Finite Group Databasebeta

← #49172431

Subgroups — #49172431

Order 37504.

1492 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22627671C211yesmaximal, characteristic, complemented
2C22×Q8×D293C211yesmaximal, characteristic, complemented
3Q8×D1172C211yesmaximal, characteristic, complemented
4D4×Dic586C213yesmaximal, complemented
5Q8×C22⋊D293C211yesmaximal, characteristic, complemented
6#34036792C213yesmaximal, complemented
7#34036787C213yesmaximal, complemented
8#34036786C211yesmaximal, characteristic, complemented
9#34036785C213yesmaximal, complemented
10#34036702C213yesmaximal, complemented
11#34036699C213yesmaximal, complemented
12#34036639C213yesmaximal, complemented
13#34036638C213yesmaximal, complemented
14#34036605C211yesmaximal, characteristic, complemented
15#548×C293C211yesmaximal, characteristic, complemented
16#22627671—28no—
17C2×#17989802C2213yes—
18C2×#17989801C2213yes—
19C2×#17989801C2213yescomplemented
20C22×Dic586C2213yes—
21C22×Dic586C2213yescomplemented
22C2×Q8×D293—28no—
23C2×Q8×D293C2211yescharacteristic
24C2×Q8×D293C2213yes—
25C2×Q8×D293C2213yes—
26C2×Q8×D293C2211yescharacteristic
27C2×Q8×D293C2211yescharacteristic, complemented
28C22×C4×D293C2213yes—
29#22627624C2213yescomplemented
30#22627623C2216yescomplemented
31#22627622C2213yescomplemented
32#22627611C2213yescomplemented
33#22627610C2216yescomplemented
34#22627609C2213yes—
35#22627606C2213yes—
36#22627606C2216yes—
37#22627604C2213yes—
38#22627604C2213yescomplemented
39#22627603C2213yes—
40#22627603C2213yescomplemented
41#22627602C2213yes—
42#22627601C2213yes—
43#22627601C2216yescomplemented
44#22627601C2216yescomplemented
45#22627600C2213yescomplemented
46#22627597C2213yes—
47#22627596C2213yes—
48#22627595C2213yes—
49#22627594C2211yescharacteristic
50#22627592C2213yes—
51Q8×Dic293C2211yescharacteristic, complemented
52C4×Dic586C2213yescomplemented
53#22627589C2213yescomplemented
54C4×D1172C2213yes—
55D4×Dic293C2216yes—
56C4×C22⋊D293C2213yes—
57C4×C22⋊D293C2213yes—
58#22627584C2216yes—
59#22627582C2213yes—
60#22627581C2213yes—
61C2×#41×C293C2211yescharacteristic, complemented
62C22×Q8×C293C2211yescharacteristic
63#124×C293C2213yescomplemented
64C22⋊Q8×C293C2213yescomplemented
65#121×C293C2213yes—
66C4×Q8×C293C2211yescharacteristic, complemented
67C4×D4×C293C2213yes—
68C23×D293C2311yescharacteristic
69#17989802—212no—
70#17989801—212no—
71#17989801—212no—
72C2×Dic586—212no—
73C2×Dic586—212no—
74C2×Dic586C2313yes—
75C2×Dic586C2316yes—
76C2×Dic586C2313yes—
77C2×Dic586C2313yes—
78C2×Dic586C2313yes—
79Q8×D293—212no—
80Q8×D293—212no—
81Q8×D293—24no—
82Q8×D293—24no—
83Q8×D293D418yescomplemented
84C2×D1172C2311yescharacteristic
85C2×D1172C2311yescharacteristic
86C2×C22⋊D293C2313yes—
87C2×C22⋊D293C2311yescharacteristic
88C2×C22⋊D293C2313yes—
89C2×C22⋊D293C2313yes—
90C22×Dic293C2316yes—
91C22×Dic293C2313yes—
92C2×C4×D293—212no—
93C2×C4×D293C2313yes—
94C2×C4×D293C2313yes—
95C2×C4×D293C2313yes—
96C2×C4×D293C2313yes—
97C2×C4×D293C2313yes—
98C4⋊Dic293C2316yes—
99C4⋊Dic293C2313yes—
100C4⋊Dic293C2313yes—
101#17989781C2316yes—
102#17989781C2313yes—
103#17989781C2313yes—
104#17989781C2313yes—
105#17989781C2313yes—
106C22⋊Dic293C2313yes—
107C22⋊Dic293C2313yes—
108C22⋊Dic293C2316yes—
109#17989779C2313yes—
110#17989779C2311yescharacteristic
111#17989779C2313yes—
112#17989779C2313yes—
113#17989779C2313yes—
114#17989779C2311yescharacteristic
115C4×Dic293C2316yes—
116C4×Dic293C2313yes—
117#41×C293—24no—
118C2×Q8×C293—24no—
119C2×Q8×C293C2313yes—
120C2×Q8×C293C2311yescharacteristic
121C2×D4×C293C2313yes—
122C2×D4×C293C2313yes—
123C22×C4×C293C2313yes—
124C22×C4×C293C2313yes—
125C4⋊C4×C293C2313yes—
126C4⋊C4×C293C2313yes—
127C22⋊C4×C293C2316yes—
128C22⋊C4×C293C2311yescharacteristic
129C22⋊C4×C293C2313yes—
130C42×C293C2313yes—
131C22×D293—22no—
132C22×D293—22no—
133C22×D293—22no—
134C22×D293C2411yescharacteristic
135C22×D293C2411yescharacteristic
136C22×D293C2411yescharacteristic
137Dic586—26no—
138Dic586—26no—
139Dic586—212no—
140Dic586—26no—
141Dic586C2×D4112yes—
142D1172—22no—
143D1172—22no—
144C22⋊D293—22no—
145C22⋊D293—26no—
146C22⋊D293—26no—
147C22⋊D293—26no—
148C2×Dic293—26no—
149C2×Dic293—212no—
150C2×Dic293C2411yescharacteristic
151C2×Dic293C2413yes—
152C2×Dic293C2413yes—
153C2×Dic293C2413yes—
154C2×Dic293C2413yes—
155C4×D293—26no—
156C4×D293—26no—
157C4×D293—26no—
158C4×D293—26no—
159C4×D293C2×D4112yes—
160C23×C293C2413yes—
161C23×C293C2411yescharacteristic
162Q8×C293—26no—
163Q8×C293C2×D414yes—
164D4×C293—26no—
165D4×C293—26no—
166C2×C4×C293—26no—
167C2×C4×C293—26no—
168C2×C4×C293C2411yescharacteristic
169C2×C4×C293C2413yes—
170C2×C4×C293C2413yes—
171C2×C4×C293C2411yescharacteristic
172C2×C4×C293C2413yes—
173C22×C293—21no—
174C22×C293—21no—
175C22×C293—23no—
176C22×C293—23no—
177C22×C293—23no—
178C22×C2932−1+412yescomplemented
179C22×C293C2511yescharacteristic
180C2×D293—28no—
181C2×D293—21no—
182C2×D293—21no—
183C2×D293—21no—
184C2×D293—21no—
185C2×D293—21no—
186C2×D293—21no—
187C2×D293—21no—
188C2×D293C22×D412yes—
189C2×D2932−1+412yescomplemented
190C4×C293—21no—
191C4×C293—23no—
192C4×C293—23no—
193C4×C293—21no—
194C4×C293C22×D416yes—
195Dic293—23no—
196Dic293—23no—
197Dic293—23no—
198Dic293—21no—
199Dic293C22×D416yes—
200C2×C293—22no—
201C2×C293—43no—
202C2×C293C23×D411yescharacteristic
203C2×C293C2×2−1+411yescharacteristic
204C2×C293C2×2−1+411yescharacteristic
205D293—22no—
206D293—22no—
207D293—41no—
208C293#348611yescharacteristic, complemented, Sylow
209#3486—2931nomaximal, Sylow
210C2×2−1+4—2931no—
211C23×Q8—2931no—
212D4×Q8—2931no—
213D4×Q8—2933no—
214D4×Q8—2931no—
215D4×Q8—2933no—
216#548—2931no—
217#548—2933no—
218#548—2933no—
219#548D29311yescharacteristic, complemented
220#536—2933no—
221#536—2933no—
222#525—2933no—
223#525—2933no—
224#519—2931no—
2252−1+4—5868no—
226C2×#41—2933no—
227C2×#41—2933no—
228C2×#41—2933no—
229C2×#41C2×D29311yescharacteristic, complemented
230C22×Q8—2931no—
231C22×Q8—2933no—
232C22×Q8—2933no—
233C22×Q8—2931no—
234C22×Q8—2931no—
235C22×Q8—2933no—
236C22×Q8—2933no—
237C22×Q8—5868no—
238C22×Q8C2×D29311yescharacteristic
239C23×C4—2933no—
240C4⋊Q8—2933no—
241C4⋊Q8—2936no—
242C4⋊Q8—2933no—
243#124—2933no—
244#124—2936no—
245#124C2×D29313yescomplemented
246#123—2933no—
247#123—2936no—
248#123—2933no—
249C22⋊Q8—2933no—
250C22⋊Q8—2933no—
251C22⋊Q8—2933no—
252C22⋊Q8—2933no—
253C22⋊Q8—2933no—
254C22⋊Q8—2933no—
255C22⋊Q8—2933no—
256C22⋊Q8—2936no—
257C22⋊Q8—2936no—
258C22⋊Q8C2×D29313yescomplemented
259#121—2933no—
260#121—2933no—
261#121—2933no—
262#121C2×D29313yes—
263#120—2933no—
264#120—2931no—
265C4×Q8—2931no—
266C4×Q8—2933no—
267C4×Q8—2933no—
268C4×Q8C2×D29311yescharacteristic, complemented
269C4×D4—2936no—
270C4×D4—2933no—
271C4×D4—2933no—
272C4×D4—2936no—
273C4×D4—2933no—
274C4×D4C2×D29313yes—
275#117—2933no—
276#117—2933no—
277C24—2931no—
278#41—24no—
279#41—58612no—
280#41—58612no—
281#41—58612no—
282C2×Q8—24no—
283C2×Q8—2938no—
284C2×Q8—2933no—
285C2×Q8—2936no—
286C2×Q8—2933no—
287C2×Q8—2933no—
288C2×Q8—2933no—
289C2×Q8—58612no—
290C2×Q8—58612no—
291C2×Q8—58612no—
292C2×Q8—58612no—
293C2×Q8—5864no—
294C2×Q8—5864no—
295C2×Q8C22×D29313yes—
296C2×Q8C22×D29311yescharacteristic
297C2×D4—2933no—
298C2×D4—2931no—
299C2×D4—2933no—
300C2×D4—2933no—
301C2×D4—2931no—
302C2×D4—2931no—
303C2×D4C22×D29313yes—
304C2×D4C22×D29313yes—
305C22×C4—2933no—
306C22×C4—2933no—
307C22×C4—2936no—
308C22×C4—2933no—
309C22×C4—2933no—
310C22×C4—2933no—
311C22×C4—2933no—
312C22×C4—58612no—
313C22×C4C22×D29313yes—
314C22×C4C22×D29313yes—
315C4⋊C4—2933no—
316C4⋊C4—2933no—
317C4⋊C4—2933no—
318C4⋊C4—2933no—
319C4⋊C4—2933no—
320C4⋊C4—2936no—
321C4⋊C4—2936no—
322C4⋊C4—2933no—
323C4⋊C4C22×D29313yes—
324C4⋊C4C22×D29313yes—
325C22⋊C4—2933no—
326C22⋊C4—2936no—
327C22⋊C4—2933no—
328C22⋊C4—2933no—
329C22⋊C4—2931no—
330C22⋊C4—2931no—
331C22⋊C4—2933no—
332C22⋊C4—2933no—
333C22⋊C4—2933no—
334C22⋊C4C22×D29316yes—
335C22⋊C4C22×D29311yescharacteristic
336C22⋊C4C22×D29313yes—
337C42—2936no—
338C42—2933no—
339C42C22×D29313yes—
340C23—2931no—
341C23—2931no—
342C23—2931no—
343C23—5862no—
344C23—5862no—
345C23—5862no—
346C23C23×D29313yes—
347C23C23×D29311yescharacteristic
348Q8—26no—
349Q8—29312no—
350Q8—58612no—
351Q8—5866no—
352Q8—5866no—
353Q8—5866no—
354Q8D4×D29314yes—
355D4—26no—
356D4—26no—
357D4—5862no—
358D4—5862no—
359D4—5866no—
360D4—5866no—
361D4—5866no—
362D4—5862no—
363C2×C4—26no—
364C2×C4—26no—
365C2×C4—2933no—
366C2×C4—29312no—
367C2×C4—2933no—
368C2×C4—2931no—
369C2×C4—2933no—
370C2×C4—2933no—
371C2×C4—5866no—
372C2×C4—5866no—
373C2×C4—5866no—
374C2×C4—58612no—
375C2×C4—5866no—
376C2×C4—5866no—
377C2×C4C23×D29313yes—
378C2×C4C23×D29311yescharacteristic
379C2×C4C23×D29313yes—
380C2×C4C23×D29311yescharacteristic
381C2×C4C23×D29313yes—
382C22—21no—
383C22—21no—
384C22—23no—
385C22—23no—
386C22—23no—
387C22—2932no—
388C22—2932no—
389C22—5868no—
390C22—5861no—
391C22—5861no—
392C22—5861no—
393C22—5861no—
394C22—5861no—
395C22—5861no—
396C22—5861no—
397C22#2262767112yescomplemented
398C22C24×D29311yescharacteristic
399C4—21no—
400C4—23no—
401C4—21no—
402C4—23no—
403C4—2936no—
404C4—5863no—
405C4—5863no—
406C4—5863no—
407C4—5861no—
408C4C2×D4×D29316yes—
409C2—22no—
410C2—43no—
411C2—5862no—
412C2—5862no—
413C2—11721no—
414C2C22×D4×D29311yescharacteristic
415C22−1+4×D29311yescharacteristic
416C2C2×#2262767111yescharacteristic
subgroupG/H
center, FrattiniC22C24×D293
commutator, socle, cosocleC22×C293C25
Fitting#548×C293C2
hypercenter#548D293
hypocenterC293#3486
LCSG ▷ C22×C293 ▷ C293
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#22627671C2semidirect
2. C22×Q8×D293C2semidirect
3. Q8×D1172C2semidirect
4. D4×Dic586C2semidirect
5. Q8×C22⋊D293C2semidirect
6. #34036792C2semidirect
7. #34036787C2semidirect
8. #34036786C2semidirect
9. #34036785C2semidirect
10. #34036702C2semidirect
11. #34036699C2semidirect
12. #34036639C2semidirect
13. #34036638C2semidirect
14. #34036605C2semidirect
15. #548×C293C2semidirect
19. C2×#17989801C22semidirect
21. C22×Dic586C22semidirect
27. C2×Q8×D293C22semidirect
29. #22627624C22semidirect
30. #22627623C22semidirect
31. #22627622C22semidirect
32. #22627611C22semidirect
33. #22627610C22semidirect
38. #22627604C22semidirect
40. #22627603C22semidirect
43. #22627601C22semidirect
44. #22627601C22semidirect
45. #22627600C22semidirect
51. Q8×Dic293C22semidirect
52. C4×Dic586C22semidirect
53. #22627589C22semidirect
61. C2×#41×C293C22semidirect
63. #124×C293C22semidirect
64. C22⋊Q8×C293C22semidirect
66. C4×Q8×C293C22semidirect
83. Q8×D293D4semidirect
178. C22×C2932−1+4semidirect
189. C2×D2932−1+4semidirect
208. C293#3486semidirect
219. #548D293semidirect
229. C2×#41C2×D293semidirect
245. #124C2×D293semidirect
258. C22⋊Q8C2×D293semidirect
268. C4×Q8C2×D293semidirect
397. C22#22627671semidirect

Maximal subgroups

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

subgroupindex
C2×#226276712
C22×Q8×D2932
Q8×D11722
D4×Dic5862
Q8×C22⋊D2932
#340367922
#340367872
#340367862
#340367852
#340367022
#340366992
#340366392
#340366382
#340366052
#548×C2932
#3486293