Finite Group Databasebeta

← #36370183

Subgroups — #36370183

Order 22144.

772 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D173⋊M16C211yesmaximal, characteristic, complemented
2C2×#19149758C211yesmaximal, characteristic, complemented
3#26012484C212yesmaximal, complemented
4#26012483C212yesmaximal, complemented
5#26012482C214yesmaximal, complemented
6#26012481C214yesmaximal, complemented
7C2×#41×D173C211yesmaximal, characteristic
8C2×#14583836—21no—
9D173⋊M16—21no—
10D173⋊M16—21no—
11D173⋊M16C2212yes—
12D173⋊M16C2212yescomplemented
13C2×#14583835—21no—
14C2×D173⋊C8C2212yescomplemented
15#19149758—21no—
16#19149758—21no—
17#19149758C2212yes—
18#19149758C2212yescomplemented
19C2×C4⋊#10621924C2212yescomplemented
20C2×C22⋊#10621924—21no—
21C2×C4×#10621924—21no—
22#19149742C2218yescomplemented
23#19149741C2218yescomplemented
24C2×#14583827C411yescharacteristic, complemented
25C2×#14583826—21no—
26C2×#14583826C411yescharacteristic, complemented
27C2×#14583825—21no—
28C2×#14583825C411yescharacteristic, complemented
29#41×D173—21no—
30#41×D173—21no—
31#41×D173—21no—
32#41×D173—21no—
33#41×D173C412yescomplemented
34#41×D173C412yescomplemented
35#41×D173C2212yes—
36#41×D173C2212yes—
37C2×Q8×D173C2211yescharacteristic
38C2×D4×D173—21no—
39C2×D4×D173C2211yescharacteristic
40C22×C4×D173—21no—
41C22×C4×D173C2211yescharacteristic
42C2×#41×C173C411yescharacteristic, complemented
43C22×#10621924—22no—
44#14583836—22no—
45#14583836—22no—
46#14583835—22no—
47#14583835—22no—
48D173⋊C8—22no—
49D173⋊C8C2314yes—
50C2×#13063443—22no—
51C4⋊#10621924—22no—
52C4⋊#10621924C2314yes—
53C22⋊#10621924—22no—
54C22⋊#10621924—22no—
55C4×#10621924—22no—
56C4×#10621924—22no—
57C23×D173—21no—
58C23×D173D411yescharacteristic
59#14583827—21no—
60#14583827—21no—
61#14583827C2×C412yes—
62#14583827C2×C412yescomplemented
63#14583826—22no—
64#14583826—22no—
65#14583826—22no—
66#14583826—22no—
67#14583826—21no—
68#14583826—21no—
69#14583826C2×C412yes—
70#14583826C2×C412yescomplemented
71#14583825—22no—
72#14583825—22no—
73#14583825—22no—
74#14583825—22no—
75#14583825—21no—
76#14583825—21no—
77#14583825C2×C412yes—
78#14583825C2×C412yescomplemented
79C2×Dic346—21no—
80C2×Dic346C2×C411yescharacteristic, complemented
81Q8×D173—21no—
82Q8×D173—21no—
83Q8×D173C2×C412yescomplemented
84Q8×D173C2312yes—
85C2×D692—21no—
86C2×D692C2×C411yescharacteristic, complemented
87C2×C22⋊D173—21no—
88C2×C22⋊D173—21no—
89C2×C22⋊D173—21no—
90D4×D173—22no—
91D4×D173—22no—
92D4×D173—22no—
93D4×D173—22no—
94D4×D173—21no—
95D4×D173—21no—
96D4×D173C2×C412yescomplemented
97D4×D173C2312yes—
98C22×Dic173—21no—
99C22×Dic173D411yescharacteristic
100C2×C4×D173—21no—
101C2×C4×D173—21no—
102C2×C4×D173—22no—
103C2×C4×D173—22no—
104C2×C4×D173—22no—
105C2×C4×D173—22no—
106C2×C4×D173—21no—
107C2×C4×D173C2×C411yescharacteristic
108C2×C4×D173C2×C411yescharacteristic
109C2×C4×D173C2×C411yescharacteristic
110C2×C4×D173D411yescharacteristic
111C2×C4×D173D411yescharacteristic
112C2×C4×D173D411yescharacteristic
113C2×C4×D173D411yescharacteristic
114C2×C4×D173D411yescharacteristic
115C2×C4×D173D411yescharacteristic
116C2×C4×D173C2311yescharacteristic
117C2×C4×D173C2311yescharacteristic
118C2×C4×D173C2311yescharacteristic
119#41×C173—21no—
120#41×C173—21no—
121#41×C173C2×C412yes—
122#41×C173C2×C412yescomplemented
123C2×Q8×C173C2×C411yescharacteristic, complemented
124C2×D4×C173—21no—
125C2×D4×C173C2×C411yescharacteristic, complemented
126C22×C4×C173—21no—
127C22×C4×C173C2×C411yescharacteristic
128C2×#10621924—24no—
129C2×#10621924—44no—
130#13063443—24no—
131C22×D173—21no—
132C22×D173—22no—
133C22×D173—22no—
134C22×D173—42no—
135C22×D173—44no—
136C22×D173C22⋊C411yescharacteristic
137C22×D173C22⋊C411yescharacteristic
138C22×D173C22⋊C411yescharacteristic
139C22×D173C2×D411yescharacteristic
140C22×D173C2×D411yescharacteristic
141C22×D173C2×D411yescharacteristic
142Dic346—21no—
143Dic346—22no—
144Dic346—22no—
145Dic346C22×C412yes—
146D692—22no—
147D692—22no—
148D692—21no—
149D692C22×C412yes—
150C22⋊D173—22no—
151C22⋊D173—22no—
152C22⋊D173—22no—
153C22⋊D173—22no—
154C22⋊D173—22no—
155C22⋊D173—22no—
156C2×Dic173—21no—
157C2×Dic173—22no—
158C2×Dic173—22no—
159C2×Dic173C22⋊C411yescharacteristic
160C2×Dic173C22⋊C411yescharacteristic
161C2×Dic173C22⋊C411yescharacteristic
162C2×Dic173C2×D411yescharacteristic
163C2×Dic173C2×D411yescharacteristic
164C2×Dic173C2×D411yescharacteristic
165C4×D173—22no—
166C4×D173—22no—
167C4×D173—22no—
168C4×D173—22no—
169C4×D173—22no—
170C4×D173—22no—
171C4×D173C22⋊C411yescharacteristic
172C4×D173C22⋊C411yescharacteristic
173C4×D173C22⋊C411yescharacteristic
174C4×D173C22⋊C411yescharacteristic
175C4×D173C22⋊C411yescharacteristic
176C4×D173C22⋊C411yescharacteristic
177C4×D173C22×C411yescharacteristic
178C4×D173C22×C411yescharacteristic
179C4×D173C22×C411yescharacteristic
180C4×D173C2×D411yescharacteristic
181C4×D173C2×D411yescharacteristic
182C4×D173C2×D411yescharacteristic
183C4×D173C2×D411yescharacteristic
184C4×D173C2×D411yescharacteristic
185C4×D173C2×D411yescharacteristic
186C4×D173C2411yescharacteristic
187C23×C173—21no—
188C23×C173C22⋊C411yescharacteristic
189Q8×C173—21no—
190Q8×C173C22×C412yes—
191D4×C173—22no—
192D4×C173—22no—
193D4×C173—21no—
194D4×C173C22×C412yes—
195C2×C4×C173—21no—
196C2×C4×C173—22no—
197C2×C4×C173—22no—
198C2×C4×C173C22⋊C411yescharacteristic
199C2×C4×C173C22⋊C411yescharacteristic
200C2×C4×C173C22⋊C411yescharacteristic
201C2×C4×C173C22×C411yescharacteristic
202C2×C4×C173C22×C411yescharacteristic
203C2×C4×C173C22×C411yescharacteristic
204C22×C173—22no—
205C22×C173—41no—
206C22×C173—42no—
207C22×C173C2×C22⋊C411yescharacteristic
208C22×C173C2×C22⋊C411yescharacteristic
209C22×C173C2×C22⋊C411yescharacteristic
210C2×D173—22no—
211C2×D173—22no—
212C2×D173—22no—
213C2×D173—22no—
214C2×D173—41no—
215C2×D173—41no—
216C2×D173—41no—
217C2×D173—44no—
218C2×D173—42no—
219C2×D173—44no—
220C2×D173C2×C22⋊C411yescharacteristic
221C2×D173C2×C22⋊C411yescharacteristic
222C2×D173C2×C22⋊C411yescharacteristic
223C2×D173C22×D411yescharacteristic
224#10621924—44no—
225C4×C173—22no—
226C4×C173C2×C22⋊C411yescharacteristic
227C4×C173C2×C22⋊C411yescharacteristic
228C4×C173C2×C22⋊C411yescharacteristic
229C4×C173C23×C411yescharacteristic
230Dic173—22no—
231Dic173C2×C22⋊C411yescharacteristic
232Dic173C2×C22⋊C411yescharacteristic
233Dic173C2×C22⋊C411yescharacteristic
234Dic173C22×D411yescharacteristic
235C2×C173—21no—
236C2×C173—21no—
237C2×C173—21no—
238C2×C173—42no—
239C2×C173C22×C22⋊C411yescharacteristic
240D173—21no—
241D173—21no—
242D173—21no—
243D173—42no—
244D173#57512yes—
245C173#289611yescharacteristic, complemented, Sylow
246#2896—1731nomaximal, Sylow
247C22×#41—1731no—
248C22×M16—1731no—
249C2×#117—1731no—
250#419—1732no—
251#418—1732no—
252#417—1734no—
253#416—1734no—
254C2×#41—3461no—
255C2×#41—3461no—
256C2×#41—3461no—
257C2×#41—3461no—
258C2×#41—3461no—
259C2×#41—3461no—
260C2×#41—1732no—
261C2×#41—1732no—
262C2×#41—1732no—
263C2×#41—1732no—
264C2×#41—1731no—
265C2×#41—1731no—
266C2×#41—1731no—
267C2×#41#1062192411yescharacteristic, complemented
268C22×Q8—1731no—
269C22×D4—3461no—
270C22×D4—1731no—
271C23×C4—3461no—
272C23×C4—1731no—
273C2×M16—3461no—
274C2×M16—3461no—
275C2×M16—3461no—
276C2×M16—3461no—
277C2×M16—1732no—
278C2×M16—1732no—
279C22×C8—1732no—
280#117—3461no—
281#117—3461no—
282#117—1732no—
283#117—1732no—
284C2×C4⋊C4—1732no—
285C2×C22⋊C4—3461no—
286C2×C42—3461no—
287#103—1738no—
288#102—1738no—
289C24—3461no—
290C24—1731no—
291#41—21no—
292#41—21no—
293#41—3462no—
294#41—3462no—
295#41—3462no—
296#41—3462no—
297#41—3462no—
298#41—3462no—
299#41—3462no—
300#41—3462no—
301#41—3461no—
302#41—3461no—
303#41—3461no—
304#41—3461no—
305#41—3461no—
306#41—3461no—
307#41—1732no—
308#41—1732no—
309#41—1732no—
310#41—1732no—
311#41—1732no—
312#41—1732no—
313#41C2×#1062192412yes—
314#41C2×#1062192412yescomplemented
315C2×Q8—3461no—
316C2×Q8—3461no—
317C2×Q8—3461no—
318C2×Q8—1732no—
319C2×Q8—1732no—
320C2×Q8—1731no—
321C2×Q8C2×#1062192411yescharacteristic, complemented
322C2×D4—21no—
323C2×D4—3461no—
324C2×D4—3462no—
325C2×D4—3462no—
326C2×D4—3462no—
327C2×D4—3462no—
328C2×D4—3461no—
329C2×D4—3461no—
330C2×D4—3461no—
331C2×D4—3461no—
332C2×D4—3461no—
333C2×D4—1732no—
334C2×D4—1732no—
335C2×D4—1731no—
336C2×D4C2×#1062192411yescharacteristic, complemented
337C22×C4—21no—
338C22×C4—3461no—
339C22×C4—3461no—
340C22×C4—3462no—
341C22×C4—3462no—
342C22×C4—3462no—
343C22×C4—3462no—
344C22×C4—3461no—
345C22×C4—3461no—
346C22×C4—3462no—
347C22×C4—1731no—
348C22×C4—1731no—
349C22×C4—1731no—
350C22×C4—1731no—
351C22×C4—1731no—
352C22×C4—1731no—
353C22×C4—1731no—
354C22×C4—1731no—
355C22×C4—1731no—
356C22×C4—1731no—
357C22×C4—1731no—
358C22×C4—1731no—
359C22×C4—1731no—
360C22×C4C2×#1062192411yescharacteristic
361M16—3462no—
362M16—3462no—
363M16—3462no—
364M16—3462no—
365C2×C8—3462no—
366C2×C8—3462no—
367C2×C8—1734no—
368C4⋊C4—3462no—
369C4⋊C4—1734no—
370C22⋊C4—3462no—
371C22⋊C4—3462no—
372C42—3462no—
373C42—3462no—
374C23—21no—
375C23—3462no—
376C23—3462no—
377C23—3461no—
378C23—1731no—
379C23—1731no—
380C23—1731no—
381C23—1731no—
382C23—1731no—
383C23—1731no—
384C23—6922no—
385C23—6924no—
386C23C22⋊#1062192411yescharacteristic
387Q8—21no—
388Q8—3462no—
389Q8—3462no—
390Q8—3461no—
391Q8—1732no—
392Q8C22×#1062192412yes—
393D4—22no—
394D4—22no—
395D4—21no—
396D4—3462no—
397D4—3462no—
398D4—3462no—
399D4—3462no—
400D4—3462no—
401D4—3462no—
402D4—3462no—
403D4—3462no—
404D4—3461no—
405D4—1732no—
406D4C22×#1062192412yes—
407C2×C4—21no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—3462no—
411C2×C4—3462no—
412C2×C4—3462no—
413C2×C4—3462no—
414C2×C4—3461no—
415C2×C4—3462no—
416C2×C4—3462no—
417C2×C4—3462no—
418C2×C4—3462no—
419C2×C4—3464no—
420C2×C4—1731no—
421C2×C4—1731no—
422C2×C4—1731no—
423C2×C4—1731no—
424C2×C4—1731no—
425C2×C4—1731no—
426C2×C4—1731no—
427C2×C4—1731no—
428C2×C4—1731no—
429C2×C4—1731no—
430C2×C4—1731no—
431C2×C4—1731no—
432C2×C4—1731no—
433C2×C4—1731no—
434C2×C4—1731no—
435C2×C4—1731no—
436C2×C4—1731no—
437C2×C4—1731no—
438C2×C4—1731no—
439C2×C4—1731no—
440C2×C4—1731no—
441C2×C4—1731no—
442C2×C4—6924no—
443C2×C4C22⋊#1062192411yescharacteristic
444C2×C4C22⋊#1062192411yescharacteristic
445C2×C4C22⋊#1062192411yescharacteristic
446C2×C4C22×#1062192411yescharacteristic
447C2×C4C22×#1062192411yescharacteristic
448C2×C4C22×#1062192411yescharacteristic
449C8—3464no—
450C22—22no—
451C22—41no—
452C22—42no—
453C22—3462no—
454C22—3462no—
455C22—3462no—
456C22—3462no—
457C22—1731no—
458C22—1731no—
459C22—1731no—
460C22—1731no—
461C22—6921no—
462C22—6921no—
463C22—6921no—
464C22—6922no—
465C22—6924no—
466C22—6924no—
467C22C2×C22⋊#1062192411yescharacteristic
468C22C2×C22⋊#1062192411yescharacteristic
469C22C2×C22⋊#1062192411yescharacteristic
470C4—22no—
471C4—3462no—
472C4—1731no—
473C4—1731no—
474C4—1731no—
475C4—1731no—
476C4—6924no—
477C4C2×C22⋊#1062192411yescharacteristic
478C4C2×C22⋊#1062192411yescharacteristic
479C4C2×C22⋊#1062192411yescharacteristic
480C4C23×#1062192411yescharacteristic
481C2—21no—
482C2—21no—
483C2—21no—
484C2—42no—
485C2—3461no—
486C2—3461no—
487C2—3461no—
488C2—1732no—
489C2—6922no—
490C2C22×C22⋊#1062192411yescharacteristic
subgroupG/H
centerC4C2×C22⋊#10621924
commutatorC4×C173C23×C4
FrattiniC4C23×#10621924
FittingC2×#41×C173C4
socleC2×C173C22×C22⋊C4
prime radicalC2×#41×D173C2
cosocleC4×D173C24
hypercenterC2×#41#10621924
hypocenterC173#2896
LCSG ▷ C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. C2×D173⋊M16C2semidirect
2. C2×#19149758C2semidirect
3. #26012484C2semidirect
4. #26012483C2semidirect
5. #26012482C2semidirect
6. #26012481C2semidirect
12. D173⋊M16C22semidirect
14. C2×D173⋊C8C22semidirect
18. #19149758C22semidirect
19. C2×C4⋊#10621924C22semidirect
22. #19149742C22semidirect
23. #19149741C22semidirect
24. C2×#14583827C4semidirect
26. C2×#14583826C4semidirect
28. C2×#14583825C4semidirect
33. #41×D173C4semidirect
34. #41×D173C4semidirect
42. C2×#41×C173C4semidirect
62. #14583827C2×C4semidirect
70. #14583826C2×C4semidirect
78. #14583825C2×C4semidirect
80. C2×Dic346C2×C4semidirect
83. Q8×D173C2×C4semidirect
86. C2×D692C2×C4semidirect
96. D4×D173C2×C4semidirect
122. #41×C173C2×C4semidirect
123. C2×Q8×C173C2×C4semidirect
125. C2×D4×C173C2×C4semidirect
245. C173#2896semidirect
267. C2×#41#10621924semidirect
314. #41C2×#10621924semidirect
321. C2×Q8C2×#10621924semidirect
336. C2×D4C2×#10621924semidirect

Maximal subgroups

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

subgroupindex
C2×D173⋊M162
C2×#191497582
#260124842
#260124832
#260124822
#260124812
C2×#41×D1732
#2896173