Finite Group Databasebeta

← #34057500

Subgroups — #34057500

Order 19072.

350 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23750357C211yesmaximal, characteristic
2Q8×Dic298C211yesmaximal, characteristic, complemented
3#23749986C211yesmaximal, characteristic, complemented
4#23749984C211yesmaximal, characteristic, complemented
5#23749970C211yesmaximal, characteristic
6#23749918C211yesmaximal, characteristic, complemented
7#23749906C211yesmaximal, characteristic, complemented
8#23749903C211yesmaximal, characteristic, complemented
9#23749849C211yesmaximal, characteristic, complemented
10#23749823C211yesmaximal, characteristic
11#23749754C211yesmaximal, characteristic, complemented
12#23749726C211yesmaximal, characteristic, complemented
13#23749723C211yesmaximal, characteristic, complemented
14Q16×Dic149C211yesmaximal, characteristic, complemented
15#464×C149C211yesmaximal, characteristic, complemented
16C2×#14558723C2211yescharacteristic
17#18011911—21no—
18#18011911C2211yescharacteristic
19#18011910—21no—
20#18011910—21no—
21#18011910C2211yescharacteristic
22#18011910C2211yescharacteristic
23#18011907C2211yescharacteristic
24#18011900C2211yescharacteristic
25#18011897—21no—
26#18011895—21no—
27#18011893—21no—
28Q8×Dic149C2211yescharacteristic
29Q8×Dic149C2211yescharacteristic
30Q8×Dic149C2211yescharacteristic
31C4×Dic298—21no—
32C4×Dic298C2211yescharacteristic
33C4×D596C2211yescharacteristic
34D4×Dic149C2211yescharacteristic
35#18011871—21no—
36#18011845C2211yescharacteristic
37#18011845C2211yescharacteristic
38#18011844C2211yescharacteristic
39#18011840C2211yescharacteristic
40#18011840C2211yescharacteristic
41#18011835C2211yescharacteristic
42#18011835C2211yescharacteristic
43#18011835C2211yescharacteristic
44#18011833C2211yescharacteristic
45#18011833C2211yescharacteristic
46#18011832C2211yescharacteristic
47#18011832C2211yescharacteristic
48#18011831C2211yescharacteristic
49#18011822C2211yescharacteristic
50#18011821C2211yescharacteristic
51C8×Dic149C2211yescharacteristic
52C2×Q16×C149C2211yescharacteristic
53C2×SD16×C149C2211yescharacteristic
54#124×C149C2211yescharacteristic
55C4×Q8×C149C2211yescharacteristic
56C4⋊C8×C149C2211yescharacteristic
57#103×C149C2211yescharacteristic
58#102×C149C2211yescharacteristic
59C2×Dic298—21no—
60C2×Dic298—21no—
61C2×Dic298—21no—
62C2×Dic298D411yescharacteristic
63C2×D596C2311yescharacteristic
64C2×C22⋊D149—21no—
65C2×C22⋊D149—21no—
66C22×Dic149—21no—
67C2×C4×D149—21no—
68#14558723—22no—
69C2×#11924746C2311yescharacteristic
70C2×#11924746C2311yescharacteristic
71C4⋊Dic149—21no—
72C4⋊Dic149—21no—
73C4⋊Dic149—21no—
74C4⋊Dic149—21no—
75C4⋊Dic149D411yescharacteristic
76C4⋊Dic149C2311yescharacteristic
77C4⋊Dic149C2311yescharacteristic
78C4⋊Dic149C2311yescharacteristic
79#14558715—21no—
80#14558715—21no—
81#14558715D411yescharacteristic
82#14558715D411yescharacteristic
83C22⋊Dic149—21no—
84#14558713—21no—
85#14558713—21no—
86#14558713—21no—
87C4×Dic149—21no—
88C4×Dic149—21no—
89C4×Dic149—21no—
90C4×Dic149C2311yescharacteristic
91C4×Dic149C2311yescharacteristic
92C2×Q8×C149C2311yescharacteristic
93C2×Q8×C149C2311yescharacteristic
94C2×D4×C149C2311yescharacteristic
95Q16×C149—22no—
96SD16×C149—22no—
97C2×C8×C149C2311yescharacteristic
98C2×C8×C149C2311yescharacteristic
99C4⋊C4×C149—21no—
100C4⋊C4×C149C2311yescharacteristic
101C22⋊C4×C149—21no—
102C22⋊C4×C149—21no—
103C42×C149—21no—
104C42×C149C2311yescharacteristic
105C22×D149—21no—
106Dic298—22no—
107Dic298—81no—
108Dic298—81no—
109Dic298—81no—
110D596—21no—
111D596—21no—
112C22⋊D149—42no—
113C22⋊D149—81no—
114C2×Dic149—21no—
115C2×Dic149—21no—
116C2×Dic149—21no—
117C2×Dic149—81no—
118C2×Dic149C2×D411yescharacteristic
119C2×Dic149C2×D411yescharacteristic
120C2×Dic149C2×D411yescharacteristic
121C2×Dic149C2×D411yescharacteristic
122C4×D149—81no—
123#11924746—21no—
124#11924746—21no—
125C23×C149—21no—
126Q8×C149—21no—
127Q8×C149—21no—
128Q8×C149—21no—
129Q8×C149#4112yes—
130D4×C149—21no—
131D4×C149—21no—
132C2×C4×C149—21no—
133C2×C4×C149—21no—
134C2×C4×C149—21no—
135C2×C4×C149C2×D411yescharacteristic
136C2×C4×C149C2×D411yescharacteristic
137C2×C4×C149C2411yescharacteristic
138C8×C149—21no—
139C8×C149—21no—
140C22×C149—41no—
141C22×C149—41no—
142C22×C149—41no—
143C22×C149C22×D411yescharacteristic
144C2×D149—41no—
145C2×D149—41no—
146C2×D149—41no—
147C4×C149—22no—
148C4×C149—21no—
149C4×C149—21no—
150C4×C149—41no—
151C4×C149—41no—
152C4×C149C2×#4111yescharacteristic
153C4×C1492−1+411yescharacteristic
154Dic149—21no—
155Dic149—21no—
156Dic149—21no—
157Dic149—41no—
158Dic149—41no—
159Dic149—41no—
160Dic149#13712yes—
161C2×C149—81no—
162C2×C149#54811yescharacteristic
163C2×C149C2×#13711yescharacteristic
164C2×C149#57611yescharacteristic
165D149—81no—
166C149#337511yescharacteristic, complemented, Sylow
167#3375—1491nomaximal, Sylow
168#559—1491no—
169Q82—1491no—
170#489—1491no—
171#489—1491no—
172#486—1491no—
173#477—1491no—
174#474—1491no—
175#474—1491no—
176#464—1491no—
177#464D14911yescharacteristic, complemented
178#460—1491no—
179#446—1491no—
180#440—1491no—
181#440—1491no—
182C4×Q16—1491no—
183C2×Q16C2×D14911yescharacteristic
184C2×SD16—1491no—
185C2×SD16C2×D14911yescharacteristic
186C4⋊Q8—2981no—
187C4⋊Q8—2981no—
188C4⋊Q8—2981no—
189C4⋊Q8—1491no—
190C4⋊Q8—1491no—
191C4⋊Q8—1491no—
192#127—1491no—
193#124—2981no—
194#124—1491no—
195#124C2×D14911yescharacteristic
196#123—2981no—
197#123—2981no—
198C4×Q8—2981no—
199C4×Q8—1491no—
200C4×Q8—1491no—
201C4×Q8—1491no—
202C4×Q8—1491no—
203C4×Q8C2×D14911yescharacteristic
204C4×D4—2981no—
205C4×D4—1491no—
206C4×D4—1491no—
207#107—1491no—
208#107—1491no—
209#107—1491no—
210C4⋊C8—1491no—
211C4⋊C8—1491no—
212C4⋊C8C2×D14911yescharacteristic
213#103—1491no—
214#103—1491no—
215#103—1491no—
216#103C2×D14911yescharacteristic
217#102—1491no—
218#102—1491no—
219#102—1491no—
220#102—1491no—
221#102—1491no—
222#102C2×D14911yescharacteristic
223#97—1491no—
224#97—1491no—
225C4×C8—1491no—
226C2×Q8—2981no—
227C2×Q8—2981no—
228C2×Q8—2981no—
229C2×Q8—1491no—
230C2×Q8C22×D14911yescharacteristic
231C2×Q8C22×D14911yescharacteristic
232C2×D4—2981no—
233C2×D4—2981no—
234C2×D4—1491no—
235C2×D4C22×D14911yescharacteristic
236C22×C4—2981no—
237C22×C4—2981no—
238Q16—22no—
239SD16—22no—
240SD16—2982no—
241C2×C8—1491no—
242C2×C8—1491no—
243C2×C8C22×D14911yescharacteristic
244C2×C8C22×D14911yescharacteristic
245C4⋊C4—21no—
246C4⋊C4—2981no—
247C4⋊C4—2981no—
248C4⋊C4—2981no—
249C4⋊C4—2981no—
250C4⋊C4—2981no—
251C4⋊C4—2981no—
252C4⋊C4—1491no—
253C4⋊C4—1491no—
254C4⋊C4—1491no—
255C4⋊C4—1491no—
256C4⋊C4—1491no—
257C4⋊C4—1491no—
258C4⋊C4C22×D14911yescharacteristic
259C22⋊C4—21no—
260C22⋊C4—21no—
261C22⋊C4—2981no—
262C22⋊C4—2981no—
263C22⋊C4—2981no—
264C22⋊C4—2981no—
265C42—21no—
266C42—2981no—
267C42—2981no—
268C42—2981no—
269C42—1491no—
270C42—1491no—
271C42C22×D14911yescharacteristic
272C23—21no—
273C23—2981no—
274Q8—21no—
275Q8—21no—
276Q8—21no—
277Q8—2982no—
278Q8—11921no—
279Q8—11921no—
280Q8—11921no—
281Q8#1455873512yes—
282D4—21no—
283D4—21no—
284D4—2981no—
285D4—2981no—
286D4—5962no—
287D4—11921no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—2981no—
292C2×C4—2981no—
293C2×C4—2981no—
294C2×C4—1491no—
295C2×C4—1491no—
296C2×C4—1491no—
297C2×C4—1491no—
298C2×C4—11921no—
299C2×C4—11921no—
300C2×C4D4×D14911yescharacteristic
301C2×C4D4×D14911yescharacteristic
302C2×C4C23×D14911yescharacteristic
303C8—21no—
304C8—21no—
305C8—2981no—
306C8—2981no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—5961no—
311C22—5961no—
312C22—5961no—
313C22C2×D4×D14911yescharacteristic
314C4—21no—
315C4—21no—
316C4—22no—
317C4—41no—
318C4—41no—
319C4—2981no—
320C4—2981no—
321C4—2981no—
322C4—5961no—
323C4—5961no—
324C4—5961no—
325C4—1492no—
326C4C2×#1455873511yescharacteristic
327C4#1801195811yescharacteristic
328C2—81no—
329C2—11921no—
330C2#2375028111yescharacteristic
331C2#137×D14911yescharacteristic
332C2#2375042711yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#464×C149C2
socleC22×C149C22×D4
prime radical#18011907C22
hypercenter#464D149
hypocenterC149#3375
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
2. Q8×Dic298C2semidirect
3. #23749986C2semidirect
4. #23749984C2semidirect
6. #23749918C2semidirect
7. #23749906C2semidirect
8. #23749903C2semidirect
9. #23749849C2semidirect
11. #23749754C2semidirect
12. #23749726C2semidirect
13. #23749723C2semidirect
14. Q16×Dic149C2semidirect
15. #464×C149C2semidirect
166. C149#3375semidirect
177. #464D149semidirect

Maximal subgroups

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

subgroupindex
#237503572
Q8×Dic2982
#237499862
#237499842
#237499702
#237499182
#237499062
#237499032
#237498492
#237498232
#237497542
#237497262
#237497232
Q16×Dic1492
#464×C1492
#3375149