Finite Group Databasebeta

← #37543037

Subgroups — #37543037

Order 23168.

712 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19153864C211yesmaximal, characteristic, complemented
2C2×#19153848C211yesmaximal, characteristic, complemented
3C2×#19153839C211yesmaximal, characteristic, complemented
4#26041859C211yesmaximal, characteristic, complemented
5#26041440C211yesmaximal, characteristic, complemented
6#26041439C211yesmaximal, characteristic, complemented
7#26041433C211yesmaximal, characteristic, complemented
8#26041427C211yesmaximal, characteristic, complemented
9#26041426C211yesmaximal, characteristic, complemented
10#26041419C211yesmaximal, characteristic, complemented
11#26041418C211yesmaximal, characteristic, complemented
12#26041279C211yesmaximal, characteristic, complemented
13#103×D181C211yesmaximal, characteristic, complemented
14#26041244C211yesmaximal, characteristic, complemented
15#451×C181C211yesmaximal, characteristic, complemented
16#19153864—24no—
17#19153864—24no—
18C2×#15668450—21no—
19C2×#15668450C2211yescharacteristic
20C2×#15668450C2211yescharacteristic
21C2×#15668450C2211yescharacteristic
22C2×#15668449—21no—
23C2×#15668449—21no—
24C2×#15668449C2211yescharacteristic
25C2×#15668449C2211yescharacteristic
26C2×Q8×D181—21no—
27C2×Q8×D181C2211yescharacteristic
28C2×Q8×D181C2211yescharacteristic
29#19153848—24no—
30#19153839—24no—
31C2×#15668439C2211yescharacteristic
32C2×#15668438C2211yescharacteristic
33C2×#15668436C2211yescharacteristic
34C2×#15668436C2211yescharacteristic
35C2×#15668436C2211yescharacteristic
36C2×D1448C2211yescharacteristic
37C2×#15668432C2211yescharacteristic
38C2×C8×D181C2211yescharacteristic
39#19153794—21no—
40#19153792C2211yescharacteristic
41#19153791—21no—
42#19153791C2211yescharacteristic
43#19153790C2211yescharacteristic
44#19153789—21no—
45#19153789—21no—
46C4⋊C4×D181C2211yescharacteristic
47#19153741C2211yescharacteristic
48#19153740C2211yescharacteristic
49#19153740C2211yescharacteristic
50#19153739C2211yescharacteristic
51#19153738C2211yescharacteristic
52#19153736C2211yescharacteristic
53C22⋊#13089678C2211yescharacteristic
54#19153728C2211yescharacteristic
55#19153728C2211yescharacteristic
56C22×Q8×C181C2211yescharacteristic
57C2×Q16×C181C2211yescharacteristic
58C2×Q16×C181C2211yescharacteristic
59C22⋊Q8×C181C2211yescharacteristic
60#103×C181C2211yescharacteristic
61#103×C181C2211yescharacteristic
62C22⋊C8×C181C2211yescharacteristic
63#15668450—28no—
64#15668450—22no—
65#15668450—22no—
66#15668450—22no—
67#15668450—22no—
68#15668450—22no—
69#15668450D414yescomplemented
70#15668449—28no—
71#15668449—28no—
72#15668449—22no—
73#15668449—22no—
74#15668449—22no—
75#15668449—22no—
76C2×Dic362—21no—
77C2×Dic362—21no—
78C2×Dic362D411yescharacteristic
79C2×Dic362C2311yescharacteristic
80Q8×D181—28no—
81Q8×D181—22no—
82Q8×D181—22no—
83Q8×D181—22no—
84Q8×D181D414yescomplemented
85C2×D724—21no—
86C2×D724—21no—
87C2×D724—21no—
88C2×D724—21no—
89C2×D724D411yescharacteristic
90C2×D724C2311yescharacteristic
91C2×D724C2311yescharacteristic
92C2×C22⋊D181—21no—
93C2×C22⋊D181—21no—
94C2×C22⋊D181D411yescharacteristic
95C2×C22⋊D181D411yescharacteristic
96C2×C4×D181—21no—
97C2×C4×D181—21no—
98C2×C4×D181—21no—
99C2×C4×D181—21no—
100C2×C4×D181—21no—
101C2×C4×D181—21no—
102C2×C4×D181—21no—
103C2×C4×D181C2311yescharacteristic
104C2×C4×D181C2311yescharacteristic
105#15668439—22no—
106#15668438—22no—
107#15668436—22no—
108#15668436—22no—
109#15668436—22no—
110D1448—22no—
111#15668432—22no—
112C2×#13089678C2311yescharacteristic
113C2×#13089678C2311yescharacteristic
114C8×D181—22no—
115C4⋊Dic181C2311yescharacteristic
116#15668428—21no—
117#15668428—21no—
118#15668426—21no—
119#15668426—21no—
120#15668426—21no—
121C2×Q8×C181—22no—
122C2×Q8×C181—22no—
123C2×Q8×C181—21no—
124C2×Q8×C181C2311yescharacteristic
125C2×Q8×C181C2311yescharacteristic
126C2×Q8×C181C2311yescharacteristic
127C22×C4×C181—21no—
128C22×C4×C181C2311yescharacteristic
129Q16×C181—22no—
130Q16×C181—22no—
131C2×C8×C181C2311yescharacteristic
132C2×C8×C181C2311yescharacteristic
133C4⋊C4×C181—21no—
134C4⋊C4×C181C2311yescharacteristic
135C22⋊C4×C181—21no—
136C22×D181—21no—
137C22×D181—21no—
138C22×D181C2×D411yescharacteristic
139C22×D181C2×D411yescharacteristic
140Dic362—24no—
141Dic362—21no—
142Dic362—22no—
143Dic362—24no—
144Dic362C2×D412yes—
145D724—24no—
146D724—24no—
147D724—22no—
148D724—21no—
149D724—21no—
150D724—21no—
151D724—42no—
152D724—81no—
153D724C2×D412yes—
154C22⋊D181—24no—
155C22⋊D181—22no—
156C22⋊D181—22no—
157C22⋊D181—81no—
158C2×Dic181—21no—
159C2×Dic181—21no—
160C2×Dic181C2×D411yescharacteristic
161C2×Dic181C2×D411yescharacteristic
162C4×D181—24no—
163C4×D181—24no—
164C4×D181—24no—
165C4×D181—24no—
166C4×D181—21no—
167C4×D181—21no—
168C4×D181—24no—
169C4×D181—42no—
170C4×D181—81no—
171C4×D181C2×D412yes—
172C4×D181C2×D412yes—
173#13089678—21no—
174#13089678—21no—
175C23×C181C2×D411yescharacteristic
176Q8×C181—24no—
177Q8×C181—21no—
178Q8×C181—21no—
179Q8×C181—21no—
180Q8×C181—21no—
181Q8×C181C2×D412yes—
182Q8×C181C2×D412yes—
183C2×C4×C181—24no—
184C2×C4×C181—21no—
185C2×C4×C181—21no—
186C2×C4×C181—21no—
187C2×C4×C181—21no—
188C2×C4×C181—21no—
189C2×C4×C181—21no—
190C2×C4×C181C2×D411yescharacteristic
191C2×C4×C181C2411yescharacteristic
192C8×C181—21no—
193C8×C181—21no—
194C22×C181—21no—
195C22×C181—21no—
196C22×C181—21no—
197C22×C181C22×D411yescharacteristic
198C2×D181—21no—
199C2×D181—21no—
200C2×D181—21no—
201C2×D181—22no—
202C2×D181—21no—
203C2×D181—21no—
204C2×D181—41no—
205C2×D181—41no—
206C2×D181—41no—
207C2×D181—41no—
208C2×D181—41no—
209C2×D181#12012yes—
210C4×C181—22no—
211C4×C181—22no—
212C4×C181—21no—
213C4×C181—41no—
214C4×C181—41no—
215C4×C181C22×D411yescharacteristic
216C4×C181C22×D411yescharacteristic
217Dic181—22no—
218Dic181—21no—
219Dic181—41no—
220Dic181#12012yes—
221C2×C181—41no—
222C2×C181C2×#12011yescharacteristic
223C2×C181#57711yescharacteristic
224C2×C181#57811yescharacteristic
225D181—22no—
226D181—41no—
227D181—42no—
228D181—81no—
229C181#301311yescharacteristic, complemented, Sylow
230#3013—1811nomaximal, Sylow
231C2×2−1+4—1811no—
232C2×#137—1811no—
233C2×#135—1811no—
234#537—1811no—
235#452—1811no—
236#452—1811no—
237#451—1811no—
238#451D18111yescharacteristic, complemented
239#449—1811no—
240#449—1811no—
241#448—1811no—
242#448—1811no—
243#417—1811no—
244C2×#103—1811no—
245#408—1811no—
2462−1+4—3624no—
2472−1+4—3624no—
248C2×#41—3621no—
249C2×#41—3621no—
250C2×#41—3621no—
251C2×#41—1811no—
252C2×#41—1811no—
253C2×#41—1811no—
254C2×#41—1811no—
255C2×#41—1811no—
256C22×Q8—3621no—
257C22×Q8—1811no—
258C22×Q8—1811no—
259C22×Q8C2×D18111yescharacteristic
260#137—3624no—
261#135—3624no—
262C2×Q16—1811no—
263C2×Q16C2×D18111yescharacteristic
264C2×Q16C2×D18111yescharacteristic
265C2×SD16—1811no—
266C2×SD16—1811no—
267C2×SD16—1811no—
268C2×SD16—1811no—
269C2×D8—1811no—
270C2×M16—1811no—
271C22×C8—1811no—
272C22⋊Q8—3621no—
273C22⋊Q8—1811no—
274C22⋊Q8C2×D18111yescharacteristic
275#121—3621no—
276#121—3621no—
277#121—3621no—
278#121—1811no—
279#121—1811no—
280C2×C4⋊C4—1811no—
281#103—1811no—
282#103—1811no—
283#103—1811no—
284#103—1811no—
285#103C2×D18111yescharacteristic
286#103C2×D18111yescharacteristic
287#102—1811no—
288#102—1811no—
289C22⋊C8—1811no—
290C22⋊C8—1811no—
291C22⋊C8—1811no—
292C22⋊C8C2×D18111yescharacteristic
293#41—3628no—
294#41—3628no—
295#41—3622no—
296#41—3622no—
297#41—3622no—
298#41—3622no—
299#41—3628no—
300#41—3622no—
301#41—3622no—
302#41—3622no—
303#41—3622no—
304#41—3622no—
305#41—1814no—
306C2×Q8—22no—
307C2×Q8—22no—
308C2×Q8—21no—
309C2×Q8—3621no—
310C2×Q8—3621no—
311C2×Q8—3628no—
312C2×Q8—3622no—
313C2×Q8—3622no—
314C2×Q8—3622no—
315C2×Q8—1814no—
316C2×Q8—1811no—
317C2×Q8—1811no—
318C2×Q8C22×D18111yescharacteristic
319C2×Q8C22×D18111yescharacteristic
320C2×Q8C22×D18111yescharacteristic
321C2×D4—3621no—
322C2×D4—3621no—
323C2×D4—3621no—
324C2×D4—3621no—
325C2×D4—3621no—
326C2×D4—3621no—
327C2×D4—1811no—
328C2×D4—1811no—
329C2×D4—1811no—
330C2×D4—1811no—
331C2×D4—1811no—
332C22×C4—21no—
333C22×C4—3621no—
334C22×C4—3621no—
335C22×C4—3621no—
336C22×C4—3621no—
337C22×C4—3621no—
338C22×C4—3621no—
339C22×C4—3621no—
340C22×C4—1811no—
341C22×C4—1811no—
342C22×C4C22×D18111yescharacteristic
343Q16—22no—
344Q16—22no—
345Q16—3622no—
346SD16—3622no—
347SD16—3622no—
348SD16—3622no—
349SD16—3622no—
350D8—3622no—
351M16—3622no—
352C2×C8—3622no—
353C2×C8—1811no—
354C2×C8—1811no—
355C2×C8C22×D18111yescharacteristic
356C2×C8C22×D18111yescharacteristic
357C4⋊C4—21no—
358C4⋊C4—3621no—
359C4⋊C4—3621no—
360C4⋊C4—1811no—
361C4⋊C4C22×D18111yescharacteristic
362C22⋊C4—21no—
363C22⋊C4—3621no—
364C22⋊C4—3621no—
365C22⋊C4—3621no—
366C23—3621no—
367C23—3621no—
368C23—1811no—
369C23—1811no—
370C23D4×D18111yescharacteristic
371Q8—24no—
372Q8—21no—
373Q8—21no—
374Q8—21no—
375Q8—21no—
376Q8—3624no—
377Q8—3624no—
378Q8—3622no—
379Q8—3621no—
380Q8—1812no—
381Q8D4×D18112yes—
382Q8D4×D18112yes—
383D4—3624no—
384D4—3624no—
385D4—3622no—
386D4—3624no—
387D4—3622no—
388D4—3622no—
389D4—3621no—
390D4—3621no—
391D4—3621no—
392D4—14481no—
393D4—14481no—
394D4—1812no—
395D4—7242no—
396C2×C4—24no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—3624no—
404C2×C4—3624no—
405C2×C4—3624no—
406C2×C4—3624no—
407C2×C4—3624no—
408C2×C4—3621no—
409C2×C4—3621no—
410C2×C4—3621no—
411C2×C4—3621no—
412C2×C4—14481no—
413C2×C4—1811no—
414C2×C4—1811no—
415C2×C4—1812no—
416C2×C4—1812no—
417C2×C4—7242no—
418C2×C4D4×D18111yescharacteristic
419C2×C4C23×D18111yescharacteristic
420C8—21no—
421C8—21no—
422C8—3621no—
423C8—3621no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—3621no—
428C22—3621no—
429C22—3621no—
430C22—3622no—
431C22—3621no—
432C22—3621no—
433C22—1812no—
434C22—7241no—
435C22—7241no—
436C22—7241no—
437C22—7241no—
438C22—7241no—
439C22C2×D4×D18111yescharacteristic
440C4—22no—
441C4—22no—
442C4—21no—
443C4—41no—
444C4—41no—
445C4—3622no—
446C4—3621no—
447C4—1812no—
448C4—7241no—
449C4C2×D4×D18111yescharacteristic
450C4C2×D4×D18111yescharacteristic
451C2—41no—
452C2—3622no—
453C2—14481no—
454C2—7241no—
455C2—7242no—
456C2#120×D18111yescharacteristic
457C2#2604208811yescharacteristic
458C2#2604209211yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#451×C181C2
socleC22×C181C22×D4
hypercenter#451D181
hypocenterC181#3013
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. C2×#19153864C2semidirect
2. C2×#19153848C2semidirect
3. C2×#19153839C2semidirect
4. #26041859C2semidirect
5. #26041440C2semidirect
6. #26041439C2semidirect
7. #26041433C2semidirect
8. #26041427C2semidirect
9. #26041426C2semidirect
10. #26041419C2semidirect
11. #26041418C2semidirect
12. #26041279C2semidirect
13. #103×D181C2semidirect
14. #26041244C2semidirect
15. #451×C181C2semidirect
69. #15668450D4semidirect
84. Q8×D181D4semidirect
229. C181#3013semidirect
238. #451D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538642
C2×#191538482
C2×#191538392
#260418592
#260414402
#260414392
#260414332
#260414272
#260414262
#260414192
#260414182
#260412792
#103×D1812
#260412442
#451×C1812
#3013181