Finite Group Databasebeta

← C2×#17204 #844990

Subgroups — C2×#17204 #844990

Order 512.

4850 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#399C211yesmaximal, characteristic, complemented
2C22×#396C211yesmaximal, characteristic, complemented
3#17204C2116yesmaximal, direct product factor
4C2×#1443C211yesmaximal, characteristic, complemented
5C2×#1441C213yesmaximal, complemented
6C2×#1439C213yesmaximal, complemented
7C2×#1439C213yesmaximal, complemented
8C2×#1438C213yesmaximal
9C24×D4C2211yescharacteristic
10C23×C4⋊C4C2213yes—
11C23×C4⋊C4C2213yes—
12C23×C22⋊C4C2213yes—
13C23×C22⋊C4C2213yes—
14C23×C22⋊C4C2213yes—
15C23×C22⋊C4C2213yes—
16C2×#399—28no—
17C2×#399C2218yescomplemented
18C2×#396—28no—
19C2×#396C2218yescomplemented
20C22×#95C2213yes—
21C22×#95C2213yes—
22C22×#95C2213yes—
23C22×#95C2213yes—
24C22×#95C2213yes—
25C22×#95C2211yescharacteristic
26#1443C2218yescomplemented
27#1441C22124yescomplemented
28#1439C22124yescomplemented
29#1439C22124yescomplemented
30#1438C22124yes—
31C26C2311yescharacteristic
32C26C2311yescharacteristic
33C23×D4—46no—
34C23×D4—46no—
35C23×D4Q818yes—
36C23×D4C2314yes—
37C24×C4C2313yes—
38C24×C4C2313yes—
39C24×C4C2313yes—
40C24×C4C2311yescharacteristic
41C24×C4C2313yes—
42C22×C4⋊C4—212no—
43C22×C4⋊C4—212no—
44C22×C4⋊C4—212no—
45C22×C4⋊C4—46no—
46C22×C4⋊C4—46no—
47C22×C4⋊C4—46no—
48C22×C4⋊C4C23112yes—
49C22×C4⋊C4C23112yes—
50C22×C22⋊C4—212no—
51C22×C22⋊C4—212no—
52C22×C22⋊C4—212no—
53C22×C22⋊C4—212no—
54C22×C22⋊C4—46no—
55C22×C22⋊C4—46no—
56C22×C22⋊C4—46no—
57C22×C22⋊C4—46no—
58C22×C22⋊C4—46no—
59C22×C22⋊C4—46no—
60C22×C22⋊C4—46no—
61C22×C22⋊C4—46no—
62C22×C22⋊C4C23112yes—
63C22×C22⋊C4C23112yes—
64C22×C22⋊C4C23112yes—
65C22×C22⋊C4C23112yes—
66#399—232no—
67#396—232no—
68C2×#95—46no—
69C2×#95—46no—
70C2×#95—46no—
71C2×#95—46no—
72C2×#95—46no—
73C2×#95D418yescomplemented
74C2×#95C23112yes—
75C2×#95C23112yes—
76C2×#95C23112yes—
77C2×#95C23112yes—
78C2×#95C23112yes—
79C2×#95C2314yes—
80C25—22no—
81C25—22no—
82C25—26no—
83C25—26no—
84C25—26no—
85C25—26no—
86C25—26no—
87C25—26no—
88C25—26no—
89C25—26no—
90C25—22no—
91C25—22no—
92C25C2×Q814yes—
93C25C2×Q814yes—
94C25C2412yes—
95C25C2412yes—
96C25C2411yescharacteristic
97C22×D4—448no—
98C22×D4—412no—
99C22×D4—412no—
100C22×D4—416no—
101C22×D4—448no—
102C22×D4C2×Q8116yes—
103C23×C4—26no—
104C23×C4—26no—
105C23×C4—26no—
106C23×C4—26no—
107C23×C4—26no—
108C23×C4—26no—
109C23×C4—26no—
110C23×C4—26no—
111C23×C4—26no—
112C23×C4—26no—
113C23×C4—26no—
114C23×C4—26no—
115C23×C4—26no—
116C23×C4—26no—
117C23×C4—26no—
118C23×C4—26no—
119C23×C4—26no—
120C23×C4—26no—
121C23×C4—26no—
122C23×C4—26no—
123C23×C4—26no—
124C23×C4—26no—
125C23×C4—26no—
126C23×C4—26no—
127C23×C4—26no—
128C23×C4—26no—
129C23×C4—26no—
130C23×C4—26no—
131C23×C4C2×D4112yes—
132C23×C4C2×Q814yes—
133C23×C4#41112yes—
134C23×C4C2416yes—
135C23×C4C2416yes—
136C23×C4C2416yes—
137C23×C4C2412yes—
138C23×C4C2416yes—
139C2×C4⋊C4—224no—
140C2×C4⋊C4—224no—
141C2×C4⋊C4—224no—
142C2×C4⋊C4—412no—
143C2×C4⋊C4—412no—
144C2×C4⋊C4—412no—
145C2×C4⋊C4—448no—
146C2×C4⋊C4—448no—
147C2×C22⋊C4—224no—
148C2×C22⋊C4—224no—
149C2×C22⋊C4—224no—
150C2×C22⋊C4—224no—
151C2×C22⋊C4—412no—
152C2×C22⋊C4—412no—
153C2×C22⋊C4—412no—
154C2×C22⋊C4—412no—
155C2×C22⋊C4—412no—
156C2×C22⋊C4—412no—
157C2×C22⋊C4—412no—
158C2×C22⋊C4—412no—
159C2×C22⋊C4—448no—
160C2×C22⋊C4—448no—
161C2×C22⋊C4—448no—
162C2×C22⋊C4—448no—
163#95—412no—
164#95—412no—
165#95—412no—
166#95—412no—
167#95—412no—
168#95C2×D4116yescomplemented
169C24—224no—
170C24—224no—
171C24—28no—
172C24—28no—
173C24—224no—
174C24—224no—
175C24—22no—
176C24—22no—
177C24—26no—
178C24—26no—
179C24—26no—
180C24—26no—
181C24—26no—
182C24—26no—
183C24—26no—
184C24—26no—
185C24—22no—
186C24—22no—
187C24—412no—
188C24—412no—
189C24—412no—
190C24—412no—
191C24—412no—
192C24—412no—
193C24—412no—
194C24—412no—
195C24—412no—
196C24—412no—
197C24—412no—
198C24—412no—
199C24—412no—
200C24—44no—
201C24—44no—
202C24—412no—
203C24—412no—
204C24—412no—
205C24—412no—
206C24—412no—
207C24C22×D412yes—
208C24C22×Q814yes—
209C24C22×Q814yes—
210C24C22×Q812yes—
211C24C2×#4116yes—
212C24H(5,𝔽2)16yes—
213C24H(5,𝔽2)16yes—
214C24H(5,𝔽2)16yes—
215C242−1+412yes—
216C24C2511yescharacteristic
217C2×D4—448no—
218C2×D4—4128no—
219C2×D4—416no—
220C2×D4—448no—
221C22×C4—224no—
222C22×C4—224no—
223C22×C4—224no—
224C22×C4—224no—
225C22×C4—224no—
226C22×C4—224no—
227C22×C4—224no—
228C22×C4—26no—
229C22×C4—26no—
230C22×C4—26no—
231C22×C4—26no—
232C22×C4—26no—
233C22×C4—26no—
234C22×C4—26no—
235C22×C4—26no—
236C22×C4—26no—
237C22×C4—26no—
238C22×C4—26no—
239C22×C4—26no—
240C22×C4—26no—
241C22×C4—26no—
242C22×C4—26no—
243C22×C4—26no—
244C22×C4—26no—
245C22×C4—26no—
246C22×C4—26no—
247C22×C4—26no—
248C22×C4—26no—
249C22×C4—26no—
250C22×C4—26no—
251C22×C4—26no—
252C22×C4—26no—
253C22×C4—26no—
254C22×C4—26no—
255C22×C4—26no—
256C22×C4—412no—
257C22×C4—412no—
258C22×C4—412no—
259C22×C4—412no—
260C22×C4—412no—
261C22×C4—412no—
262C22×C4—412no—
263C22×C4—412no—
264C22×C4—44no—
265C22×C4—412no—
266C22×C4—412no—
267C22×C4—412no—
268C22×C4—412no—
269C22×C4—412no—
270C22×C4—412no—
271C22×C4—412no—
272C22×C4—412no—
273C22×C4—412no—
274C22×C4—412no—
275C22×C4—412no—
276C22×C4—412no—
277C22×C4—412no—
278C22×C4—412no—
279C22×C4—412no—
280C22×C4C22×D4112yes—
281C22×C4C22×Q814yes—
282C22×C4C2×#41112yes—
283C4⋊C4—448no—
284C4⋊C4—448no—
285C22⋊C4—448no—
286C22⋊C4—448no—
287C22⋊C4—448no—
288C22⋊C4—448no—
289C23—212no—
290C23—212no—
291C23—24no—
292C23—24no—
293C23—212no—
294C23—212no—
295C23—416no—
296C23—416no—
297C23—46no—
298C23—46no—
299C23—46no—
300C23—46no—
301C23—46no—
302C23—46no—
303C23—46no—
304C23—46no—
305C23—448no—
306C23—448no—
307C23—46no—
308C23—46no—
309C23—46no—
310C23—46no—
311C23—46no—
312C23—42no—
313C23—42no—
314C23—46no—
315C23—46no—
316C23—46no—
317C23—46no—
318C23—46no—
319C23—448no—
320C23—448no—
321C23—88no—
322C23—88no—
323C23—824no—
324C23—824no—
325C23—824no—
326C23—824no—
327C23—88no—
328C23—88no—
329C23#535112yes—
330C23#538112yes—
331C23#540112yes—
332C23#542112yes—
333C23#543112yes—
334C23#546112yes—
335C23D4×Q814yes—
336C23#553112yes—
337C23#554112yes—
338C23#560112yes—
339C23#56014yes—
340C23#560112yes—
341C23#562112yes—
342C23C23×D411yescharacteristic
343C23C23×Q811yescharacteristic
344C23C22×#4113yes—
345C23C2×H(5,𝔽2)13yes—
346C23C2×H(5,𝔽2)13yes—
347C23C2×H(5,𝔽2)13yes—
348C23C2×2−1+411yescharacteristic
349D4—464no—
350C2×C4—212no—
351C2×C4—212no—
352C2×C4—212no—
353C2×C4—212no—
354C2×C4—212no—
355C2×C4—212no—
356C2×C4—212no—
357C2×C4—416no—
358C2×C4—46no—
359C2×C4—46no—
360C2×C4—46no—
361C2×C4—46no—
362C2×C4—46no—
363C2×C4—448no—
364C2×C4—448no—
365C2×C4—46no—
366C2×C4—46no—
367C2×C4—46no—
368C2×C4—42no—
369C2×C4—46no—
370C2×C4—46no—
371C2×C4—46no—
372C2×C4—46no—
373C2×C4—46no—
374C2×C4—46no—
375C2×C4—46no—
376C2×C4—46no—
377C2×C4—46no—
378C2×C4—46no—
379C2×C4—46no—
380C2×C4—46no—
381C2×C4—46no—
382C2×C4—46no—
383C2×C4—46no—
384C2×C4—824no—
385C2×C4—824no—
386C22—44no—
387C22—44no—
388C22—412no—
389C22—412no—
390C22—412no—
391C22—412no—
392C22—82no—
393C22—82no—
394C22—832no—
395C22—832no—
396C22—86no—
397C22—86no—
398C22—82no—
399C22—82no—
400C22—86no—
401C22—86no—
402C22#2692124yes—
403C22#2703124yes—
404C22#2780124yes—
405C22#280118yes—
406C22#281818yes—
407C22#2829124yes—
408C22#284218yes—
409C22C2×#53513yes—
410C22C2×#53813yes—
411C22C2×#54013yes—
412C22C2×#54213yes—
413C22C2×#54313yes—
414C22C2×#54613yes—
415C22C2×D4×Q811yescharacteristic
416C22C2×#55313yes—
417C22C2×#55413yes—
418C22C2×#56011yescharacteristic
419C22C2×#56013yes—
420C22C2×#56013yes—
421C22C2×#56213yes—
422C4—44no—
423C4—412no—
424C4—412no—
425C4—86no—
426C4—86no—
427C2—84no—
428C2—84no—
429C2#17204116yesdirect product factor
430C2C2×#282913yes—
431C2C2×#284211yescharacteristic
432C2C2×#281811yescharacteristic
433C2C2×#269213yes—
434C2C2×#270313yes—
435C2C2×#278013yes—
436C2C2×#280111yescharacteristic
subgroupG/H
center, socleC25C24
commutator, Frattini, cosocleC24C25
prime radicalC24×D4C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C25 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#399C2semidirect
2. C22×#396C2semidirect
3. #17204C2direct (N × H)
4. C2×#1443C2semidirect
5. C2×#1441C2semidirect
6. C2×#1439C2semidirect
7. C2×#1439C2semidirect
17. C2×#399C22semidirect
19. C2×#396C22semidirect
26. #1443C22semidirect
27. #1441C22semidirect
28. #1439C22semidirect
29. #1439C22semidirect
73. C2×#95D4semidirect
168. #95C2×D4semidirect
429. C2#17204direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#3992
C22×#3962
#172042
C2×#14432
C2×#14412
C2×#14392
C2×#14382