Finite Group Databasebeta

← #110341

Subgroups — #110341

Order 512.

768 conjugacy classes of subgroups.

The subgroup lattice has 437 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#101C211yesmaximal, characteristic
2C2×#1479C211yesmaximal, characteristic
3C22×Q8⋊C8C211yesmaximal, characteristic
4C2×#1451C211yesmaximal, characteristic
5C2×#1283C211yesmaximal, characteristic
6C2×#1278C211yesmaximal, characteristic
7#7873C211yesmaximal, characteristic
8C22×C4×Q8C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C22×C4⋊C8C2211yescharacteristic
12C22×#101C412yes—
13C2×C4×M16C2211yescharacteristic
14C2×C4×M16C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16C4×#101—22no—
17C4×#101—22no—
18C2×#343C412yes—
19#1479—22no—
20#1479C414yes—
21C2×Q8⋊C8—22no—
22C2×Q8⋊C8—21no—
23C2×Q8⋊C8—21no—
24C2×Q8⋊C8—22no—
25C2×Q8⋊C8C414yes—
26#1451—22no—
27#1451—22no—
28C2×#322—21no—
29C2×#322—21no—
30#1283—22no—
31#1278—22no—
32#1277—22no—
33C23×Q8C2×C411yescharacteristic
34C22×M16C2×C412yes—
35C22×M16C2×C412yes—
36C23×C8—21no—
37C23×C8—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×Q8—21no—
45C2×C4×Q8—21no—
46C2×C4×Q8—21no—
47C2×C4×Q8—21no—
48C2×C4×Q8C2×C412yes—
49C2×C4×Q8C2×C412yes—
50C22×C4⋊C4—21no—
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C42—21no—
53C22×C42C2311yescharacteristic
54#431—21no—
55#431—21no—
56#431—21no—
57#431—21no—
58#431—21no—
59#431C2×C412yes—
60#431C2×C412yes—
61#431D412yes—
62C2×C4⋊C8—21no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8C2×C412yes—
69C2×C4⋊C8D412yes—
70C2×#101—24no—
71C2×#101—24no—
72C2×#101—24no—
73C2×C22⋊C8—21no—
74C2×C22⋊C8—21no—
75C4×M16—21no—
76C4×M16—21no—
77C4×M16—21no—
78C4×M16—21no—
79C4×M16—21no—
80C4×M16—21no—
81C4×M16D412yes—
82C4×M16D412yes—
83C2×#97—21no—
84C2×#97—21no—
85C2×C4×C8—21no—
86C2×C4×C8—21no—
87C2×C4×C8—21no—
88C2×C4×C8—21no—
89C2×C4×C8—21no—
90C2×C4×C8—21no—
91C2×C4×C8—21no—
92C2×C4×C8C2×C412yes—
93C2×C4×C8D411yescharacteristic
94C2×C4×C8D411yescharacteristic
95C2×C4×C8D412yes—
96#343—24no—
97Q8⋊C8—28no—
98Q8⋊C8—24no—
99Q8⋊C8—24no—
100#322—24no—
101#322—24no—
102C22×Q8—21no—
103C22×Q8—22no—
104C22×Q8—21no—
105C22×Q8—21no—
106C22×Q8—21no—
107C22×Q8—21no—
108C22×Q8—22no—
109C22×Q8—22no—
110C22×Q8C4212yes—
111C22×Q8M1614yes—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4C22×C411yescharacteristic
115C23×C4#4111yescharacteristic
116C23×C4#4111yescharacteristic
117C2×M16—22no—
118C2×M16—22no—
119C2×M16—22no—
120C2×M16—22no—
121C2×M16—22no—
122C2×M16—22no—
123C2×M16—22no—
124C2×M16—22no—
125C2×M16—22no—
126C2×M16—22no—
127C2×M16C22⋊C414yes—
128C2×M16M1614yes—
129C22×C8—21no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—21no—
136C22×C8—22no—
137C22×C8—44no—
138C22×C8—41no—
139C22×C8—41no—
140C22×C8—41no—
141C22×C8—82no—
142C4×Q8—22no—
143C4×Q8—22no—
144C4×Q8—22no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8—22no—
150C4×Q8—41no—
151C4×Q8—41no—
152C4×Q8—41no—
153C4×Q8—41no—
154C4×Q8—41no—
155C4×Q8—41no—
156C4×Q8—41no—
157C4×Q8—41no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4C4212yes—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—41no—
173C2×C42—41no—
174C2×C42—41no—
175C2×C42—41no—
176C2×C42C22×C411yescharacteristic
177C2×C42C22×C411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42#4111yescharacteristic
185C2×C42#4111yescharacteristic
186C2×C42#4111yescharacteristic
187C2×C42#4111yescharacteristic
188C4⋊C8—22no—
189C4⋊C8—22no—
190C4⋊C8—22no—
191C4⋊C8—22no—
192C4⋊C8—41no—
193C4⋊C8—41no—
194C4⋊C8—41no—
195C4⋊C8C22⋊C414yes—
196#101—216no—
197C22⋊C8—22no—
198C22⋊C8—22no—
199C22⋊C8—22no—
200C22⋊C8—41no—
201#97—22no—
202#97—22no—
203#97—41no—
204#97—41no—
205C4×C8—22no—
206C4×C8—22no—
207C4×C8—22no—
208C4×C8—21no—
209C4×C8—21no—
210C4×C8—22no—
211C4×C8—22no—
212C4×C8—22no—
213C4×C8—22no—
214C4×C8—41no—
215C4×C8—41no—
216C4×C8—41no—
217C4×C8C22⋊C412yes—
218C4×C8C22⋊C412yes—
219C24#11711yescharacteristic
220C2×Q8—22no—
221C2×Q8—24no—
222C2×Q8—22no—
223C2×Q8—22no—
224C2×Q8—22no—
225C2×Q8—22no—
226C2×Q8—24no—
227C2×Q8—24no—
228C2×Q8—24no—
229C2×Q8—24no—
230C2×Q8—41no—
231C2×Q8—41no—
232C2×Q8—42no—
233C2×Q8—42no—
234C2×Q8—42no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8#9714yes—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—24no—
242C22×C4—22no—
243C22×C4—42no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—42no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—82no—
250C22×C4C2×C4211yescharacteristic
251C22×C4C2×C22⋊C411yescharacteristic
252C22×C4C4×D411yescharacteristic
253C22×C4C4×D411yescharacteristic
254C22×C4C4×D411yescharacteristic
255C22×C4C4×D411yescharacteristic
256C22×C4C4×D411yescharacteristic
257C22×C4C4×D411yescharacteristic
258C22×C4C4×D411yescharacteristic
259C22×C4C4×D411yescharacteristic
260C22×C4#12111yescharacteristic
261C22×C4#12111yescharacteristic
262C22×C4#12311yescharacteristic
263C22×C4#12311yescharacteristic
264C22×C4#12411yescharacteristic
265C22×C4#12411yescharacteristic
266C22×C4#12611yescharacteristic
267C22×C4#12611yescharacteristic
268C22×C4C2×M1612yes—
269C22×C4C2×M1612yes—
270C22×C4#13112yes—
271C22×C4#13112yes—
272M16—24no—
273M16—24no—
274M16—24no—
275M16—24no—
276M16—42no—
277M16—42no—
278M16—42no—
279M16—42no—
280C2×C8—24no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—24no—
284C2×C8—24no—
285C2×C8—24no—
286C2×C8—44no—
287C2×C8—44no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—82no—
298C2×C8—82no—
299C2×C8—82no—
300C2×C8—82no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—41no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—41no—
308C4⋊C4—42no—
309C42—42no—
310C42—42no—
311C42—42no—
312C42—42no—
313C42C2×C22⋊C411yescharacteristic
314C42C2×C22⋊C411yescharacteristic
315C42#11711yescharacteristic
316C42#11711yescharacteristic
317C42C4×D411yescharacteristic
318C42C4×D411yescharacteristic
319C42C4×D411yescharacteristic
320C42C4×D411yescharacteristic
321C42#12011yescharacteristic
322C42#12111yescharacteristic
323C42#12111yescharacteristic
324C42#12111yescharacteristic
325C42#12111yescharacteristic
326C42#12311yescharacteristic
327C42#12411yescharacteristic
328C42#12711yescharacteristic
329C23—22no—
330C23—22no—
331C23C4×C22⋊C411yescharacteristic
332C23#38811yescharacteristic
333C23#38811yescharacteristic
334C23#43212yes—
335C23#43312yes—
336Q8—24no—
337Q8—24no—
338Q8—44no—
339Q8—44no—
340Q8—42no—
341Q8—42no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—24no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4C4×C22⋊C411yescharacteristic
365C2×C4C4×C22⋊C411yescharacteristic
366C2×C4#38611yescharacteristic
367C2×C4#38611yescharacteristic
368C2×C4#38611yescharacteristic
369C2×C4#38811yescharacteristic
370C2×C4#38811yescharacteristic
371C2×C4#38811yescharacteristic
372C2×C4#38811yescharacteristic
373C2×C4#39011yescharacteristic
374C2×C4#39011yescharacteristic
375C2×C4#39011yescharacteristic
376C2×C4C2×#9711yescharacteristic
377C2×C4C4×M1611yescharacteristic
378C2×C4#40511yescharacteristic
379C2×C4#40511yescharacteristic
380C2×C4#40712yes—
381C2×C4#40812yes—
382C2×C4#43512yes—
383C2×C4#43512yes—
384C2×C4#43512yes—
385C2×C4#43512yes—
386C2×C4#43512yes—
387C2×C4#43512yes—
388C2×C4#43612yes—
389C2×C4#43612yes—
390C2×C4#47112yes—
391C2×C4#47312yes—
392C8—24no—
393C8—42no—
394C8—44no—
395C8—44no—
396C8—44no—
397C8—82no—
398C8—82no—
399C22—24no—
400C22—24no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22#143711yescharacteristic
406C22#174911yescharacteristic
407C22#175011yescharacteristic
408C22#192412yes—
409C22#192412yes—
410C22#193012yes—
411C22#193212yes—
412C4—21no—
413C4—21no—
414C4—21no—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—44no—
419C4—82no—
420C4#175211yescharacteristic
421C4#175211yescharacteristic
422C4#175311yescharacteristic
423C4#175311yescharacteristic
424C4#187012yes—
425C4#190812yes—
426C4#195512yes—
427C4#197711yescharacteristic
428C4#197911yescharacteristic
429C2—22no—
430C2—21no—
431C2—21no—
432C2#955114yes—
433C2#1092211yescharacteristic
434C2#1094211yescharacteristic
435C2#1094611yescharacteristic
subgroupG/H
centerC22×C4#121
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
socleC23#388
prime radicalC24#117
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ 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
C2×C4×#1012
C2×#14792
C22×Q8⋊C82
C2×#14512
C2×#12832
C2×#12782
#78732