Finite Group Databasebeta

← #33071374

Subgroups — #33071374

Order 18688.

674 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620853C211yesmaximal, characteristic, complemented
2#22619746C211yesmaximal, characteristic, complemented
3#22619042C211yesmaximal, characteristic, complemented
4#22609363C212yesmaximal, complemented
5#22609356C212yesmaximal, complemented
6#22609277C212yesmaximal, complemented
7#22609270C212yesmaximal, complemented
8#22608701C211yesmaximal, characteristic, complemented
9#22608608C211yesmaximal, characteristic, complemented
10C2×#17988332C211yesmaximal, characteristic, complemented
11C73×#3035C211yesmaximal, characteristic
12C2×#14556591C2211yescharacteristic
13#17988959C2212yes—
14#17988939C2212yes—
15#17988882—21no—
16#17988877—21no—
17#17988872C2212yes—
18#17988867C2212yes—
19#17988792—21no—
20#17988792—21no—
21#17988725—22no—
22Q8×#10596831C2212yes—
23D4×#10596831—22no—
24D4×#10596831C2212yes—
25#17988640—21no—
26#17988640C2211yescharacteristic
27#17988630—21no—
28#17988611C2211yescharacteristic
29C2×C4⋊#10596831—21no—
30C2×C4⋊#10596831C2211yescharacteristic
31#17988588—21no—
32#17988337C2212yes—
33#17988337C2212yes—
34#17988333C2212yes—
35#17988332C2212yes—
36#17988332C2212yes—
37#17988331C2212yes—
38C2×#135×C73—21no—
39#532×C73C2211yescharacteristic
40C4×#41×C73C2211yescharacteristic
41#464×C73—21no—
42#463×C73—21no—
43#462×C73C2211yescharacteristic
44#461×C73C2211yescharacteristic
45#460×C73C2211yescharacteristic
46#459×C73C2211yescharacteristic
47C2×C4⋊C8×C73C2211yescharacteristic
48#416×C73—21no—
49C22×Dic146—21no—
50C2×#11924016—22no—
51C2×#11924016—22no—
52C2×#11924012—21no—
53C22×#10596831—22no—
54C22×#10596831—21no—
55#14556591—21no—
56#14556591—21no—
57#14556591—21no—
58#14556591C2312yes—
59C2×C4⋊Dic73—21no—
60C2×C4⋊Dic73—21no—
61#14556524—22no—
62#14556522—22no—
63C4⋊#10596831—22no—
64C4⋊#10596831—21no—
65C4⋊#10596831—21no—
66C4⋊#10596831D411yescharacteristic
67C4⋊#10596831D411yescharacteristic
68C4⋊#10596831C2312yes—
69C4⋊#10596831C2312yes—
70#14556515—22no—
71#14556514—22no—
72#14556514—22no—
73#14556512—22no—
74C22⋊#10596831—22no—
75C22⋊#10596831D412yes—
76C4×#10596831—22no—
77C4×#10596831C2312yes—
78C2×#41×C73—21no—
79C2×#41×C73D411yescharacteristic
80#135×C73—44no—
81C2×Q16×C73—21no—
82C2×SD16×C73—21no—
83C2×SD16×C73—21no—
84C2×D8×C73—21no—
85C22×C8×C73—21no—
86C4⋊Q8×C73C2311yescharacteristic
87#127×C73C2311yescharacteristic
88#124×C73—21no—
89#121×C73—21no—
90#121×C73—21no—
91C4×Q8×C73D411yescharacteristic
92C4×Q8×C73C2311yescharacteristic
93C4×D4×C73—21no—
94C4×D4×C73—21no—
95C4×D4×C73—21no—
96C4×D4×C73—21no—
97C4×D4×C73D411yescharacteristic
98C4×D4×C73C2311yescharacteristic
99#117×C73—21no—
100#117×C73D411yescharacteristic
101C2×C42×C73—21no—
102C2×C42×C73C2311yescharacteristic
103C4⋊C8×C73—21no—
104C4⋊C8×C73C2311yescharacteristic
105C4⋊C8×C73C2311yescharacteristic
106#103×C73—21no—
107#103×C73—21no—
108#102×C73—21no—
109#102×C73—21no—
110C2×Dic146—22no—
111C2×Dic146—41no—
112C2×Dic146—81no—
113C2×Dic146—81no—
114C22×Dic73—41no—
115#11924016—82no—
116#11924016—82no—
117#11924012—42no—
118C2×#10596831—22no—
119C2×#10596831—22no—
120C2×#10596831—82no—
121C2×#10596831—81no—
122C2×#10596831C2×D412yes—
123C2×#10596831C2×D412yes—
124C4⋊Dic73—22no—
125C4⋊Dic73—22no—
126C4⋊Dic73—41no—
127C4⋊Dic73—41no—
128#41×C73—22no—
129#41×C73—42no—
130#41×C73—42no—
131#41×C73SD1614yescomplemented
132C2×Q8×C73—21no—
133C2×Q8×C73C2×D411yescharacteristic
134C2×D4×C73—21no—
135C2×D4×C73—21no—
136C2×D4×C73—21no—
137C2×D4×C73—21no—
138C2×D4×C73—41no—
139C2×D4×C73C2×D411yescharacteristic
140C22×C4×C73—21no—
141C22×C4×C73—21no—
142C22×C4×C73—21no—
143C22×C4×C73—41no—
144C22×C4×C73C2×D411yescharacteristic
145C22×C4×C73C2×D411yescharacteristic
146C22×C4×C73#4111yescharacteristic
147Q16×C73—42no—
148SD16×C73—42no—
149SD16×C73—42no—
150D8×C73—42no—
151C2×C8×C73—21no—
152C2×C8×C73—21no—
153C2×C8×C73—42no—
154C4⋊C4×C73—21no—
155C4⋊C4×C73—21no—
156C4⋊C4×C73—21no—
157C4⋊C4×C73—21no—
158C4⋊C4×C73C2×D411yescharacteristic
159C4⋊C4×C73C2×D411yescharacteristic
160C22⋊C4×C73—21no—
161C22⋊C4×C73—21no—
162C22⋊C4×C73—21no—
163C22⋊C4×C73—41no—
164C22⋊C4×C73—41no—
165C42×C73—21no—
166C42×C73—21no—
167C42×C73—21no—
168C42×C73C2×D411yescharacteristic
169C42×C73C2×D411yescharacteristic
170C42×C73#4111yescharacteristic
171C42×C73C2411yescharacteristic
172C23×C73—21no—
173C23×C73—41no—
174C23×C73#12111yescharacteristic
175Q8×C73—21no—
176Q8×C73—41no—
177Q8×C73—41no—
178Q8×C73C2×SD1612yes—
179D4×C73—24no—
180D4×C73—21no—
181D4×C73—41no—
182D4×C73—41no—
183D4×C73—41no—
184D4×C73—41no—
185D4×C73—81no—
186D4×C73—82no—
187D4×C73C2×SD1612yes—
188C2×C4×C73—21no—
189C2×C4×C73—21no—
190C2×C4×C73—21no—
191C2×C4×C73—24no—
192C2×C4×C73—21no—
193C2×C4×C73—21no—
194C2×C4×C73—41no—
195C2×C4×C73—41no—
196C2×C4×C73—41no—
197C2×C4×C73—41no—
198C2×C4×C73—81no—
199C2×C4×C73—82no—
200C2×C4×C73#12111yescharacteristic
201C2×C4×C73#12111yescharacteristic
202C2×C4×C73#12111yescharacteristic
203C2×C4×C73C2×SD1612yes—
204C2×C4×C73#13512yes—
205C2×C4×C73C22×D411yescharacteristic
206C2×C4×C73C22×D411yescharacteristic
207C2×C4×C73C2×#4111yescharacteristic
208C2×Dic73—42no—
209C2×Dic73—82no—
210Dic146—42no—
211Dic146—42no—
212Dic146—82no—
213C8×C73—41no—
214C8×C73—41no—
215#10596831—22no—
216#10596831—22no—
217#10596831—42no—
218#10596831—42no—
219C22×C73—21no—
220C22×C73—21no—
221C22×C73—22no—
222C22×C73—41no—
223C22×C73—41no—
224C22×C73—81no—
225C22×C73—81no—
226C22×C73—81no—
227C22×C73#46512yes—
228C22×C73C2×#12111yescharacteristic
229C4×C73—21no—
230C4×C73—22no—
231C4×C73—21no—
232C4×C73—21no—
233C4×C73—41no—
234C4×C73—41no—
235C4×C73—81no—
236C4×C73#46512yes—
237C4×C73C22×SD1611yescharacteristic
238C4×C73C2×#13511yescharacteristic
239C4×C73#57611yescharacteristic
240C4×C73#57711yescharacteristic
241Dic73—82no—
242#19408—731nomaximal, Sylow
243C2×C73—22no—
244C2×C73—42no—
245C2×C73—161no—
246C2×C73#304511yescharacteristic
247C2×C73C2×#46511yescharacteristic
248C2×C73#306911yescharacteristic
249#3206—731no—
250#3079—731no—
251#3035D7311yescharacteristic
252#2988—731no—
253#1680—732no—
254#1679—732no—
255#1666—732no—
256#1665—732no—
257#1570—731no—
258#1547—731no—
259C2×#332—731no—
260C73#1940811yescharacteristic, complemented, Sylow
261C2×#135—21no—
262#532C2×D7311yescharacteristic
263C2×C4⋊Q8—731no—
264C4×#41C2×D7311yescharacteristic
265#492—732no—
266#487—732no—
267#477—1461no—
268#476—1461no—
269#475—732no—
270#474—732no—
271#464—21no—
272#463—21no—
273#462C2×D7311yescharacteristic
274#461—1461no—
275#461—1461no—
276#461C2×D7311yescharacteristic
277#460C2×D7311yescharacteristic
278#459C2×D7311yescharacteristic
279#448—1462no—
280C8×Q8—732no—
281C8×D4—732no—
282C8×D4—1462no—
283#431—731no—
284#431—1461no—
285#428—1461no—
286#423—731no—
287C2×C4⋊C8—731no—
288C2×C4⋊C8—1461no—
289C2×C4⋊C8C2×D7311yescharacteristic
290#417—1461no—
291#416—21no—
292#334—732no—
293#334—732no—
294#332—732no—
295#332—732no—
296#332—732no—
297#331—732no—
298C2×#41—21no—
299C2×#41C22⋊D7311yescharacteristic
300C22×Q8—1461no—
301#135—44no—
302C2×Q16—21no—
303C2×SD16—21no—
304C2×SD16—21no—
305C2×SD16—1462no—
306C2×SD16—1462no—
307C2×D8—21no—
308C2×M16—1461no—
309C22×C8—21no—
310C22×C8—1462no—
311C22×C8—1461no—
312C4⋊Q8—732no—
313C4⋊Q8—1461no—
314C4⋊Q8—1461no—
315C4⋊Q8—1461no—
316C4⋊Q8C22×D7311yescharacteristic
317#127C22×D7311yescharacteristic
318#124—21no—
319#121—21no—
320#121—21no—
321C4×Q8C22⋊D7311yescharacteristic
322C4×Q8C22×D7311yescharacteristic
323C4×D4—21no—
324C4×D4—21no—
325C4×D4—21no—
326C4×D4—21no—
327C4×D4C22⋊D7311yescharacteristic
328C4×D4C22×D7311yescharacteristic
329#117—21no—
330#117C22⋊D7311yescharacteristic
331C2×C4⋊C4—1461no—
332C2×C4⋊C4—1461no—
333C2×C42—21no—
334C2×C42C22×D7311yescharacteristic
335#107—1462no—
336#106—1462no—
337C4⋊C8—21no—
338C4⋊C8—731no—
339C4⋊C8—731no—
340C4⋊C8—732no—
341C4⋊C8—732no—
342C4⋊C8—1462no—
343C4⋊C8—1461no—
344C4⋊C8—1461no—
345C4⋊C8C22×D7311yescharacteristic
346C4⋊C8C22×D7311yescharacteristic
347#103—21no—
348#103—21no—
349#103—1462no—
350#103—1462no—
351#103—1462no—
352#102—21no—
353#102—21no—
354#102—1462no—
355C22⋊C8—732no—
356C22⋊C8—1462no—
357C4×C8—732no—
358C4×C8—1462no—
359#41—22no—
360#41—42no—
361#41—42no—
362#41#1192401614yescomplemented
363C2×Q8—21no—
364C2×Q8—2921no—
365C2×Q8—5841no—
366C2×Q8—5841no—
367C2×Q8—1462no—
368C2×Q8C2×C22⋊D7311yescharacteristic
369C2×D4—21no—
370C2×D4—21no—
371C2×D4—21no—
372C2×D4—21no—
373C2×D4—41no—
374C2×D4C2×C22⋊D7311yescharacteristic
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—41no—
379C22×C4—2921no—
380C22×C4D4×D7311yescharacteristic
381C22×C4C2×C22⋊D7311yescharacteristic
382C22×C4#1192402711yescharacteristic
383Q16—42no—
384SD16—42no—
385SD16—42no—
386SD16—5842no—
387SD16—5842no—
388D8—42no—
389M16—2922no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—42no—
393C2×C8—5841no—
394C2×C8—5842no—
395C2×C8—732no—
396C2×C8—732no—
397C2×C8—1462no—
398C2×C8—1462no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—2921no—
404C4⋊C4—2921no—
405C4⋊C4—1462no—
406C4⋊C4—1462no—
407C4⋊C4C2×C22⋊D7311yescharacteristic
408C4⋊C4C2×C22⋊D7311yescharacteristic
409C22⋊C4—21no—
410C22⋊C4—21no—
411C22⋊C4—21no—
412C22⋊C4—41no—
413C22⋊C4—41no—
414C42—21no—
415C42—21no—
416C42—21no—
417C42D4×D7311yescharacteristic
418C42C2×C22⋊D7311yescharacteristic
419C42#1192402711yescharacteristic
420C42C23×D7311yescharacteristic
421C23—21no—
422C23—41no—
423C23#1455656311yescharacteristic
424Q8—21no—
425Q8—41no—
426Q8—41no—
427Q8—2922no—
428Q8—2922no—
429Q8—5842no—
430Q8C2×#1192401612yes—
431D4—24no—
432D4—21no—
433D4—41no—
434D4—41no—
435D4—41no—
436D4—41no—
437D4—81no—
438D4—82no—
439D4C2×#1192401612yes—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—24no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—81no—
451C2×C4—82no—
452C2×C4—2922no—
453C2×C4—5842no—
454C2×C4#1455656311yescharacteristic
455C2×C4#1455656311yescharacteristic
456C2×C4#1455656311yescharacteristic
457C2×C4C2×#1192401612yes—
458C2×C4#1455661112yes—
459C2×C4C2×D4×D7311yescharacteristic
460C2×C4C22×C22⋊D7311yescharacteristic
461C2×C4C2×#1192402711yescharacteristic
462C8—41no—
463C8—41no—
464C8—2922no—
465C8—2922no—
466C8—1462no—
467C8—1462no—
468C22—21no—
469C22—21no—
470C22—22no—
471C22—41no—
472C22—41no—
473C22—81no—
474C22—81no—
475C22—81no—
476C22#1798881412yes—
477C22C2×#1455656311yescharacteristic
478C4—21no—
479C4—21no—
480C4—21no—
481C4—22no—
482C4—41no—
483C4—41no—
484C4—81no—
485C4—5842no—
486C4#1798881412yes—
487C4C22×#1192401611yescharacteristic
488C4C2×#1455661111yescharacteristic
489C4#1798938911yescharacteristic
490C4#1798939411yescharacteristic
491C2—22no—
492C2—42no—
493C2—161no—
494C2#2261946811yescharacteristic
495C2C2×#1798881411yescharacteristic
496C2#2261967011yescharacteristic
subgroupG/H
centerC22C2×#14556563
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
Fitting, prime radicalC73×#3035C2
socleC22×C73C2×#121
hypercenter#3035D73
hypocenterC73#19408
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3035

Complements

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

NH ≅ G/Ntype
1. #22620853C2semidirect
2. #22619746C2semidirect
3. #22619042C2semidirect
4. #22609363C2semidirect
5. #22609356C2semidirect
6. #22609277C2semidirect
7. #22609270C2semidirect
8. #22608701C2semidirect
9. #22608608C2semidirect
10. C2×#17988332C2semidirect
131. #41×C73SD16semidirect
260. C73#19408semidirect
362. #41#11924016semidirect

Maximal subgroups

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

subgroupindex
#226208532
#226197462
#226190422
#226093632
#226093562
#226092772
#226092702
#226087012
#226086082
C2×#179883322
C73×#30352
#1940873