C22⋊C8—41no—
188C22⋊C8—41no—
189C22⋊C8—41no—
190#97—22no—
191#97—41no—
192#97—41no—
193#97—41no—
194C4×C8—21no—
195C4×C8—21no—
196C4×C8—22no—
197C4×C8—22no—
198C4×C8—21no—
199C4×C8—21no—
200C4×C8—41no—
201C4×C8—41no—
202#95—22no—
203#95—41no—
204#95—41no—
205#95—41no—
206C24C4×D411yescharacteristic
207C2×Q8—24no—
208C2×Q8—24no—
209C2×Q8—24no—
210C2×Q8—22no—
211C2×Q8—22no—
212C2×Q8—44no—
213C2×Q8—42no—
214C2×Q8—42no—
215C2×Q8—42no—
216C2×Q8—44no—
217C2×Q8—44no—
218C2×Q8—41no—
219C2×Q8—41no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—44no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—42no—
231C22×C4—82no—
232C22×C4C2×C22⋊C411yescharacteristic
233C22×C4C2×C4⋊C411yescharacteristic
234C22×C4#11711yescharacteristic
235C22×C4C4×D411yescharacteristic
236C22×C4C4×D411yescharacteristic
237C22×C4C4×D411yescharacteristic
238C22×C4#12011yescharacteristic
239C22×C4#12111yescharacteristic
240C22×C4#12111yescharacteristic
241C22×C4#12111yescharacteristic
242C22×C4C22⋊Q811yescharacteristic
243C22×C4C22⋊Q811yescharacteristic
244C22×C4#12311yescharacteristic
245C22×C4#12311yescharacteristic
246C22×C4#12411yescharacteristic
247C22×C4#12411yescharacteristic
248C22×C4#12611yescharacteristic
249C22×C4#12711yescharacteristic
250M16—44no—
251M16—82no—
252M16—82no—
253M16—82no—
254C2×C8—22no—
255C2×C8—22no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—42no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C22⋊C4—82no—
273C42—42no—
274C42—42no—
275C42—42no—
276C42—42no—
277C42—82no—
278C42#12011yescharacteristic
279C42#12011yescharacteristic
280C42#12111yescharacteristic
281C42#12111yescharacteristic
282C42#12111yescharacteristic
283C42#12111yescharacteristic
284C42#12111yescharacteristic
285C42C22⋊Q811yescharacteristic
286C42C22⋊Q811yescharacteristic
287C42C22⋊Q811yescharacteristic
288C42C22⋊Q811yescharacteristic
289C42#12311yescharacteristic
290C42#12311yescharacteristic
291C42#12311yescharacteristic
292C42#12411yescharacteristic
293C42#12611yescharacteristic
294C23—22no—
295C23—41no—
296C23—41no—
297C23#38511yescharacteristic
298C23#38811yescharacteristic
299C23#39011yescharacteristic
300Q8—24no—
301Q8—24no—
302Q8—42no—
303Q8—42no—
304Q8—44no—
305Q8—44no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—24no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—42no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4#38011yescharacteristic
337C2×C4#38511yescharacteristic
338C2×C4#38611yescharacteristic
339C2×C4#38811yescharacteristic
340C2×C4#39211yescharacteristic
341C2×C4#39311yescharacteristic
342C2×C4#39411yescharacteristic
343C2×C4#39411yescharacteristic
344C2×C4#39611yescharacteristic
345C2×C4#39611yescharacteristic
346C2×C4#39711yescharacteristic
347C2×C4#39911yescharacteristic
348C8—42no—
349C8—42no—
350C8—82no—
351C8—82no—
352C8—82no—
353C8—82no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—44no—
363C22#143911yescharacteristic
364C22#197612yes—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—44no—
376C4—82no—
377C4—162no—
378C4#189112yes—
379C4#208212yes—
380C2—22no—
381C2—41no—
382C2—41no—
383C2#1107111yescharacteristic
384C2#1112611yescharacteristic
385C2#1113411yescharacteristic
subgroupG/H
center, socleC22#1439
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24C4×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#207352
#178152
#174492
#173652
#81162
#79762
#79362