Finite Group Databasebeta

← #16936634

Subgroups — #16936634

Order 4352.

676 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763010C211yesmaximal, characteristic
2C2×#11763009C211yesmaximal, characteristic
3#14534154C211yesmaximal, characteristic, complemented
4#14529590C211yesmaximal, characteristic
5C2×#11762541C211yesmaximal, characteristic, complemented
6#14526668C211yesmaximal, characteristic, complemented
7C2×C17×Q8⋊C8C211yesmaximal, characteristic, complemented
8C22×#63219—21no—
9C22×#63230—21no—
10C2×#10587560C2211yescharacteristic
11C2×C4×Dic34C2211yescharacteristic
12#11763010—22no—
13#11763010—22no—
14#11763009—22no—
15#11763009—22no—
16C2×C4⋊#3622C2211yescharacteristic
17C2×#10587492—21no—
18C2×#10587492—22no—
19C2×#10587480—21no—
20#11762742—22no—
21#11762716—21no—
22#11762715—21no—
23#11762707—21no—
24#11762705—21no—
25#11762679C2211yescharacteristic
26#11762541—22no—
27C2×C4×Q8×C17C2211yescharacteristic
28C2×C17×C4⋊C8C2211yescharacteristic
29C2×C4×C8×C17C2211yescharacteristic
30C17×Q8⋊C8—22no—
31C22×Q8×C17D411yescharacteristic
32C22×D68—21no—
33C22×D68—21no—
34C22×D68—21no—
35C22×Dic34D411yescharacteristic
36C22×C8×C17—21no—
37C22×C8×C17—21no—
38C4×Q8×C17—21no—
39C4×Q8×C17—21no—
40C4×Q8×C17—21no—
41C4×Q8×C17—21no—
42C2×C4⋊C4×C17—21no—
43C2×C4⋊C4×C17D411yescharacteristic
44C2×C42×C17—21no—
45C2×C42×C17C2311yescharacteristic
46C2×#63230—24no—
47C2×#63230—42no—
48C2×#63230—42no—
49C2×C4⋊Dic17—21no—
50C2×C4⋊Dic17—21no—
51C2×C4⋊Dic17D411yescharacteristic
52C2×C4⋊Dic17D412yes—
53C2×#63225—22no—
54C2×#63225—21no—
55C2×C4×Dic17—21no—
56C22×#3622—21no—
57C2×#63219—24no—
58C2×#63219—42no—
59C2×#63219—42no—
60#10587560—21no—
61#10587560—21no—
62#10587560—21no—
63#10587560D412yes—
64C4×Dic34—21no—
65C4×Dic34—21no—
66C4×Dic34—21no—
67C4×Dic34—21no—
68C17×C4⋊C8—21no—
69C17×C4⋊C8D412yes—
70C4×C8×C17—21no—
71C4×C8×C17—21no—
72#10587502—22no—
73#10587502—22no—
74#10587502—22no—
75#10587492—24no—
76#10587492—44no—
77#10587480—24no—
78C4⋊#3622—21no—
79C4⋊#3622D412yes—
80C23×D17—41no—
81C2×Q8×C17—21no—
82C2×Q8×C17—21no—
83C2×Q8×C17—21no—
84C2×Q8×C17—21no—
85C2×Q8×C17—21no—
86C2×Q8×C17SD1612yescomplemented
87C22×C4×C17—21no—
88C22×C4×C17—21no—
89C22×C4×C17C2×D411yescharacteristic
90C22×C4×C17C2×D411yescharacteristic
91C22×C4×C17C2×D411yescharacteristic
92C22×Dic17—21no—
93C22×Dic17—22no—
94C22×Dic17—21no—
95C2×D68—22no—
96C2×D68—22no—
97C2×D68—22no—
98C2×D68—42no—
99C2×D68—42no—
100C2×D68—41no—
101C2×D68—41no—
102C2×D68—41no—
103C2×D68—41no—
104C2×D68—41no—
105C2×D68—41no—
106C2×D68—41no—
107C2×D68—41no—
108C2×D68—41no—
109C2×D68—41no—
110C2×D68—41no—
111C2×Dic34—21no—
112C2×Dic34—21no—
113C2×Dic34—21no—
114C2×Dic34—21no—
115C2×Dic34—21no—
116C2×Dic34SD1612yescomplemented
117C2×C8×C17—22no—
118C2×C8×C17—22no—
119C2×C8×C17—41no—
120C2×C8×C17—41no—
121C2×C8×C17—41no—
122C2×C8×C17—41no—
123C4⋊C4×C17—21no—
124C4⋊C4×C17—21no—
125C4⋊C4×C17—41no—
126C4⋊C4×C17—41no—
127C42×C17—41no—
128C42×C17—41no—
129C42×C17C2×D411yescharacteristic
130C42×C17C2×D411yescharacteristic
131C42×C17C2×D411yescharacteristic
132C42×C17#4111yescharacteristic
133#63230—48no—
134C4⋊Dic17—21no—
135C4⋊Dic17—21no—
136C4⋊Dic17—41no—
137C4⋊Dic17—41no—
138C4⋊Dic17—41no—
139C4⋊Dic17—41no—
140C4⋊Dic17—42no—
141#63225—41no—
142#63225—41no—
143#63225—82no—
144C4×Dic17—41no—
145C4×Dic17—41no—
146C2×#3622—22no—
147C2×#3622—41no—
148C2×#3622—41no—
149#63219—48no—
150#10230—171nomaximal, Sylow
151C23×C17#12011yescharacteristic
152C22×D17—42no—
153C22×D17—81no—
154C22×D17—81no—
155C22×D17—81no—
156C22×D17—81no—
157C22×D17—81no—
158C22×D17—81no—
159Q8×C17—22no—
160Q8×C17—22no—
161Q8×C17—41no—
162Q8×C17—41no—
163C2×C4×C17—21no—
164C2×C4×C17—21no—
165C2×C4×C17—22no—
166C2×C4×C17—41no—
167C2×C4×C17—41no—
168C2×C4×C17—41no—
169C2×C4×C17—41no—
170C2×C4×C17—41no—
171C2×C4×C17#12011yescharacteristic
172C2×C4×C17#12011yescharacteristic
173C2×C4×C17#12111yescharacteristic
174C2×C4×C17#12111yescharacteristic
175C2×C4×C17#12111yescharacteristic
176C2×C4×C17#12711yescharacteristic
177C2×C4×C17C2×SD1611yescharacteristic
178C2×C4×C17C2×SD1611yescharacteristic
179C2×C4×C17#13511yescharacteristic
180C2×C4×C17#13511yescharacteristic
181C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
182C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
183C2×C4×C17#13711yescharacteristic
184C2×C4×C17#13711yescharacteristic
185C2×Dic17—22no—
186C2×Dic17—41no—
187C2×Dic17—41no—
188C2×Dic17—41no—
189C2×Dic17—42no—
190C2×Dic17—42no—
191C2×Dic17—41no—
192C2×Dic17—42no—
193C2×Dic17—41no—
194D68—44no—
195D68—42no—
196D68—42no—
197D68—42no—
198D68—42no—
199D68—82no—
200D68—81no—
201D68—81no—
202D68—81no—
203D68—81no—
204Dic34—22no—
205Dic34—22no—
206Dic34—41no—
207Dic34—41no—
208C8×C17—42no—
209C8×C17—42no—
210#3622—42no—
211C2×#460—171no—
212C2×#460—171no—
213#2582—171no—
214#1957—171no—
215C2×Q8⋊C8—171no—
216C2×Q8⋊C8D1711yescharacteristic, complemented
217#1296—171no—
218C22×C17#39211yescharacteristic
219C22×C17#44811yescharacteristic
220C22×C17#44811yescharacteristic
221C22×C17#44911yescharacteristic
222C22×C17#44911yescharacteristic
223C22×C17#45311yescharacteristic
224C22×C17#45411yescharacteristic
225C2×D17—82no—
226C2×D17—82no—
227C2×D17—82no—
228C2×D17—161no—
229C2×D17—161no—
230C2×D17—161no—
231C2×D17—161no—
232C4×C17—21no—
233C4×C17—21no—
234C4×C17—42no—
235C4×C17—81no—
236C4×C17#45011yescharacteristic
237C4×C17#45011yescharacteristic
238C4×C17#45211yescharacteristic
239C4×C17#45211yescharacteristic
240C4×C17#46511yescharacteristic
241C4×C17#46511yescharacteristic
242C4×C17#46811yescharacteristic
243C4×C17#46811yescharacteristic
244Dic17—42no—
245Dic17—82no—
246Dic17—81no—
247C22×SD16—341no—
248C22×SD16—341no—
249C2×#127—171no—
250C2×C4×Q8—171no—
251C2×C4×Q8C2×D1711yescharacteristic
252#460—342no—
253#460—342no—
254#460—342no—
255#460—342no—
256C2×C4⋊C8—171no—
257C2×C4⋊C8C2×D1711yescharacteristic
258C2×#102—341no—
259C2×#102—341no—
260C2×#102—342no—
261C2×C4×C8C2×D1711yescharacteristic
262#398—342no—
263#391—341no—
264#391—341no—
265#389—341no—
266#389—341no—
267#384—171no—
268Q8⋊C8—22no—
269Q8⋊C8—342no—
270C2×C17#162011yescharacteristic
271C2×C17#162311yescharacteristic
272C2×C17#162611yescharacteristic
273C2×C17#162611yescharacteristic
274C2×C17#200011yescharacteristic
275C2×C17#200011yescharacteristic
276C2×C17#200611yescharacteristic
277D17—162no—
278C22×Q8—171no—
279C22×Q8D6811yescharacteristic
280C22×D4—341no—
281C22×D4—341no—
282C22×D4—341no—
283C2×SD16—344no—
284C2×SD16—344no—
285C2×SD16—682no—
286C2×SD16—682no—
287C2×SD16—682no—
288C2×SD16—682no—
289C22×C8—21no—
290C22×C8—21no—
291C22×C8—341no—
292#127—172no—
293#127—341no—
294#127—341no—
295#127—341no—
296C4×Q8—21no—
297C4×Q8—21no—
298C4×Q8—21no—
299C4×Q8—21no—
300C4×Q8—341no—
301C4×Q8—341no—
302C4×Q8—341no—
303C4×Q8—341no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—171no—
306C2×C4⋊C4—172no—
307C2×C4⋊C4—341no—
308C2×C4⋊C4—341no—
309C2×C4⋊C4—341no—
310C2×C4⋊C4—342no—
311C2×C4⋊C4D6811yescharacteristic
312C2×C42—21no—
313C2×C42—341no—
314C2×C42C22×D1711yescharacteristic
315C4⋊C8—21no—
316C4⋊C8—172no—
317C4⋊C8—341no—
318C4⋊C8C22⋊D1712yes—
319#102—344no—
320#102—344no—
321#102—684no—
322C4×C8—21no—
323C4×C8—21no—
324#95—342no—
325#95—342no—
326#95—342no—
327C17#1023011yescharacteristic, complemented, Sylow
328C24—681no—
329C2×Q8—21no—
330C2×Q8—21no—
331C2×Q8—21no—
332C2×Q8—21no—
333C2×Q8—21no—
334C2×Q8—172no—
335C2×Q8—341no—
336C2×Q8—341no—
337C2×Q8—341no—
338C2×Q8—341no—
339C2×Q8—341no—
340C2×Q8#6321912yescomplemented
341C2×D4—342no—
342C2×D4—342no—
343C2×D4—342no—
344C2×D4—682no—
345C2×D4—682no—
346C2×D4—681no—
347C2×D4—681no—
348C2×D4—681no—
349C2×D4—681no—
350C2×D4—681no—
351C2×D4—681no—
352C2×D4—681no—
353C2×D4—681no—
354C2×D4—681no—
355C2×D4—681no—
356C2×D4—681no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—341no—
360C22×C4—342no—
361C22×C4—341no—
362C22×C4C2×D6811yescharacteristic
363C22×C4D4×D1711yescharacteristic
364C22×C4D4×D1711yescharacteristic
365SD16—688no—
366SD16—688no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—342no—
374C2×C8—681no—
375C2×C8—681no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—341no—
381C4⋊C4—341no—
382C4⋊C4—682no—
383C4⋊C4—681no—
384C4⋊C4—681no—
385C4⋊C4—681no—
386C4⋊C4—681no—
387C4⋊C4—681no—
388C4⋊C4—681no—
389C4⋊C4—1362no—
390C42—41no—
391C42—41no—
392C42—681no—
393C42—681no—
394C42#6325211yescharacteristic
395C42D4×D1711yescharacteristic
396C42D4×D1711yescharacteristic
397C42C2×C22⋊D1711yescharacteristic
398C23—682no—
399C23—1361no—
400C23—1361no—
401C23—1361no—
402C23—1361no—
403C23—1361no—
404C23—1361no—
405C23#1058756811yescharacteristic
406Q8—22no—
407Q8—22no—
408Q8—41no—
409Q8—41no—
410Q8—342no—
411Q8—342no—
412Q8—681no—
413Q8—681no—
414D4—684no—
415D4—682no—
416D4—682no—
417D4—682no—
418D4—682no—
419D4—1362no—
420D4—1361no—
421D4—1361no—
422D4—1361no—
423D4—1361no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—22no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—342no—
433C2×C4—681no—
434C2×C4—681no—
435C2×C4—682no—
436C2×C4—682no—
437C2×C4—681no—
438C2×C4—682no—
439C2×C4—681no—
440C2×C4—681no—
441C2×C4#1058756811yescharacteristic
442C2×C4#1058757011yescharacteristic
443C2×C4#1058757011yescharacteristic
444C2×C4C2×#6321911yescharacteristic
445C2×C4#1058759011yescharacteristic
446C2×C4#1058759411yescharacteristic
447C2×C4#1058759511yescharacteristic
448C2×C4SD16×D1711yescharacteristic
449C2×C4#1058760011yescharacteristic
450C2×C4#1058760411yescharacteristic
451C2×C4#1058760511yescharacteristic
452C2×C4#1058761611yescharacteristic
453C2×C4#1058762311yescharacteristic
454C2×C4#1058762611yescharacteristic
455C8—42no—
456C8—42no—
457C8—682no—
458C22—2721no—
459C22—2721no—
460C22—2721no—
461C22—2721no—
462C22—1362no—
463C22—1362no—
464C22—1362no—
465C22#1176271811yescharacteristic
466C22#1176294911yescharacteristic
467C22#1176295011yescharacteristic
468C22#1176295611yescharacteristic
469C22#1176295711yescharacteristic
470C22#1176297811yescharacteristic
471C22#1176298311yescharacteristic
472C4—21no—
473C4—21no—
474C4—42no—
475C4—81no—
476C4—682no—
477C4—1361no—
478C4—1362no—
479C4#1176295811yescharacteristic
480C4#1176296211yescharacteristic
481C4#1176297011yescharacteristic
482C4#1176297311yescharacteristic
483C4#1176303811yescharacteristic
484C4#1176304111yescharacteristic
485C4#1176305511yescharacteristic
486C4#1176305811yescharacteristic
487C2—2722no—
488C2#1452784511yescharacteristic
489C2#1452786411yescharacteristic
490C2#1452788211yescharacteristic
491C2#1452788311yescharacteristic
492C2#1452978611yescharacteristic
493C2#1452978911yescharacteristic
494C2#1452982711yescharacteristic
subgroupG/H
centerC23#10587568
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC2×C17×Q8⋊C8C2
socleC23×C17#120
prime radicalC2×#10587560C22
hypercenterC2×Q8⋊C8D17
hypocenterC17#10230
LCSG ▷ C2×C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×Q8⋊C8

Complements

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

NH ≅ G/Ntype
3. #14534154C2semidirect
5. C2×#11762541C2semidirect
6. #14526668C2semidirect
7. C2×C17×Q8⋊C8C2semidirect
86. C2×Q8×C17SD16semidirect
116. C2×Dic34SD16semidirect
216. C2×Q8⋊C8D17semidirect
327. C17#10230semidirect
340. C2×Q8#63219semidirect

Maximal subgroups

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

subgroupindex
C2×#117630102
C2×#117630092
#145341542
#145295902
C2×#117625412
#145266682
C2×C17×Q8⋊C82
#1023017