Finite Group Databasebeta

← #33055780

Subgroups — #33055780

Order 18688.

648 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22618897C211yesmaximal, characteristic
2C4×#14556622C211yesmaximal, characteristic
3#22608838C212yesmaximal, complemented
4#22608831C212yesmaximal, complemented
5#22608826C212yesmaximal, complemented
6#22608821C212yesmaximal, complemented
7#22608681C211yesmaximal, characteristic, complemented
8#22608326C211yesmaximal, characteristic, complemented
9#22608316C211yesmaximal, characteristic
10C8×#11924012C211yesmaximal, characteristic, complemented
11C2×#446×C73C211yesmaximal, characteristic, complemented
12C2×#14556622C411yescharacteristic
13C4×#11924028C2211yescharacteristic
14#17988717C2211yescharacteristic
15#17988717C2211yescharacteristic
16#17988692C412yes—
17#17988685C412yes—
18C4×#11924018C2212yes—
19C4×#11924015C2212yes—
20#17988666C2211yescharacteristic
21#17988666C2211yescharacteristic
22#17988658—21no—
23#17988658—21no—
24#17988640C2211yescharacteristic
25#17988640C2211yescharacteristic
26#17988638—21no—
27#17988627C411yescharacteristic
28#17988596C411yescharacteristic
29#17988586C411yescharacteristic
30C4×#11924012C2211yescharacteristic
31C4×#11924011—21no—
32#17988339C2212yes—
33#17988337C2212yes—
34Q8⋊#10596831C2212yes—
35Q8⋊#10596831C2212yes—
36#17988317C2212yes—
37#17988314C2212yes—
38#17988309C2212yes—
39C8×#10596831C2212yes—
40C2×C4×Q8×C73C2211yescharacteristic
41#446×C73—21no—
42#446×C73—21no—
43#446×C73C2212yes—
44#446×C73C2212yes—
45C2×C4⋊C8×C73—21no—
46C2×C4⋊C8×C73C2211yescharacteristic
47C2×#97×C73—21no—
48C2×C4×C8×C73C2211yescharacteristic
49C2×#11924028C2×C411yescharacteristic
50#14556622—24no—
51C2×#11924018C2×C412yes—
52C2×#11924015C2×C412yes—
53C2×#11924012C2×C411yescharacteristic
54C2×#11924012C2×C411yescharacteristic
55C2×#11924011—21no—
56C2×#11924011—21no—
57C4×Dic146C2×C411yescharacteristic
58C4×Dic146C2311yescharacteristic
59C4×D292C2×C411yescharacteristic
60C4×D292C2311yescharacteristic
61C4×C22⋊D73—21no—
62C4×C22⋊D73—21no—
63#14556550C2×C411yescharacteristic
64#14556548—21no—
65C42×D73—21no—
66#14556524C2×C412yes—
67#14556522C2×C412yes—
68C4⋊#10596831—21no—
69C4⋊#10596831C2311yescharacteristic
70C4⋊#10596831C2311yescharacteristic
71#14556520—21no—
72#14556520—21no—
73#14556515C2×C412yes—
74#14556515C2×C412yes—
75#14556514C2×C412yes—
76#14556511C2×C412yes—
77C22⋊#10596831—21no—
78C22⋊#10596831—21no—
79#14556503—21no—
80#14556503—21no—
81#14556502C2×C412yes—
82#14556501—21no—
83#14556501—21no—
84C4×#10596831—21no—
85C4×#10596831C2312yes—
86C4×#10596831C2312yes—
87C22×Q8×C73C2×C411yescharacteristic
88C22×C8×C73—21no—
89C22×C8×C73D411yescharacteristic
90C22×C8×C73D411yescharacteristic
91C4×Q8×C73—21no—
92C4×Q8×C73—21no—
93C4×Q8×C73C2×C412yes—
94C4×Q8×C73C2312yes—
95C2×C4⋊C4×C73—21no—
96C2×C4⋊C4×C73C2×C411yescharacteristic
97C2×C42×C73—21no—
98C2×C42×C73C2311yescharacteristic
99C4⋊C8×C73—22no—
100C4⋊C8×C73—22no—
101C4⋊C8×C73—21no—
102C4⋊C8×C73C2312yes—
103#97×C73—22no—
104#97×C73—22no—
105C4×C8×C73D411yescharacteristic
106C4×C8×C73D411yescharacteristic
107C4×C8×C73C2311yescharacteristic
108C4×C8×C73C2311yescharacteristic
109#11924028—22no—
110#11924028—22no—
111C2×Dic146C22×C411yescharacteristic
112C2×D292C22×C411yescharacteristic
113C2×C22⋊D73—21no—
114C2×C4×D73—21no—
115C2×C4×D73—21no—
116C2×C4×D73—21no—
117#11924018—24no—
118#11924015—24no—
119#11924012—22no—
120#11924012M1614yescomplemented
121#11924011—42no—
122#11924011—42no—
123C2×#10596831—21no—
124C2×#10596831—21no—
125C2×#10596831—22no—
126C2×#10596831C22×C412yes—
127C2×#10596831C22×C412yes—
128C4⋊Dic73C22×C411yescharacteristic
129C4⋊Dic73C22×C411yescharacteristic
130#11924007—21no—
131#11924007—21no—
132C22⋊Dic73—21no—
133#11924005—21no—
134#11924005—21no—
135C4×Dic73—21no—
136C4×Dic73—21no—
137C4×Dic73—21no—
138C2×Q8×C73—21no—
139C2×Q8×C73—22no—
140C2×Q8×C73—22no—
141C2×Q8×C73C22×C412yes—
142C22×C4×C73—21no—
143C22×C4×C73—21no—
144C22×C4×C73C22×C411yescharacteristic
145C22×C4×C73C2×D411yescharacteristic
146C22×C4×C73#4111yescharacteristic
147C2×C8×C73—22no—
148C2×C8×C73—22no—
149C2×C8×C73—41no—
150C2×C8×C73—42no—
151C2×C8×C73C2×D411yescharacteristic
152C2×C8×C73C2×D411yescharacteristic
153C2×C8×C73C2×D411yescharacteristic
154C2×C8×C73C2×D411yescharacteristic
155C4⋊C4×C73—22no—
156C4⋊C4×C73—21no—
157C4⋊C4×C73—22no—
158C4⋊C4×C73C22×C412yes—
159C42×C73—22no—
160C42×C73—22no—
161C42×C73C22×C411yescharacteristic
162C42×C73C2×D411yescharacteristic
163C42×C73#4111yescharacteristic
164C42×C73C2411yescharacteristic
165C23×C73C4×D411yescharacteristic
166C22×D73—21no—
167Q8×C73—22no—
168Q8×C73—22no—
169Q8×C73—42no—
170C2×C4×C73—21no—
171C2×C4×C73—21no—
172C2×C4×C73—22no—
173C2×C4×C73—22no—
174C2×C4×C73—41no—
175C2×C4×C73—42no—
176C2×C4×C73—42no—
177C2×C4×C73C4×D411yescharacteristic
178C2×C4×C73C4×D411yescharacteristic
179C2×C4×C73C4×D411yescharacteristic
180C2×C4×C73C2×M1612yes—
181C2×C4×C73#13112yes—
182C2×C4×C73C23×C411yescharacteristic
183C2×C4×C73C22×D411yescharacteristic
184C2×C4×C73C2×#4111yescharacteristic
185C22⋊D73—42no—
186C2×Dic73—21no—
187C2×Dic73—21no—
188C2×Dic73—21no—
189D292—21no—
190D292—21no—
191C4×D73—42no—
192C4×D73—41no—
193C4×D73—41no—
194C4×D73—42no—
195Dic146—21no—
196Dic146—21no—
197C8×C73—21no—
198C8×C73—21no—
199C8×C73—21no—
200C8×C73—42no—
201C8×C73#13712yescomplemented
202#10596831—22no—
203#10596831—42no—
204#10596831—41no—
205#10596831—41no—
206#10596831C2×M1614yes—
207C22×C73—21no—
208C22×C73—21no—
209C22×C73#43612yes—
210C22×C73C2×C4×D411yescharacteristic
211C2×D73—41no—
212C2×D73—41no—
213C2×D73—41no—
214C4×C73—21no—
215C4×C73—21no—
216C4×C73—21no—
217C4×C73—42no—
218C4×C73—42no—
219C4×C73#43612yes—
220C4×C73C22×M1611yescharacteristic
221C4×C73C2×#13111yescharacteristic
222C4×C73C2×#13711yescharacteristic
223C4×C73#57511yescharacteristic
224Dic73—41no—
225Dic73—41no—
226Dic73—41no—
227#17892—731nomaximal, Sylow
228C2×C73—22no—
229C2×C73C2×#43611yescharacteristic
230C2×C73C4×#13711yescharacteristic
231C2×C73#295511yescharacteristic
232D73—81no—
233#2965—731no—
234C2×#446D7311yescharacteristic, complemented
235C4×#137—731no—
236#1593—732no—
237#1592—732no—
238#1591—732no—
239#1590—732no—
240#1565—731no—
241#1479—731no—
242#1477—731no—
243C8×M16—731no—
244C73#1789211yescharacteristic, complemented, Sylow
245C2×#137—731no—
246C4×#41—731no—
247C2×C4×Q8C2×D7311yescharacteristic
248#446—21no—
249#446—21no—
250#446—731no—
251#446—731no—
252#446C2×D7312yes—
253#446C2×D7312yes—
254#441—732no—
255#440—732no—
256C4×Q16—732no—
257C4×SD16—732no—
258#436—731no—
259#436—731no—
260#435—1461no—
261#435—1461no—
262#431—731no—
263#431—731no—
264#431—1461no—
265#428—731no—
266C2×C4⋊C8—21no—
267C2×C4⋊C8C2×D7311yescharacteristic
268#419—731no—
269#417—731no—
270C4×M16—731no—
271C4×M16—1461no—
272C2×#97—21no—
273C2×C4×C8C2×D7311yescharacteristic
274#335—732no—
275#334—732no—
276Q8⋊C8—732no—
277Q8⋊C8—732no—
278Q8⋊C8—732no—
279D4⋊C8—732no—
280#322—732no—
281C82—732no—
282C2×#41—731no—
283C22×Q8C4×D7311yescharacteristic
284#137—1464no—
285C2×Q16—732no—
286C2×SD16—732no—
287C2×M16—731no—
288C2×M16—731no—
289C2×M16—1461no—
290C2×M16—1461no—
291C22×C8—21no—
292C22×C8C22⋊D7311yescharacteristic
293C22×C8C22⋊D7311yescharacteristic
294C4×Q8—21no—
295C4×Q8—21no—
296C4×Q8—731no—
297C4×Q8—731no—
298C4×Q8C4×D7312yes—
299C4×Q8C22×D7312yes—
300C4×D4—731no—
301C4×D4—731no—
302C4×D4—1461no—
303C4×D4—1461no—
304#117—731no—
305#117—1461no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4C4×D7311yescharacteristic
308C2×C42—21no—
309C2×C42—1461no—
310C2×C42C22×D7311yescharacteristic
311#107—732no—
312#106—732no—
313C4⋊C8—22no—
314C4⋊C8—22no—
315C4⋊C8—21no—
316C4⋊C8—731no—
317C4⋊C8—731no—
318C4⋊C8—1461no—
319C4⋊C8—1461no—
320C4⋊C8—1461no—
321C4⋊C8C22×D7312yes—
322#103—732no—
323#103—732no—
324#103—732no—
325#102—732no—
326C22⋊C8—1461no—
327C22⋊C8—1461no—
328C22⋊C8—1461no—
329C22⋊C8—1461no—
330#97—22no—
331#97—22no—
332#97—732no—
333#97—1461no—
334#97—1461no—
335C4×C8—732no—
336C4×C8—732no—
337C4×C8—1461no—
338C4×C8C22⋊D7311yescharacteristic
339C4×C8C22⋊D7311yescharacteristic
340C4×C8C22×D7311yescharacteristic
341C4×C8C22×D7311yescharacteristic
342#41—1462no—
343#41—1462no—
344C2×Q8—21no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—731no—
348C2×Q8C2×C4×D7312yes—
349C2×D4—731no—
350C2×D4—1461no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—1461no—
354C22×C4—1461no—
355C22×C4—1461no—
356C22×C4C2×C4×D7311yescharacteristic
357C22×C4C2×C22⋊D7311yescharacteristic
358C22×C4#1192402811yescharacteristic
359Q16—1464no—
360SD16—1464no—
361M16—2922no—
362M16—2922no—
363M16—734no—
364M16—1462no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—41no—
368C2×C8—42no—
369C2×C8—732no—
370C2×C8—732no—
371C2×C8—1461no—
372C2×C8—1461no—
373C2×C8—1462no—
374C2×C8C2×C22⋊D7311yescharacteristic
375C2×C8C2×C22⋊D7311yescharacteristic
376C2×C8C2×C22⋊D7311yescharacteristic
377C2×C8C2×C22⋊D7311yescharacteristic
378C4⋊C4—22no—
379C4⋊C4—21no—
380C4⋊C4—22no—
381C4⋊C4—731no—
382C4⋊C4—731no—
383C4⋊C4—1461no—
384C4⋊C4—1461no—
385C4⋊C4C2×C4×D7312yes—
386C22⋊C4—1461no—
387C22⋊C4—1461no—
388C22⋊C4—1461no—
389C42—22no—
390C42—22no—
391C42—1461no—
392C42—1461no—
393C42—1461no—
394C42C2×C4×D7311yescharacteristic
395C42C2×C22⋊D7311yescharacteristic
396C42#1192402811yescharacteristic
397C42C23×D7311yescharacteristic
398C23—1461no—
399C23C4×C22⋊D7311yescharacteristic
400Q8—22no—
401Q8—22no—
402Q8—42no—
403Q8—1461no—
404Q8—1461no—
405D4—2922no—
406D4—1461no—
407D4—1461no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—2922no—
416C2×C4—2921no—
417C2×C4—2921no—
418C2×C4—2922no—
419C2×C4—1461no—
420C2×C4—1461no—
421C2×C4—1461no—
422C2×C4C4×C22⋊D7311yescharacteristic
423C2×C4C4×C22⋊D7311yescharacteristic
424C2×C4C4×C22⋊D7311yescharacteristic
425C2×C4M16×D7312yes—
426C2×C4#1455659912yes—
427C2×C4C22×C4×D7311yescharacteristic
428C2×C4C22×C22⋊D7311yescharacteristic
429C2×C4C2×#1192402811yescharacteristic
430C8—21no—
431C8—21no—
432C8—21no—
433C8—42no—
434C8—2922no—
435C8—2921no—
436C8—2921no—
437C8—734no—
438C8—1462no—
439C8#1455662212yescomplemented
440C22—21no—
441C22—21no—
442C22—2921no—
443C22—2921no—
444C22—2921no—
445C22#1798866312yes—
446C22C2×C4×C22⋊D7311yescharacteristic
447C4—21no—
448C4—21no—
449C4—21no—
450C4—42no—
451C4—42no—
452C4—2921no—
453C4—2921no—
454C4—2921no—
455C4#1798866312yes—
456C4C2×M16×D7311yescharacteristic
457C4C2×#1455659911yescharacteristic
458C4C2×#1455662211yescharacteristic
459C4#1798938511yescharacteristic
460C2—22no—
461C2—5841no—
462C2C2×#1798866311yescharacteristic
463C2C4×#1455662211yescharacteristic
464C2#2261883211yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D73
commutatorC2×C4×C73C23×C4
FrattiniC42C23×D73
FittingC2×#446×C73C2
socleC22×C73C2×C4×D4
prime radicalC2×#11924028C2×C4
cosocleC42×C73C24
hypercenterC2×#446D73
hypocenterC73#17892
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ C2×#446

Complements

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

NH ≅ G/Ntype
3. #22608838C2semidirect
4. #22608831C2semidirect
5. #22608826C2semidirect
6. #22608821C2semidirect
7. #22608681C2semidirect
8. #22608326C2semidirect
10. C8×#11924012C2semidirect
11. C2×#446×C73C2semidirect
120. #11924012M16semidirect
201. C8×C73#137semidirect
234. C2×#446D73semidirect
244. C73#17892semidirect
439. C8#14556622semidirect

Maximal subgroups

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

subgroupindex
#226188972
C4×#145566222
#226088382
#226088312
#226088262
#226088212
#226086812
#226083262
#226083162
C8×#119240122
C2×#446×C732
#1789273