Finite Group Databasebeta

← #33089211

Subgroups — #33089211

Order 18688.

800 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14556611C211yesmaximal, characteristic
2#22618575C211yesmaximal, characteristic, complemented
3C4×#14556599C211yesmaximal, characteristic, complemented
4#22610582C212yesmaximal, complemented
5#22610533C212yesmaximal, complemented
6#22610255C212yesmaximal
7#22610207C212yesmaximal, complemented
8C4×#14556526C211yesmaximal, characteristic, complemented
9C4×#14556519C211yesmaximal, characteristic, complemented
10#22609677C211yesmaximal, characteristic, complemented
11C4×#104×C73C211yesmaximal, characteristic, complemented
12C2×#14556611C2211yescharacteristic
13C2×#14556599C2212yes—
14C4×#11924028C2211yescharacteristic
15C4×#11924018C411yescharacteristic, complemented
16C4×#11924018C411yescharacteristic, complemented
17C4×#11924015C411yescharacteristic, complemented
18C4×#11924015C411yescharacteristic, complemented
19C4×#11924016C411yescharacteristic, complemented
20C4×#11924016C411yescharacteristic, complemented
21C4×#11924013C411yescharacteristic, complemented
22C4×#11924013C411yescharacteristic, complemented
23#17988648—22no—
24#17988640—21no—
25#17988640—21no—
26C2×#14556526C2212yes—
27#17988623C2211yescharacteristic
28#17988614C2212yes—
29C2×#14556519C2212yes—
30#17988583C2211yescharacteristic
31#17988581C2211yescharacteristic
32#17988554C2212yes—
33#17988545C2212yes—
34#17988544—22no—
35C4×#11924012C2211yescharacteristic
36C4×#11924011—21no—
37C2×C4×#10596831C2211yescharacteristic
38C2×C4×#10596831C2211yescharacteristic
39C4×C8×D73—21no—
40#17988351C2212yes—
41#17988347C2212yes—
42#17988342C2212yes—
43#17988342C2212yes—
44C4×#41×C73C2211yescharacteristic
45C2×#104×C73C2212yes—
46C4×M16×C73C2211yescharacteristic
47C43×C73C2211yescharacteristic
48#339×C73C2212yes—
49C2×#11924028C2311yescharacteristic
50#14556611—24no—
51C2×#11924018C2×C411yescharacteristic
52C2×#11924016C2×C411yescharacteristic
53C2×#11924015C2×C411yescharacteristic
54C2×#11924013C2×C411yescharacteristic
55#14556599—24no—
56#14556599—24no—
57C2×#11924012C2312yes—
58C2×#11924011—22no—
59C22×#10596831C2311yescharacteristic
60C22×#10596831C2311yescharacteristic
61C22×#10596831C2312yes—
62C2×C8×D73—22no—
63C4×Dic146C2×C411yescharacteristic
64C4×Dic146C2×C411yescharacteristic
65C4×D292C2×C411yescharacteristic
66C4×D292C2×C411yescharacteristic
67C4×C22⋊D73—21no—
68C4×C22⋊D73—21no—
69#14556550C2311yescharacteristic
70#14556548—21no—
71C42×D73—21no—
72#14556526—24no—
73#14556524C2×C411yescharacteristic
74#14556524C2×C411yescharacteristic
75#14556522C2×C411yescharacteristic
76#14556522C2×C411yescharacteristic
77C4⋊#10596831—22no—
78C4⋊#10596831—22no—
79#14556519—24no—
80#14556515C2×C411yescharacteristic
81#14556515C2×C411yescharacteristic
82#14556514C2×C411yescharacteristic
83#14556514C2×C411yescharacteristic
84#14556512C2×C411yescharacteristic
85#14556512C2×C411yescharacteristic
86#14556511C2×C411yescharacteristic
87#14556511C2×C411yescharacteristic
88C22⋊#10596831—22no—
89C22⋊#10596831—22no—
90#14556502—21no—
91#14556501—22no—
92C4×#10596831—21no—
93C4×#10596831C2×C411yescharacteristic
94C4×#10596831C2×C411yescharacteristic
95C4×#10596831C2×C411yescharacteristic
96C4×#10596831C2×C411yescharacteristic
97C4×#10596831D411yescharacteristic
98C4×#10596831D411yescharacteristic
99C4×#10596831D412yes—
100C8×Dic73—22no—
101C2×#41×C73C2311yescharacteristic
102C2×M16×C73C2312yes—
103C4×Q8×C73C2×C411yescharacteristic
104C4×Q8×C73C2×C411yescharacteristic
105C4×D4×C73—21no—
106C4×D4×C73—21no—
107C4×D4×C73C2×C411yescharacteristic
108C4×D4×C73C2×C411yescharacteristic
109#117×C73—21no—
110#117×C73C2311yescharacteristic
111C2×C42×C73—21no—
112C2×C42×C73—21no—
113C2×C42×C73—21no—
114C2×C42×C73C2312yes—
115C2×C42×C73C2311yescharacteristic
116#104×C73—24no—
117#97×C73—21no—
118C4×C8×C73—21no—
119#11924028—22no—
120#11924028#4114yes—
121C2×Dic146C22×C411yescharacteristic
122C2×D292C22×C411yescharacteristic
123C2×C22⋊D73—21no—
124C2×C4×D73—21no—
125C2×C4×D73—21no—
126C2×C4×D73—21no—
127#11924018—22no—
128#11924016—22no—
129#11924015—22no—
130#11924013—22no—
131#11924012—22no—
132#11924012—22no—
133#11924011—28no—
134C2×#10596831—22no—
135C2×#10596831—22no—
136C2×#10596831—22no—
137C2×#10596831—22no—
138C2×#10596831—22no—
139C2×#10596831C22×C411yescharacteristic
140C2×#10596831C22×C411yescharacteristic
141C2×#10596831C22×C411yescharacteristic
142C2×#10596831C22×C411yescharacteristic
143C2×#10596831C2×D412yes—
144C2×#10596831C2×D412yes—
145C8×D73—28no—
146C4⋊Dic73C22×C411yescharacteristic
147C4⋊Dic73C22×C411yescharacteristic
148#11924007—21no—
149#11924007—21no—
150C22⋊Dic73—21no—
151#11924005—21no—
152#11924005—21no—
153C4×Dic73—21no—
154C4×Dic73—21no—
155C4×Dic73—21no—
156#41×C73—22no—
157#41×C73—22no—
158C2×Q8×C73C22×C411yescharacteristic
159C2×D4×C73—21no—
160C2×D4×C73C22×C411yescharacteristic
161C22×C4×C73—22no—
162C22×C4×C73—21no—
163C22×C4×C73—21no—
164C22×C4×C73—21no—
165C22×C4×C73#4111yescharacteristic
166C22×C4×C73#4111yescharacteristic
167C22×C4×C73C2411yescharacteristic
168M16×C73—22no—
169M16×C73#4114yes—
170C2×C8×C73—22no—
171C4⋊C4×C73—21no—
172C4⋊C4×C73—21no—
173C4⋊C4×C73C22×C411yescharacteristic
174C4⋊C4×C73C22×C411yescharacteristic
175C22⋊C4×C73—21no—
176C22⋊C4×C73—21no—
177C22⋊C4×C73—21no—
178C42×C73—24no—
179C42×C73—21no—
180C42×C73—21no—
181C42×C73—21no—
182C42×C73—22no—
183C42×C73—22no—
184C42×C73—42no—
185C42×C73C22×C411yescharacteristic
186C42×C73C22×C411yescharacteristic
187C42×C73C2×D411yescharacteristic
188C42×C73C2×D411yescharacteristic
189C23×C73—21no—
190C23×C73C2×#4111yescharacteristic
191C22×D73—21no—
192Q8×C73—21no—
193Q8×C73—21no—
194D4×C73—21no—
195D4×C73—21no—
196D4×C73—42no—
197C2×C4×C73—22no—
198C2×C4×C73—22no—
199C2×C4×C73—24no—
200C2×C4×C73—21no—
201C2×C4×C73—21no—
202C2×C4×C73—21no—
203C2×C4×C73—42no—
204C2×C4×C73—42no—
205C2×C4×C73—42no—
206C2×C4×C73—42no—
207C2×C4×C73—42no—
208C2×C4×C73C4×D411yescharacteristic
209C2×C4×C73C4×D411yescharacteristic
210C2×C4×C73C4×D411yescharacteristic
211C2×C4×C73C4×D411yescharacteristic
212C2×C4×C73C23×C411yescharacteristic
213C2×C4×C73C22×D411yescharacteristic
214C2×C4×C73C2×#4112yes—
215C2×C4×C73C2×#4112yes—
216C22⋊D73—24no—
217C2×Dic73—21no—
218C2×Dic73—21no—
219C2×Dic73—21no—
220D292—21no—
221D292#11712yes—
222C4×D73—24no—
223C4×D73—42no—
224C4×D73—42no—
225Dic146—21no—
226Dic146#11712yes—
227C8×C73—24no—
228#10596831—21no—
229#10596831—21no—
230#10596831—21no—
231#10596831—21no—
232#10596831—22no—
233#10596831—22no—
234#10596831—42no—
235C22×C73—21no—
236C22×C73—21no—
237C22×C73—41no—
238C22×C73—41no—
239C22×C73—41no—
240C22×C73C2×C4×D411yescharacteristic
241C22×C73#52912yes—
242C2×D73—22no—
243C2×D73—41no—
244C2×D73—41no—
245C4×C73—21no—
246C4×C73—21no—
247C4×C73—44no—
248C4×C73—41no—
249C4×C73—41no—
250C4×C73—41no—
251C4×C73#44312yes—
252C4×C73#44412yes—
253C4×C73C2×#11711yescharacteristic
254C4×C73C4×#4111yescharacteristic
255C4×C73#52512yes—
256Dic73—22no—
257Dic73—41no—
258Dic73—41no—
259#21137—731nomaximal, Sylow
260C2×C73—22no—
261C2×C73—81no—
262C2×C73#229011yescharacteristic
263C2×C73C2×#44311yescharacteristic
264C2×C73C2×#44411yescharacteristic
265D73—42no—
266C4×#135—731no—
267#2922—731no—
268C4×#131—731no—
269#1929—732no—
270#1922—732no—
271#1876—732no—
272#1866—732no—
273C4×#108—731no—
274C4×#104—731no—
275C4×#104D7311yescharacteristic, complemented
276#1744—731no—
277C73#2113711yescharacteristic, complemented, Sylow
278C2×#135—731no—
279C2×#131—732no—
280C4×#41—731no—
281C4×#41C2×D7311yescharacteristic
282C4×Q16—731no—
283C4×Q16—731no—
284C4×SD16—731no—
285C4×SD16—731no—
286C4×SD16—731no—
287C4×SD16—731no—
288C4×D8—731no—
289C4×D8—731no—
290#433—1462no—
291#431—1461no—
292#431—1461no—
293C2×#108—732no—
294#427—731no—
295#424—732no—
296C2×#104—732no—
297C2×#104C2×D7312yes—
298#416—731no—
299#416—731no—
300#408—732no—
301#405—732no—
302#405—1462no—
303C4×M16—731no—
304C4×M16—1461no—
305C4×M16C2×D7311yescharacteristic
306C2×C4×C8—731no—
307C2×C4×C8—731no—
308C2×C4×C8—1461no—
309C43C2×D7311yescharacteristic
310#341—732no—
311#339—732no—
312#339C2×D7312yes—
313#336—732no—
314#336—732no—
315C2×#41—731no—
316C2×#41C22×D7311yescharacteristic
317#135—1464no—
318C2×Q16—731no—
319C2×SD16—731no—
320C2×SD16—731no—
321C2×D8—731no—
322#131—1464no—
323#131—1464no—
324C2×M16—732no—
325C2×M16—1462no—
326C2×M16C22×D7312yes—
327C22×C8—732no—
328C22×C8—731no—
329C22×C8—731no—
330C22×C8—1462no—
331C4×Q8—731no—
332C4×Q8—731no—
333C4×Q8C4×D7311yescharacteristic
334C4×Q8C4×D7311yescharacteristic
335C4×D4—21no—
336C4×D4—21no—
337C4×D4—731no—
338C4×D4—731no—
339C4×D4—1461no—
340C4×D4—1461no—
341C4×D4C4×D7311yescharacteristic
342C4×D4C4×D7311yescharacteristic
343#117—21no—
344#117—731no—
345#117—1461no—
346#117C22×D7311yescharacteristic
347C2×C42—21no—
348C2×C42—21no—
349C2×C42—21no—
350C2×C42—1461no—
351C2×C42C22×D7312yes—
352C2×C42C22×D7311yescharacteristic
353#108—1464no—
354#107—731no—
355#107—731no—
356#106—731no—
357#106—731no—
358C4⋊C8—1462no—
359C4⋊C8—1462no—
360#104—24no—
361#104—1464no—
362#103—731no—
363#103—731no—
364#103—731no—
365#103—731no—
366#102—731no—
367#102—731no—
368#102—731no—
369#102—731no—
370C22⋊C8—1462no—
371C22⋊C8—1462no—
372#97—21no—
373#97—1462no—
374#97—1461no—
375C4×C8—21no—
376C4×C8—731no—
377C4×C8—731no—
378C4×C8—731no—
379C4×C8—731no—
380C4×C8—732no—
381C4×C8—731no—
382C4×C8—731no—
383C4×C8—1462no—
384C4×C8—1461no—
385#41—22no—
386#41—22no—
387#41—734no—
388#41—1462no—
389C2×Q8—731no—
390C2×Q8C2×C4×D7311yescharacteristic
391C2×D4—21no—
392C2×D4—731no—
393C2×D4—1461no—
394C2×D4C2×C4×D7311yescharacteristic
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—1461no—
400C22×C4—1461no—
401C22×C4—1461no—
402C22×C4#1192402711yescharacteristic
403C22×C4#1192402711yescharacteristic
404C22×C4C23×D7311yescharacteristic
405Q16—1462no—
406SD16—1462no—
407SD16—1462no—
408D8—1462no—
409M16—22no—
410M16—1468no—
411M16—1462no—
412M16—1462no—
413M16#1192402814yes—
414C2×C8—22no—
415C2×C8—731no—
416C2×C8—731no—
417C2×C8—731no—
418C2×C8—731no—
419C2×C8—732no—
420C2×C8—732no—
421C2×C8—1468no—
422C2×C8—1462no—
423C2×C8—1462no—
424C2×C8—1462no—
425C2×C8—1462no—
426C2×C8—1462no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—731no—
430C4⋊C4—731no—
431C4⋊C4—1461no—
432C4⋊C4—1461no—
433C4⋊C4C2×C4×D7311yescharacteristic
434C4⋊C4C2×C4×D7311yescharacteristic
435C22⋊C4—21no—
436C22⋊C4—21no—
437C22⋊C4—21no—
438C22⋊C4—1461no—
439C22⋊C4—1461no—
440C22⋊C4—1461no—
441C42—24no—
442C42—21no—
443C42—21no—
444C42—21no—
445C42—22no—
446C42—22no—
447C42—42no—
448C42—1461no—
449C42—1461no—
450C42—1461no—
451C42C2×C4×D7311yescharacteristic
452C42C2×C4×D7311yescharacteristic
453C42D4×D7311yescharacteristic
454C42D4×D7311yescharacteristic
455C23—21no—
456C23—1461no—
457C23C2×#1192402711yescharacteristic
458Q8—21no—
459Q8—21no—
460Q8—732no—
461Q8—1461no—
462D4—21no—
463D4—21no—
464D4—42no—
465D4—732no—
466D4—1464no—
467D4—1461no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—2922no—
480C2×C4—2922no—
481C2×C4—1461no—
482C2×C4—1461no—
483C2×C4—1461no—
484C2×C4—1464no—
485C2×C4#1455655111yescharacteristic
486C2×C4#1455655111yescharacteristic
487C2×C4#1455655111yescharacteristic
488C2×C4#1455655111yescharacteristic
489C2×C4C22×C4×D7311yescharacteristic
490C2×C4C2×D4×D7311yescharacteristic
491C2×C4#41×D7312yes—
492C2×C4C2×#1192402812yes—
493C8—24no—
494C8—2922no—
495C8—1461no—
496C8—1461no—
497C8—1461no—
498C8—1461no—
499C8—1462no—
500C8—1462no—
501C22—21no—
502C22—21no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—2921no—
507C22—2921no—
508C22—1462no—
509C22C2×#1455655111yescharacteristic
510C22#1798911912yes—
511C4—21no—
512C4—21no—
513C4—44no—
514C4—41no—
515C4—41no—
516C4—41no—
517C4—2921no—
518C4—2921no—
519C4—1462no—
520C4#1798870112yes—
521C4#1798870612yes—
522C4#117×D7311yescharacteristic
523C4C4×#1192402811yescharacteristic
524C4#1798909612yes—
525C2—22no—
526C2—81no—
527C2—2922no—
528C2#2261237711yescharacteristic
529C2C2×#1798870111yescharacteristic
530C2C2×#1798870611yescharacteristic
subgroupG/H
centerC42D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC4×#104×C73C2
socleC22×C73C2×C4×D4
prime radicalC2×#14556611C22
cosocleC22×C4×C73C24
hypercenterC4×#104D73
hypocenterC73#21137
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C42 ◁ C2×C42 ◁ C4×#104

Complements

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

NH ≅ G/Ntype
2. #22618575C2semidirect
3. C4×#14556599C2semidirect
4. #22610582C2semidirect
5. #22610533C2semidirect
7. #22610207C2semidirect
8. C4×#14556526C2semidirect
9. C4×#14556519C2semidirect
10. #22609677C2semidirect
11. C4×#104×C73C2semidirect
15. C4×#11924018C4semidirect
16. C4×#11924018C4semidirect
17. C4×#11924015C4semidirect
18. C4×#11924015C4semidirect
19. C4×#11924016C4semidirect
20. C4×#11924016C4semidirect
21. C4×#11924013C4semidirect
22. C4×#11924013C4semidirect
275. C4×#104D73semidirect
277. C73#21137semidirect

Maximal subgroups

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

subgroupindex
C4×#145566112
#226185752
C4×#145565992
#226105822
#226105332
#226102552
#226102072
C4×#145565262
C4×#145565192
#226096772
C4×#104×C732
#2113773