Finite Group Databasebeta

← #32989339

Subgroups — #32989339

Order 18688.

708 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22610738C211yesmaximal, characteristic, complemented
2#22610689C211yesmaximal, characteristic, complemented
3#22609931C211yesmaximal, characteristic, complemented
4C2×#17988331C211yesmaximal, characteristic, complemented
5C2×D4⋊#10596831C211yesmaximal, characteristic, complemented
6#22607772C211yesmaximal, characteristic, complemented
7C73×#2541C211yesmaximal, characteristic
8C2×#14556524—21no—
9C2×C4⋊#10596831C2211yescharacteristic
10C2×C4⋊#10596831C2211yescharacteristic
11C2×#14556512—21no—
12C2×#14556512—22no—
13C2×C4×#10596831C2211yescharacteristic
14#17988457C2211yescharacteristic
15#17988349—21no—
16#17988349—21no—
17#17988331—22no—
18D4⋊#10596831—22no—
19C2×#127×C73—21no—
20C2×#127×C73C2211yescharacteristic
21C2×#121×C73—21no—
22C2×#121×C73—21no—
23C2×C4×D4×C73C2211yescharacteristic
24#392×C73—22no—
25#390×C73—21no—
26C4×C4⋊C4×C73C2211yescharacteristic
27C22×#10596831—21no—
28C22×#10596831—21no—
29C22×#10596831—21no—
30C2×C4⋊Dic73—21no—
31C2×C4⋊Dic73—21no—
32C2×C4⋊Dic73—21no—
33#14556524—42no—
34C4⋊#10596831—21no—
35C4⋊#10596831—21no—
36C4⋊#10596831—21no—
37C4⋊#10596831D412yescomplemented
38#14556512—44no—
39#14556512—42no—
40C4×#10596831—21no—
41C4×#10596831—21no—
42C22×D4×C73—22no—
43C22×D4×C73—22no—
44C22×D4×C73—21no—
45C22×D4×C73—21no—
46C22×D4×C73D411yescharacteristic
47C22×D4×C73D412yes—
48C23×C4×C73—21no—
49#127×C73—21no—
50#127×C73—21no—
51#127×C73—21no—
52#127×C73—21no—
53#127×C73—42no—
54#127×C73—42no—
55#121×C73—42no—
56#121×C73—42no—
57#121×C73—42no—
58#121×C73—42no—
59C4×D4×C73—21no—
60C4×D4×C73—21no—
61C4×D4×C73—21no—
62C4×D4×C73—21no—
63C2×C4⋊C4×C73D411yescharacteristic
64C2×C4⋊C4×C73D411yescharacteristic
65C2×C4⋊C4×C73D411yescharacteristic
66C2×C22⋊C4×C73—22no—
67C2×C22⋊C4×C73—22no—
68C2×C22⋊C4×C73—21no—
69C2×C42×C73—21no—
70C2×C42×C73C2311yescharacteristic
71#95×C73—21no—
72C22×Dic73—41no—
73C2×#10596831—22no—
74C2×#10596831—22no—
75C2×#10596831—41no—
76C2×#10596831—41no—
77C2×#10596831—41no—
78C2×#10596831—41no—
79C2×#10596831—41no—
80C2×#10596831—41no—
81C2×#10596831—41no—
82C4⋊Dic73—42no—
83C4⋊Dic73—41no—
84C4⋊Dic73—41no—
85C4⋊Dic73—41no—
86C4⋊Dic73—41no—
87C24×C73—21no—
88C24×C73—22no—
89C2×D4×C73—21no—
90C2×D4×C73—22no—
91C2×D4×C73—21no—
92C2×D4×C73—22no—
93C2×D4×C73—21no—
94C2×D4×C73—21no—
95C2×D4×C73—21no—
96C2×D4×C73—42no—
97C2×D4×C73—42no—
98C2×D4×C73—44no—
99C2×D4×C73—44no—
100C2×D4×C73—44no—
101C2×D4×C73—44no—
102C2×D4×C73—41no—
103C2×D4×C73—41no—
104C2×D4×C73—41no—
105C2×D4×C73—41no—
106C2×D4×C73—41no—
107C2×D4×C73—41no—
108C2×D4×C73—41no—
109C2×D4×C73—41no—
110C2×D4×C73—41no—
111C2×D4×C73—41no—
112C2×D4×C73—41no—
113C2×D4×C73—41no—
114C2×D4×C73—82no—
115C2×D4×C73—82no—
116C2×D4×C73D812yes—
117C22×C4×C73—21no—
118C22×C4×C73—21no—
119C22×C4×C73—21no—
120C22×C4×C73—41no—
121C22×C4×C73—41no—
122C22×C4×C73—41no—
123C22×C4×C73—41no—
124C22×C4×C73—41no—
125C22×C4×C73—41no—
126C22×C4×C73C2×D411yescharacteristic
127C22×C4×C73C2×D411yescharacteristic
128C22×C4×C73C2×D411yescharacteristic
129C4⋊C4×C73—21no—
130C4⋊C4×C73—21no—
131C4⋊C4×C73—21no—
132C4⋊C4×C73—21no—
133C4⋊C4×C73D812yes—
134C4⋊C4×C73SD1612yes—
135C22⋊C4×C73—44no—
136C22⋊C4×C73—44no—
137C22⋊C4×C73—41no—
138C22⋊C4×C73—41no—
139C42×C73—41no—
140C42×C73—41no—
141C42×C73C2×D411yescharacteristic
142C42×C73#4111yescharacteristic
143C42×C73#4111yescharacteristic
144C42×C73#4111yescharacteristic
145C23×C73—22no—
146C23×C73—41no—
147C23×C73—41no—
148C23×C73—41no—
149C23×C73—41no—
150C23×C73—41no—
151C23×C73—41no—
152C23×C73—42no—
153C23×C73—42no—
154C23×C73—42no—
155C23×C73—42no—
156C23×C73—42no—
157C23×C73—42no—
158C23×C73—42no—
159C23×C73#12011yescharacteristic
160D4×C73—22no—
161D4×C73—22no—
162D4×C73—44no—
163D4×C73—41no—
164D4×C73—41no—
165D4×C73—82no—
166D4×C73—88no—
167D4×C73—88no—
168D4×C73—81no—
169D4×C73—81no—
170D4×C73—81no—
171D4×C73—81no—
172D4×C73—81no—
173D4×C73—81no—
174D4×C73—81no—
175D4×C73—81no—
176C2×C4×C73—21no—
177C2×C4×C73—21no—
178C2×C4×C73—22no—
179C2×C4×C73—22no—
180C2×C4×C73—41no—
181C2×C4×C73—41no—
182C2×C4×C73—41no—
183C2×C4×C73—41no—
184C2×C4×C73—41no—
185C2×C4×C73—41no—
186C2×C4×C73—41no—
187C2×C4×C73—81no—
188C2×C4×C73—81no—
189C2×C4×C73—81no—
190C2×C4×C73—81no—
191C2×C4×C73#12111yescharacteristic
192C2×C4×C73#12111yescharacteristic
193C2×C4×C73#12111yescharacteristic
194C2×C4×C73#12311yescharacteristic
195C2×C4×C73#12311yescharacteristic
196C2×C4×C73#12411yescharacteristic
197C2×C4×C73C2×D811yescharacteristic
198C2×C4×C73C2×D811yescharacteristic
199C2×C4×C73C2×SD1611yescharacteristic
200C2×C4×C73#13511yescharacteristic
201C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
202C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
203C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
204C2×C4×C73#13711yescharacteristic
205C2×Dic73—81no—
206C2×Dic73—81no—
207C2×Dic73—81no—
208#10596831—42no—
209#10596831—42no—
210#10596831—81no—
211C22×C73—42no—
212C22×C73—42no—
213C22×C73—42no—
214C22×C73—81no—
215C22×C73—81no—
216C22×C73—81no—
217C22×C73—81no—
218C22×C73—82no—
219C22×C73—82no—
220C22×C73—82no—
221C22×C73—82no—
222C22×C73—82no—
223C22×C73—82no—
224C22×C73—82no—
225C22×C73#39411yescharacteristic
226C22×C73#44711yescharacteristic
227C22×C73#44711yescharacteristic
228C22×C73#45011yescharacteristic
229C22×C73#45211yescharacteristic
230C22×C73#45311yescharacteristic
231C22×C73#45511yescharacteristic
232C4×C73—21no—
233C4×C73—21no—
234C4×C73—42no—
235C4×C73—42no—
236C4×C73—81no—
237C4×C73#45911yescharacteristic
238C4×C73#46011yescharacteristic
239C4×C73#46611yescharacteristic
240C4×C73#46811yescharacteristic
241C4×C73#48011yescharacteristic
242C4×C73#48211yescharacteristic
243C4×C73#48611yescharacteristic
244C4×C73#49011yescharacteristic
245Dic73—161no—
246#10408—731nomaximal, Sylow
247C2×C73—82no—
248C2×C73—162no—
249C2×C73#161911yescharacteristic
250C2×C73#163111yescharacteristic
251C2×C73#163911yescharacteristic
252C2×C73#164111yescharacteristic
253C2×C73#203311yescharacteristic
254C2×C73#203711yescharacteristic
255C2×C73#204111yescharacteristic
256#2541D7311yescharacteristic
257#1957—731no—
258#1947—731no—
259#1802—731no—
260C2×#331—731no—
261C2×D4⋊C8—731no—
262#1279—731no—
263C73#1040811yescharacteristic, complemented, Sylow
264C2×#127—21no—
265C2×#127C2×D7311yescharacteristic
266C2×#121—21no—
267C2×#121—21no—
268C2×C4×D4C2×D7311yescharacteristic
269C2×#107—1461no—
270C2×C4⋊C8—731no—
271C2×C4⋊C8—731no—
272C2×#102—1461no—
273C2×#102—1462no—
274C2×C4×C8—731no—
275#392—22no—
276#390—21no—
277#384—731no—
278C4×C4⋊C4C2×D7311yescharacteristic
279#340—1461no—
280#340—1461no—
281#331—1462no—
282D4⋊C8—1462no—
283C22×D4—22no—
284C22×D4—22no—
285C22×D4—21no—
286C22×D4—21no—
287C22×D4C22⋊D7311yescharacteristic
288C22×D4C22⋊D7312yes—
289C23×C4—21no—
290C22×C8—1461no—
291C22×C8—1461no—
292C22×C8—1461no—
293#127—21no—
294#127—21no—
295#127—21no—
296#127—21no—
297#127—42no—
298#127—42no—
299#121—42no—
300#121—42no—
301#121—42no—
302#121—42no—
303C4×D4—21no—
304C4×D4—21no—
305C4×D4—21no—
306C4×D4—21no—
307C2×C4⋊C4—1461no—
308C2×C4⋊C4—1461no—
309C2×C4⋊C4—1461no—
310C2×C4⋊C4C22⋊D7311yescharacteristic
311C2×C4⋊C4C22⋊D7311yescharacteristic
312C2×C4⋊C4C22⋊D7311yescharacteristic
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—21no—
316C2×C42—21no—
317C2×C42C22×D7311yescharacteristic
318#107—2922no—
319C4⋊C8—732no—
320C4⋊C8—1461no—
321C4⋊C8—1461no—
322C4⋊C8—1461no—
323#102—2922no—
324#102—2924no—
325C4×C8—1461no—
326C4×C8—1461no—
327#95—21no—
328C24—21no—
329C24—22no—
330C2×D4—21no—
331C2×D4—22no—
332C2×D4—21no—
333C2×D4—22no—
334C2×D4—21no—
335C2×D4—21no—
336C2×D4—21no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4—44no—
340C2×D4—44no—
341C2×D4—44no—
342C2×D4—44no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—41no—
349C2×D4—41no—
350C2×D4—41no—
351C2×D4—41no—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—82no—
356C2×D4—82no—
357C2×D4#1192401312yes—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—2921no—
368C22×C4C2×C22⋊D7311yescharacteristic
369C22×C4C2×C22⋊D7311yescharacteristic
370C22×C4C2×C22⋊D7311yescharacteristic
371C2×C8—2921no—
372C2×C8—2921no—
373C2×C8—2921no—
374C2×C8—2921no—
375C2×C8—2921no—
376C2×C8—2921no—
377C2×C8—2921no—
378C2×C8—1462no—
379C2×C8—1462no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—2922no—
385C4⋊C4—2921no—
386C4⋊C4—2921no—
387C4⋊C4—2921no—
388C4⋊C4—2921no—
389C4⋊C4#1192401312yes—
390C4⋊C4#1192401612yes—
391C22⋊C4—44no—
392C22⋊C4—44no—
393C22⋊C4—41no—
394C22⋊C4—41no—
395C42—41no—
396C42—41no—
397C42D4×D7311yescharacteristic
398C42#1192402711yescharacteristic
399C42#1192402711yescharacteristic
400C42#1192402711yescharacteristic
401C23—22no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23—41no—
408C23—42no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23#1455656011yescharacteristic
416D4—22no—
417D4—22no—
418D4—44no—
419D4—41no—
420D4—41no—
421D4—82no—
422D4—88no—
423D4—88no—
424D4—81no—
425D4—81no—
426D4—81no—
427D4—81no—
428D4—81no—
429D4—81no—
430D4—81no—
431D4—81no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—5841no—
448C2×C4—5841no—
449C2×C4—5841no—
450C2×C4#1455656211yescharacteristic
451C2×C4#1455656311yescharacteristic
452C2×C4#1455656311yescharacteristic
453C2×C4#1455657211yescharacteristic
454C2×C4#1455657211yescharacteristic
455C2×C4#1455657811yescharacteristic
456C2×C4C2×#1192401311yescharacteristic
457C2×C4C2×#1192401311yescharacteristic
458C2×C4C2×#1192401611yescharacteristic
459C2×C4#1455661111yescharacteristic
460C2×C4#1455661711yescharacteristic
461C2×C4#1455661711yescharacteristic
462C2×C4#1455661711yescharacteristic
463C2×C4#1455662111yescharacteristic
464C8—2922no—
465C8—2922no—
466C8—5841no—
467C22—42no—
468C22—42no—
469C22—42no—
470C22—81no—
471C22—81no—
472C22—81no—
473C22—81no—
474C22—82no—
475C22—82no—
476C22—82no—
477C22—82no—
478C22—82no—
479C22—82no—
480C22—82no—
481C22#1798850111yescharacteristic
482C22#1798872111yescharacteristic
483C22#1798872111yescharacteristic
484C22#1798873811yescharacteristic
485C22#1798874911yescharacteristic
486C22#1798875511yescharacteristic
487C22#1798876511yescharacteristic
488C4—21no—
489C4—21no—
490C4—42no—
491C4—42no—
492C4—81no—
493C4—11681no—
494C4#1798877911yescharacteristic
495C4#1798878411yescharacteristic
496C4#1798882111yescharacteristic
497C4#1798883111yescharacteristic
498C4#1798889811yescharacteristic
499C4#1798890811yescharacteristic
500C4#1798893311yescharacteristic
501C4#1798895111yescharacteristic
502C2—82no—
503C2—162no—
504C2#2260898511yescharacteristic
505C2#2260905811yescharacteristic
506C2#2260911011yescharacteristic
507C2#2260912411yescharacteristic
508C2#2261111011yescharacteristic
509C2#2261113811yescharacteristic
510C2#2261116211yescharacteristic
subgroupG/H
centerC23#14556560
commutator, cosocleC2×C42×C73C23
FrattiniC2×C42C22×D73
Fitting, prime radicalC73×#2541C2
socleC23×C73#120
hypercenter#2541D73
hypocenterC73#10408
LCSG ▷ C2×C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2541

Complements

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

NH ≅ G/Ntype
1. #22610738C2semidirect
2. #22610689C2semidirect
3. #22609931C2semidirect
4. C2×#17988331C2semidirect
5. C2×D4⋊#10596831C2semidirect
6. #22607772C2semidirect
37. C4⋊#10596831D4semidirect
263. C73#10408semidirect

Maximal subgroups

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

subgroupindex
#226107382
#226106892
#226099312
C2×#179883312
C2×D4⋊#105968312
#226077722
C73×#25412
#1040873