Finite Group Databasebeta

← #32982249

Subgroups — #32982249

Order 18688.

724 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22619931C211yesmaximal, characteristic
2#22610759C211yesmaximal, characteristic, complemented
3#22608425C211yesmaximal, characteristic, complemented
4#22608356C211yesmaximal, characteristic, complemented
5#22607834C211yesmaximal, characteristic, complemented
6#22607764C211yesmaximal, characteristic, complemented
7C42×D4×C73C211yesmaximal, characteristic, complemented
8#17989137C2211yescharacteristic
9#17988900—21no—
10#17988867C412yescomplemented
11#17988782C412yescomplemented
12#17988721—21no—
13C4×#11924013—22no—
14#17988640C2211yescharacteristic
15#17988640C2211yescharacteristic
16#17988611C2211yescharacteristic
17C2×#14556519—21no—
18C2×#14556519—21no—
19#17988320—22no—
20D4⋊#10596831—22no—
21C2×C4×D4×C73—21no—
22C2×C4×D4×C73C2211yescharacteristic
23C22×C42×C73—21no—
24C4×C4⋊C4×C73—21no—
25C4×C4⋊C4×C73C2211yescharacteristic
26C4×C22⋊C4×C73—21no—
27C4×C22⋊C4×C73—22no—
28C43×C73C2211yescharacteristic
29C2×#11924028—21no—
30C2×#11924013—22no—
31C2×#11924012—21no—
32#14556591C2×C411yescharacteristic, complemented
33#14556588C2×C411yescharacteristic, complemented
34#14556579—21no—
35#14556565—21no—
36#14556565—21no—
37C4×D292—21no—
38C4×D292—21no—
39#14556524—22no—
40C4⋊#10596831—21no—
41C4⋊#10596831—21no—
42C4⋊#10596831—21no—
43C4⋊#10596831C2×C412yescomplemented
44#14556519—42no—
45#14556519—42no—
46#14556512—22no—
47#14556511—22no—
48C22⋊#10596831—21no—
49C22⋊#10596831—21no—
50C4×#10596831—21no—
51C4×#10596831—21no—
52C22×D4×C73D411yescharacteristic
53C23×C4×C73—22no—
54C23×C4×C73—21no—
55C4×D4×C73—28no—
56C4×D4×C73—21no—
57C4×D4×C73—21no—
58C4×D4×C73C2×C412yes—
59C4×D4×C73D412yes—
60C2×C4⋊C4×C73—21no—
61C2×C4⋊C4×C73D411yescharacteristic
62C2×C22⋊C4×C73—22no—
63C2×C22⋊C4×C73—21no—
64C2×C42×C73—21no—
65C2×C42×C73—22no—
66C2×C42×C73—21no—
67C2×C42×C73—21no—
68C2×C42×C73—42no—
69C2×C42×C73—42no—
70C2×C42×C73D411yescharacteristic
71C2×C42×C73D411yescharacteristic
72C2×C42×C73C2311yescharacteristic
73#95×C73—22no—
74#95×C73D412yes—
75#11924028—41no—
76#11924028—41no—
77#11924028—41no—
78#11924028—41no—
79C2×Dic146—21no—
80C2×D292—21no—
81C2×D292—21no—
82C2×D292—21no—
83C2×C22⋊D73—41no—
84C2×C4×D73—41no—
85#11924013—44no—
86#11924012—41no—
87#11924012—41no—
88C2×#10596831—22no—
89C2×#10596831—41no—
90C2×#10596831—41no—
91C4⋊Dic73—21no—
92C4⋊Dic73—21no—
93#11924005—41no—
94#11924005—41no—
95C24×C73—21no—
96C2×D4×C73—24no—
97C2×D4×C73—21no—
98C2×D4×C73C22⋊C412yes—
99C22×C4×C73—21no—
100C22×C4×C73—28no—
101C22×C4×C73—21no—
102C22×C4×C73—21no—
103C22×C4×C73—22no—
104C22×C4×C73—22no—
105C22×C4×C73—22no—
106C22×C4×C73—44no—
107C22×C4×C73—44no—
108C22×C4×C73—41no—
109C22×C4×C73—41no—
110C22×C4×C73—41no—
111C22×C4×C73—41no—
112C22×C4×C73C2×D411yescharacteristic
113C22×C4×C73C2×D411yescharacteristic
114C22×C4×C73C2×D411yescharacteristic
115C4⋊C4×C73—24no—
116C4⋊C4×C73—21no—
117C4⋊C4×C73C22⋊C412yes—
118C22⋊C4×C73—28no—
119C22⋊C4×C73—22no—
120C22⋊C4×C73—22no—
121C42×C73—24no—
122C42×C73—21no—
123C42×C73—21no—
124C42×C73—21no—
125C42×C73—42no—
126C42×C73—44no—
127C42×C73—41no—
128C42×C73—41no—
129C42×C73D812yescomplemented
130C42×C73C22×C411yescharacteristic
131C42×C73C2×D411yescharacteristic
132C42×C73#4111yescharacteristic
133C42×C73#4111yescharacteristic
134C23×C73—24no—
135C23×C73—22no—
136C23×C73—42no—
137C23×C73—41no—
138C23×C73—41no—
139C23×C73#12011yescharacteristic
140C22×D73—41no—
141D4×C73—24no—
142D4×C73—22no—
143D4×C73#10414yescomplemented
144C2×C4×C73—22no—
145C2×C4×C73—22no—
146C2×C4×C73—22no—
147C2×C4×C73—22no—
148C2×C4×C73—22no—
149C2×C4×C73—24no—
150C2×C4×C73—22no—
151C2×C4×C73—41no—
152C2×C4×C73—416no—
153C2×C4×C73—41no—
154C2×C4×C73—42no—
155C2×C4×C73—42no—
156C2×C4×C73—41no—
157C2×C4×C73—41no—
158C2×C4×C73—41no—
159C2×C4×C73—42no—
160C2×C4×C73—42no—
161C2×C4×C73—42no—
162C2×C4×C73—42no—
163C2×C4×C73C2×C22⋊C411yescharacteristic
164C2×C4×C73C4×D411yescharacteristic
165C2×C4×C73C4×D411yescharacteristic
166C2×C4×C73#12111yescharacteristic
167C2×C4×C73#12111yescharacteristic
168C2×C4×C73#12311yescharacteristic
169C2×C4×C73C2×D811yescharacteristic
170C2×C4×C73#13511yescharacteristic
171C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
172C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
173C22⋊D73—81no—
174C22⋊D73—81no—
175C2×Dic73—41no—
176D292—42no—
177D292—41no—
178D292—41no—
179D292—81no—
180C4×D73—81no—
181C4×D73—81no—
182Dic146—41no—
183Dic146—41no—
184#10596831—42no—
185#10596831—81no—
186#10596831—81no—
187C22×C73—21no—
188C22×C73—21no—
189C22×C73—24no—
190C22×C73—44no—
191C22×C73—44no—
192C22×C73—42no—
193C22×C73—42no—
194C22×C73#38611yescharacteristic
195C22×C73#44711yescharacteristic
196C22×C73#44911yescharacteristic
197C2×D73—81no—
198C2×D73—81no—
199C2×D73—81no—
200C4×C73—24no—
201C4×C73—21no—
202C4×C73—21no—
203C4×C73—21no—
204C4×C73—42no—
205C4×C73—44no—
206C4×C73—42no—
207C4×C73C2×#10412yes—
208C4×C73C4×D811yescharacteristic
209C4×C73#44211yescharacteristic
210C4×C73#45911yescharacteristic
211C4×C73#46311yescharacteristic
212Dic73—81no—
213#9233—731nomaximal, Sylow
214C2×C73—21no—
215C2×C73—21no—
216C2×C73—44no—
217C2×C73#187911yescharacteristic
218C2×C73#189611yescharacteristic
219C2×C73#190611yescharacteristic
220D73—161no—
221#3100—731no—
222C42×D4D7311yescharacteristic, complemented
223#1962—731no—
224#1502—731no—
225#1486—731no—
226#1303—731no—
227#1276—731no—
228C73#923311yescharacteristic, complemented, Sylow
229#532—731no—
230C2×C4×D4—21no—
231C2×C4×D4C2×D7311yescharacteristic
232C22×C42—21no—
233#480—1461no—
234#474—732no—
235#459—732no—
236#447—1461no—
237C4×D8—1462no—
238#431—731no—
239#431—731no—
240#423—731no—
241C2×#104—1461no—
242C2×#104—1461no—
243C4×C4⋊C4—21no—
244C4×C4⋊C4C2×D7311yescharacteristic
245C4×C22⋊C4—21no—
246C4×C22⋊C4—22no—
247C43C2×D7311yescharacteristic
248#327—1462no—
249D4⋊C8—1462no—
250C2×#41—1461no—
251C22×D4C22⋊D7311yescharacteristic
252C23×C4—22no—
253C23×C4—21no—
254C2×D8—1462no—
255C2×M16—1461no—
256C4⋊Q8—731no—
257#127—731no—
258#124—1461no—
259#121—1461no—
260#121—1461no—
261C4×D4—28no—
262C4×D4—21no—
263C4×D4—21no—
264C4×D4—1461no—
265C4×D4—1461no—
266C4×D4C4×D7312yes—
267C4×D4D29212yes—
268C2×C4⋊C4—21no—
269C2×C4⋊C4C22⋊D7311yescharacteristic
270C2×C22⋊C4—22no—
271C2×C22⋊C4—21no—
272C2×C42—21no—
273C2×C42—22no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—42no—
277C2×C42—42no—
278C2×C42C22⋊D7311yescharacteristic
279C2×C42C22⋊D7311yescharacteristic
280C2×C42C22×D7311yescharacteristic
281#107—1462no—
282C4⋊C8—732no—
283C4⋊C8—1461no—
284C4⋊C8—1461no—
285C4⋊C8—1461no—
286#104—2922no—
287#104—2922no—
288#102—1462no—
289#102—1462no—
290C22⋊C8—1461no—
291C22⋊C8—1461no—
292C4×C8—1461no—
293C4×C8—1461no—
294#95—22no—
295#95C22⋊D7312yes—
296C24—21no—
297#41—2921no—
298#41—2921no—
299#41—2921no—
300#41—2921no—
301C2×Q8—1461no—
302C2×D4—24no—
303C2×D4—21no—
304C2×D4—2921no—
305C2×D4—1461no—
306C2×D4—1461no—
307C2×D4—1461no—
308C2×D4#1192400512yes—
309C22×C4—21no—
310C22×C4—28no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—44no—
317C22×C4—44no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—2921no—
323C22×C4C2×C22⋊D7311yescharacteristic
324C22×C4C2×C22⋊D7311yescharacteristic
325C22×C4C2×C22⋊D7311yescharacteristic
326D8—2924no—
327M16—2921no—
328M16—2921no—
329C2×C8—2921no—
330C2×C8—2921no—
331C2×C8—1462no—
332C4⋊C4—24no—
333C4⋊C4—21no—
334C4⋊C4—1461no—
335C4⋊C4—1461no—
336C4⋊C4#1192400512yes—
337C22⋊C4—28no—
338C22⋊C4—22no—
339C22⋊C4—22no—
340C22⋊C4—2921no—
341C22⋊C4—2921no—
342C42—24no—
343C42—21no—
344C42—21no—
345C42—21no—
346C42—42no—
347C42—44no—
348C42—41no—
349C42—41no—
350C42#1192401312yescomplemented
351C42C2×C4×D7311yescharacteristic
352C42C2×D29211yescharacteristic
353C42#1192402811yescharacteristic
354C42#1192402811yescharacteristic
355C23—24no—
356C23—22no—
357C23—42no—
358C23—41no—
359C23—41no—
360C23—2921no—
361C23#1455656011yescharacteristic
362Q8—2921no—
363Q8—2921no—
364D4—24no—
365D4—22no—
366D4—2921no—
367D4—2921no—
368D4—2922no—
369D4—5841no—
370D4—5841no—
371D4—5841no—
372D4#1455651914yescomplemented
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—24no—
379C2×C4—22no—
380C2×C4—41no—
381C2×C4—416no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—2921no—
393C2×C4—5841no—
394C2×C4—5841no—
395C2×C4C2×#1192400511yescharacteristic
396C2×C4C4×C22⋊D7311yescharacteristic
397C2×C4C4×C22⋊D7311yescharacteristic
398C2×C4#1455656511yescharacteristic
399C2×C4#1455656511yescharacteristic
400C2×C4#1455657411yescharacteristic
401C2×C4C2×#1192401311yescharacteristic
402C2×C4#1455661111yescharacteristic
403C2×C4#1455661611yescharacteristic
404C2×C4#1455661711yescharacteristic
405C8—2922no—
406C8—5841no—
407C8—5841no—
408C22—21no—
409C22—21no—
410C22—24no—
411C22—44no—
412C22—44no—
413C22—42no—
414C22—42no—
415C22—5841no—
416C22—5841no—
417C22—5841no—
418C22#1798846611yescharacteristic
419C22#1798872111yescharacteristic
420C22#1798873211yescharacteristic
421C4—24no—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—44no—
426C4—42no—
427C4—42no—
428C4—5841no—
429C4C2×#1455651912yes—
430C4C4×#1192401311yescharacteristic
431C4#1798869711yescharacteristic
432C4#1798878211yescharacteristic
433C4#1798880211yescharacteristic
434C2—21no—
435C2—21no—
436C2—44no—
437C2—11681no—
438C2#2261027911yescharacteristic
439C2#2261037411yescharacteristic
440C2#2261043211yescharacteristic
subgroupG/H
centerC2×C4#14556565
commutatorC42×C73C22×C4
FrattiniC2×C42C22×D73
FittingC42×D4×C73C2
socleC22×C73#386
prime radical#22619931C2
cosocleC2×C42×C73C23
hypercenterC42×D4D73
hypocenterC73#9233
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C42×D4

Complements

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

NH ≅ G/Ntype
2. #22610759C2semidirect
3. #22608425C2semidirect
4. #22608356C2semidirect
5. #22607834C2semidirect
6. #22607764C2semidirect
7. C42×D4×C73C2semidirect
10. #17988867C4semidirect
11. #17988782C4semidirect
32. #14556591C2×C4semidirect
33. #14556588C2×C4semidirect
43. C4⋊#10596831C2×C4semidirect
129. C42×C73D8semidirect
143. D4×C73#104semidirect
222. C42×D4D73semidirect
228. C73#9233semidirect
350. C42#11924013semidirect
372. D4#14556519semidirect

Maximal subgroups

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

subgroupindex
#226199312
#226107592
#226084252
#226083562
#226078342
#226077642
C42×D4×C732
#923373