Finite Group Databasebeta

← #33071470

Subgroups — #33071470

Order 18688.

694 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22621049C211yesmaximal, characteristic, complemented
2#22619418C211yesmaximal, characteristic, complemented
3#22619042C211yesmaximal, characteristic, complemented
4#22609373C212yesmaximal, complemented
5#22609366C212yesmaximal, complemented
6#22609323C212yesmaximal, complemented
7#22609318C212yesmaximal, complemented
8#22608705C211yesmaximal, characteristic, complemented
9#22608608C211yesmaximal, characteristic, complemented
10C2×#17988334C211yesmaximal, characteristic, complemented
11C73×#3035C211yesmaximal, characteristic
12C2×#14556621—21no—
13C2×#14556591C2211yescharacteristic
14#17988990C2212yes—
15#17988933C2212yes—
16#17988898—22no—
17#17988867—21no—
18#17988867—21no—
19#17988797—21no—
20#17988797C2212yes—
21#17988792—21no—
22#17988792C2212yes—
23Q8×#10596831C2212yes—
24D4×#10596831—22no—
25D4×#10596831C2212yes—
26#17988640—21no—
27#17988640C2211yescharacteristic
28#17988611C2211yescharacteristic
29C2×C4⋊#10596831—21no—
30C2×C4⋊#10596831C2211yescharacteristic
31#17988595—21no—
32#17988339C2212yes—
33#17988339C2212yes—
34#17988334C2212yes—
35#17988334C2212yes—
36#17988333C2212yes—
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—
50#14556621—44no—
51C2×#11924018—22no—
52C2×#11924016—22no—
53C2×#11924012—21no—
54C22×#10596831—22no—
55C22×#10596831—21no—
56#14556591—21no—
57#14556591—21no—
58#14556591—21no—
59#14556591C2312yes—
60C2×C4⋊Dic73—21no—
61C2×C4⋊Dic73—21no—
62#14556524—22no—
63#14556524—22no—
64C4⋊#10596831—22no—
65C4⋊#10596831—21no—
66C4⋊#10596831—21no—
67C4⋊#10596831D411yescharacteristic
68C4⋊#10596831D411yescharacteristic
69C4⋊#10596831C2312yes—
70C4⋊#10596831C2312yes—
71#14556514—22no—
72#14556514—22no—
73#14556512—22no—
74#14556512—22no—
75C22⋊#10596831—22no—
76C22⋊#10596831D412yes—
77C4×#10596831—22no—
78C4×#10596831C2312yes—
79C2×#41×C73—21no—
80C2×#41×C73D411yescharacteristic
81#135×C73—44no—
82C2×Q16×C73—21no—
83C2×SD16×C73—21no—
84C2×SD16×C73—21no—
85C2×D8×C73—21no—
86C22×C8×C73—21no—
87C4⋊Q8×C73C2311yescharacteristic
88#127×C73C2311yescharacteristic
89#124×C73—21no—
90#121×C73—21no—
91#121×C73—21no—
92C4×Q8×C73D411yescharacteristic
93C4×Q8×C73C2311yescharacteristic
94C4×D4×C73—21no—
95C4×D4×C73—21no—
96C4×D4×C73—21no—
97C4×D4×C73—21no—
98C4×D4×C73D411yescharacteristic
99C4×D4×C73C2311yescharacteristic
100#117×C73—21no—
101#117×C73D411yescharacteristic
102C2×C42×C73—21no—
103C2×C42×C73C2311yescharacteristic
104C4⋊C8×C73—21no—
105C4⋊C8×C73C2311yescharacteristic
106C4⋊C8×C73C2311yescharacteristic
107#103×C73—21no—
108#103×C73—21no—
109#102×C73—21no—
110#102×C73—21no—
111C2×Dic146—22no—
112C2×Dic146—42no—
113C2×Dic146—42no—
114C2×Dic146—41no—
115C22×Dic73—41no—
116#11924018—44no—
117#11924016—44no—
118#11924012—42no—
119C2×#10596831—22no—
120C2×#10596831—22no—
121C2×#10596831—82no—
122C2×#10596831—81no—
123C2×#10596831C2×D412yes—
124C2×#10596831C2×D412yes—
125C4⋊Dic73—22no—
126C4⋊Dic73—22no—
127C4⋊Dic73—41no—
128C4⋊Dic73—41no—
129#41×C73—22no—
130#41×C73—42no—
131#41×C73—42no—
132#41×C73D814yes—
133C2×Q8×C73—21no—
134C2×Q8×C73C2×D411yescharacteristic
135C2×D4×C73—21no—
136C2×D4×C73—21no—
137C2×D4×C73—21no—
138C2×D4×C73—21no—
139C2×D4×C73—41no—
140C2×D4×C73C2×D411yescharacteristic
141C22×C4×C73—21no—
142C22×C4×C73—21no—
143C22×C4×C73—21no—
144C22×C4×C73—41no—
145C22×C4×C73C2×D411yescharacteristic
146C22×C4×C73C2×D411yescharacteristic
147C22×C4×C73#4111yescharacteristic
148Q16×C73—42no—
149SD16×C73—42no—
150SD16×C73—42no—
151D8×C73—42no—
152C2×C8×C73—21no—
153C2×C8×C73—21no—
154C2×C8×C73—42no—
155C4⋊C4×C73—21no—
156C4⋊C4×C73—21no—
157C4⋊C4×C73—21no—
158C4⋊C4×C73—21no—
159C4⋊C4×C73C2×D411yescharacteristic
160C4⋊C4×C73C2×D411yescharacteristic
161C22⋊C4×C73—21no—
162C22⋊C4×C73—21no—
163C22⋊C4×C73—21no—
164C22⋊C4×C73—41no—
165C22⋊C4×C73—41no—
166C42×C73—21no—
167C42×C73—21no—
168C42×C73—21no—
169C42×C73C2×D411yescharacteristic
170C42×C73C2×D411yescharacteristic
171C42×C73#4111yescharacteristic
172C42×C73C2411yescharacteristic
173C23×C73—21no—
174C23×C73—41no—
175C23×C73#12111yescharacteristic
176Q8×C73—21no—
177Q8×C73—41no—
178Q8×C73—41no—
179Q8×C73C2×D812yes—
180D4×C73—24no—
181D4×C73—21no—
182D4×C73—41no—
183D4×C73—41no—
184D4×C73—41no—
185D4×C73—41no—
186D4×C73—81no—
187D4×C73—82no—
188D4×C73C2×D812yes—
189C2×C4×C73—21no—
190C2×C4×C73—21no—
191C2×C4×C73—21no—
192C2×C4×C73—24no—
193C2×C4×C73—21no—
194C2×C4×C73—21no—
195C2×C4×C73—41no—
196C2×C4×C73—41no—
197C2×C4×C73—41no—
198C2×C4×C73—41no—
199C2×C4×C73—81no—
200C2×C4×C73—82no—
201C2×C4×C73#12111yescharacteristic
202C2×C4×C73#12111yescharacteristic
203C2×C4×C73#12111yescharacteristic
204C2×C4×C73C2×D812yes—
205C2×C4×C73#13512yes—
206C2×C4×C73C22×D411yescharacteristic
207C2×C4×C73C22×D411yescharacteristic
208C2×C4×C73C2×#4111yescharacteristic
209C2×Dic73—42no—
210C2×Dic73—82no—
211Dic146—42no—
212Dic146—42no—
213Dic146—82no—
214C8×C73—41no—
215C8×C73—41no—
216#10596831—22no—
217#10596831—22no—
218#10596831—42no—
219#10596831—42no—
220C22×C73—21no—
221C22×C73—21no—
222C22×C73—22no—
223C22×C73—41no—
224C22×C73—41no—
225C22×C73—81no—
226C22×C73—81no—
227C22×C73—81no—
228C22×C73#46612yes—
229C22×C73C2×#12111yescharacteristic
230C4×C73—21no—
231C4×C73—22no—
232C4×C73—21no—
233C4×C73—21no—
234C4×C73—41no—
235C4×C73—41no—
236C4×C73—81no—
237C4×C73#46612yes—
238C4×C73C22×D811yescharacteristic
239C4×C73C2×#13511yescharacteristic
240C4×C73#57711yescharacteristic
241C4×C73#57811yescharacteristic
242Dic73—82no—
243#19416—731nomaximal, Sylow
244C2×C73—22no—
245C2×C73—42no—
246C2×C73—161no—
247C2×C73#304611yescharacteristic
248C2×C73C2×#46611yescharacteristic
249C2×C73#306911yescharacteristic
250#3226—731no—
251#3039—731no—
252#3035D7311yescharacteristic
253#2988—731no—
254#1682—732no—
255#1681—732no—
256#1674—732no—
257#1673—732no—
258#1571—731no—
259#1547—731no—
260C2×#332—731no—
261C73#1941611yescharacteristic, complemented, Sylow
262C2×#137—1461no—
263C2×#135—21no—
264#532C2×D7311yescharacteristic
265C2×C4⋊Q8—731no—
266C4×#41C2×D7311yescharacteristic
267#500—732no—
268#486—732no—
269#480—1462no—
270#474—1461no—
271#474—1461no—
272#464—21no—
273#463—21no—
274#462—732no—
275#462—1461no—
276#462C2×D7311yescharacteristic
277#461—732no—
278#461—1461no—
279#461C2×D7311yescharacteristic
280#460C2×D7311yescharacteristic
281#459C2×D7311yescharacteristic
282C8×Q8—732no—
283C8×D4—732no—
284C8×D4—1462no—
285#431—731no—
286#431—1461no—
287#423—731no—
288C2×C4⋊C8—731no—
289C2×C4⋊C8—1461no—
290C2×C4⋊C8C2×D7311yescharacteristic
291#419—1461no—
292#416—21no—
293#335—732no—
294#335—732no—
295#332—732no—
296#332—732no—
297#332—732no—
298#331—732no—
299C2×#41—21no—
300C2×#41C22⋊D7311yescharacteristic
301C22×Q8—1461no—
302#137—2924no—
303#135—44no—
304C2×Q16—21no—
305C2×Q16—1462no—
306C2×SD16—21no—
307C2×SD16—21no—
308C2×SD16—1462no—
309C2×D8—21no—
310C2×M16—1461no—
311C22×C8—21no—
312C22×C8—1462no—
313C22×C8—1461no—
314C4⋊Q8—732no—
315C4⋊Q8—1461no—
316C4⋊Q8—1461no—
317C4⋊Q8—1461no—
318C4⋊Q8C22×D7311yescharacteristic
319#127C22×D7311yescharacteristic
320#124—21no—
321#121—21no—
322#121—21no—
323C4×Q8C22⋊D7311yescharacteristic
324C4×Q8C22×D7311yescharacteristic
325C4×D4—21no—
326C4×D4—21no—
327C4×D4—21no—
328C4×D4—21no—
329C4×D4C22⋊D7311yescharacteristic
330C4×D4C22×D7311yescharacteristic
331#117—21no—
332#117C22⋊D7311yescharacteristic
333C2×C4⋊C4—1461no—
334C2×C4⋊C4—1461no—
335C2×C42—21no—
336C2×C42C22×D7311yescharacteristic
337#107—1462no—
338#107—1462no—
339C4⋊C8—21no—
340C4⋊C8—731no—
341C4⋊C8—731no—
342C4⋊C8—732no—
343C4⋊C8—732no—
344C4⋊C8—1462no—
345C4⋊C8—1461no—
346C4⋊C8—1461no—
347C4⋊C8C22×D7311yescharacteristic
348C4⋊C8C22×D7311yescharacteristic
349#103—21no—
350#103—21no—
351#103—1462no—
352#103—1462no—
353#102—21no—
354#102—21no—
355#102—1462no—
356#102—1462no—
357C22⋊C8—732no—
358C22⋊C8—1462no—
359C4×C8—732no—
360C4×C8—1462no—
361#41—22no—
362#41—42no—
363#41—42no—
364#41#1192401314yes—
365C2×Q8—21no—
366C2×Q8—2921no—
367C2×Q8—2922no—
368C2×Q8—2922no—
369C2×Q8—1462no—
370C2×Q8C2×C22⋊D7311yescharacteristic
371C2×D4—21no—
372C2×D4—21no—
373C2×D4—21no—
374C2×D4—21no—
375C2×D4—41no—
376C2×D4C2×C22⋊D7311yescharacteristic
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—41no—
381C22×C4—2921no—
382C22×C4D4×D7311yescharacteristic
383C22×C4C2×C22⋊D7311yescharacteristic
384C22×C4#1192402711yescharacteristic
385Q16—42no—
386Q16—2924no—
387SD16—42no—
388SD16—42no—
389SD16—2924no—
390D8—42no—
391M16—2922no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—42no—
395C2×C8—5841no—
396C2×C8—5842no—
397C2×C8—732no—
398C2×C8—732no—
399C2×C8—1462no—
400C2×C8—1462no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—2921no—
406C4⋊C4—2921no—
407C4⋊C4—1462no—
408C4⋊C4—1462no—
409C4⋊C4C2×C22⋊D7311yescharacteristic
410C4⋊C4C2×C22⋊D7311yescharacteristic
411C22⋊C4—21no—
412C22⋊C4—21no—
413C22⋊C4—21no—
414C22⋊C4—41no—
415C22⋊C4—41no—
416C42—21no—
417C42—21no—
418C42—21no—
419C42D4×D7311yescharacteristic
420C42C2×C22⋊D7311yescharacteristic
421C42#1192402711yescharacteristic
422C42C23×D7311yescharacteristic
423C23—21no—
424C23—41no—
425C23#1455656311yescharacteristic
426Q8—21no—
427Q8—41no—
428Q8—41no—
429Q8—2922no—
430Q8—2922no—
431Q8—5842no—
432Q8C2×#1192401312yes—
433D4—24no—
434D4—21no—
435D4—41no—
436D4—41no—
437D4—41no—
438D4—41no—
439D4—81no—
440D4—82no—
441D4C2×#1192401312yes—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—24no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—81no—
453C2×C4—82no—
454C2×C4—2922no—
455C2×C4—5842no—
456C2×C4#1455656311yescharacteristic
457C2×C4#1455656311yescharacteristic
458C2×C4#1455656311yescharacteristic
459C2×C4C2×#1192401312yes—
460C2×C4#1455661112yes—
461C2×C4C2×D4×D7311yescharacteristic
462C2×C4C22×C22⋊D7311yescharacteristic
463C2×C4C2×#1192402711yescharacteristic
464C8—41no—
465C8—41no—
466C8—2922no—
467C8—2922no—
468C8—1462no—
469C8—1462no—
470C22—21no—
471C22—21no—
472C22—22no—
473C22—41no—
474C22—41no—
475C22—81no—
476C22—81no—
477C22—81no—
478C22#1798882112yes—
479C22C2×#1455656311yescharacteristic
480C4—21no—
481C4—21no—
482C4—21no—
483C4—22no—
484C4—41no—
485C4—41no—
486C4—81no—
487C4—5842no—
488C4#1798882112yes—
489C4C22×#1192401311yescharacteristic
490C4C2×#1455661111yescharacteristic
491C4#1798939411yescharacteristic
492C4#1798940111yescharacteristic
493C2—22no—
494C2—42no—
495C2—161no—
496C2#2261947911yescharacteristic
497C2C2×#1798882111yescharacteristic
498C2#2261967011yescharacteristic
subgroupG/H
centerC22C2×#14556563
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
Fitting, prime radicalC73×#3035C2
socleC22×C73C2×#121
hypercenter#3035D73
hypocenterC73#19416
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. #22621049C2semidirect
2. #22619418C2semidirect
3. #22619042C2semidirect
4. #22609373C2semidirect
5. #22609366C2semidirect
6. #22609323C2semidirect
7. #22609318C2semidirect
8. #22608705C2semidirect
9. #22608608C2semidirect
10. C2×#17988334C2semidirect
261. C73#19416semidirect

Maximal subgroups

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

subgroupindex
#226210492
#226194182
#226190422
#226093732
#226093662
#226093232
#226093182
#226087052
#226086082
C2×#179883342
C73×#30352
#1941673