Finite Group Databasebeta

← #34057601

Subgroups — #34057601

Order 19072.

340 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×Dic298C211yesmaximal, characteristic, complemented
2#23750035C212yesmaximal, complemented
3#23750028C211yesmaximal, characteristic, complemented
4#23749929C212yesmaximal, complemented
5#23749917C211yesmaximal, characteristic, complemented
6#23749915C211yesmaximal, characteristic
7#23749905C212yesmaximal
8#23749756C211yesmaximal, characteristic, complemented
9#23749725C212yesmaximal
10C4×#14558724C211yesmaximal, characteristic
11#554×C149C211yesmaximal, characteristic
12C2×#14558724C2211yescharacteristic
13#18011911C2212yes—
14#18011911C2211yescharacteristic
15#18011910—21no—
16#18011910—22no—
17Q8×Dic149—21no—
18C4×Dic298C2211yescharacteristic
19C4×Dic298C2212yes—
20#18011844C2212yes—
21#18011844C2212yes—
22#18011844C2212yes—
23#18011842C2212yes—
24#18011842C2211yescharacteristic
25C4⋊#11924746C2212yes—
26C4⋊#11924746C2211yescharacteristic
27#18011834C2212yes—
28#18011834C2211yescharacteristic
29#18011832C2211yescharacteristic
30#18011832C2212yes—
31#18011822C2212yes—
32C4×#11924746C2211yescharacteristic
33C4⋊Q8×C149C2211yescharacteristic
34#125×C149C2212yes—
35C22⋊Q8×C149—21no—
36C22⋊Q8×C149—22no—
37C4×Q8×C149C2211yescharacteristic
38C4×D4×C149C2212yes—
39C4×D4×C149C2211yescharacteristic
40C2×C4⋊C4×C149—21no—
41C2×Dic298—21no—
42C2×Dic298—22no—
43C2×Dic298C2311yescharacteristic
44#14558724Q814yescomplemented
45C2×#11924746C2311yescharacteristic
46C2×#11924746C2312yes—
47C2×#11924746C2311yescharacteristic
48C4⋊Dic149—22no—
49C4⋊Dic149—21no—
50C4⋊Dic149C2311yescharacteristic
51C4⋊Dic149C2312yes—
52#14558715—21no—
53#14558715—22no—
54C4×Dic149—22no—
55C4×Dic149—21no—
56C2×Q8×C149—21no—
57C2×Q8×C149D411yescharacteristic
58C2×D4×C149C2311yescharacteristic
59C22×C4×C149—21no—
60C22×C4×C149—22no—
61C4⋊C4×C149—22no—
62C4⋊C4×C149—21no—
63C4⋊C4×C149—22no—
64C4⋊C4×C149D411yescharacteristic
65C4⋊C4×C149D412yes—
66C4⋊C4×C149C2311yescharacteristic
67C4⋊C4×C149C2312yes—
68C22⋊C4×C149—21no—
69C22⋊C4×C149—22no—
70C42×C149C2312yes—
71C42×C149C2311yescharacteristic
72Dic298—21no—
73Dic298—82no—
74Dic298—81no—
75Dic298C2×Q812yes—
76C2×Dic149—21no—
77C2×Dic149—22no—
78C2×Dic149—21no—
79#11924746—21no—
80#11924746—22no—
81#11924746C2×Q812yes—
82C23×C149—21no—
83Q8×C149—22no—
84Q8×C149—81no—
85D4×C149—21no—
86D4×C149C2×Q812yes—
87C2×C4×C149—21no—
88C2×C4×C149—22no—
89C2×C4×C149—42no—
90C2×C4×C149—82no—
91C2×C4×C149C2×D411yescharacteristic
92C2×C4×C149C2×D412yes—
93C2×C4×C149C2×D412yes—
94C2×C4×C149C2×D411yescharacteristic
95C2×C4×C149C2411yescharacteristic
96C22×C149—22no—
97C22×C149—41no—
98C22×C149—41no—
99C22×C149C22×D411yescharacteristic
100C4×C149—22no—
101C4×C149—22no—
102C4×C149—21no—
103C4×C149—41no—
104C4×C149—42no—
105C4×C149#13712yes—
106C4×C149C22×Q811yescharacteristic
107C4×C1492−1+411yescharacteristic
108Dic149—22no—
109Dic149—41no—
110Dic149—42no—
111C2×C149—42no—
112C2×C149D4×Q811yescharacteristic
113C2×C149C2×#13711yescharacteristic
114C2×C149#57611yescharacteristic
115C149#338511yescharacteristic, complemented, Sylow
116#3385—1491nomaximal, Sylow
117Q82—1491no—
118#554D14911yescharacteristic
119#501—1492no—
120#500—1491no—
121#479—1492no—
122#477—1491no—
123#476—1491no—
124#474—1492no—
125#446—1491no—
126#440—1492no—
127C4×SD16—1491no—
128C2×SD16—1491no—
129C4⋊Q8—2981no—
130C4⋊Q8—2982no—
131C4⋊Q8—1492no—
132C4⋊Q8—1491no—
133C4⋊Q8C2×D14911yescharacteristic
134#125C2×D14912yes—
135C22⋊Q8—21no—
136C22⋊Q8—22no—
137C4×Q8—2981no—
138C4×Q8—1491no—
139C4×Q8—1492no—
140C4×Q8C2×D14911yescharacteristic
141C4×D4C2×D14911yescharacteristic
142C4×D4C2×D14912yes—
143C2×C4⋊C4—21no—
144#107—1492no—
145#107—1492no—
146#107—1492no—
147#106—1491no—
148#106—1492no—
149C4⋊C8—1492no—
150C4⋊C8—1491no—
151#103—1491no—
152#103—1492no—
153#102—1491no—
154#102—1492no—
155#97—1492no—
156C4×C8—1491no—
157C2×Q8—21no—
158C2×Q8—2981no—
159C2×Q8—2982no—
160C2×Q8—1491no—
161C2×Q8C22⋊D14911yescharacteristic
162C2×D4C22×D14911yescharacteristic
163C22×C4—21no—
164C22×C4—22no—
165SD16—1494no—
166C2×C8—1491no—
167C2×C8—1492no—
168C2×C8—1491no—
169C4⋊C4—22no—
170C4⋊C4—21no—
171C4⋊C4—22no—
172C4⋊C4—2982no—
173C4⋊C4—2981no—
174C4⋊C4—2981no—
175C4⋊C4—2982no—
176C4⋊C4—1491no—
177C4⋊C4—1492no—
178C4⋊C4C22⋊D14911yescharacteristic
179C4⋊C4C22⋊D14912yes—
180C4⋊C4C22×D14911yescharacteristic
181C4⋊C4C22×D14912yes—
182C22⋊C4—21no—
183C22⋊C4—22no—
184C42—2982no—
185C42—2981no—
186C42C22×D14911yescharacteristic
187C42C22×D14912yes—
188C23—21no—
189Q8—22no—
190Q8—81no—
191Q8—2981no—
192Q8—1492no—
193Q8—11922no—
194Q8—11921no—
195D4—21no—
196D4Q8×D14912yes—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—42no—
200C2×C4—82no—
201C2×C4—2981no—
202C2×C4—2982no—
203C2×C4—2981no—
204C2×C4C2×C22⋊D14912yes—
205C2×C4C2×C22⋊D14911yescharacteristic
206C2×C4C2×C22⋊D14912yes—
207C2×C4C2×C22⋊D14911yescharacteristic
208C2×C4C23×D14911yescharacteristic
209C8—2981no—
210C8—2982no—
211C8—1492no—
212C22—22no—
213C22—41no—
214C22—41no—
215C22C22×C22⋊D14911yescharacteristic
216C4—22no—
217C4—22no—
218C4—21no—
219C4—41no—
220C4—42no—
221C4—2982no—
222C4—5961no—
223C4—5962no—
224C4#1801194112yes—
225C4C2×Q8×D14911yescharacteristic
226C4#1801195911yescharacteristic
227C2—42no—
228C2Q8×C22⋊D14911yescharacteristic
229C2C2×#1801194111yescharacteristic
230C2#2375042811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#554×C149C2
socleC22×C149C22×D4
prime radicalC2×D4×C149C23
hypercenter#554D149
hypocenterC149#3385
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #554

Complements

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

NH ≅ G/Ntype
1. Q8×Dic298C2semidirect
2. #23750035C2semidirect
3. #23750028C2semidirect
4. #23749929C2semidirect
5. #23749917C2semidirect
8. #23749756C2semidirect
44. #14558724Q8semidirect
115. C149#3385semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic2982
#237500352
#237500282
#237499292
#237499172
#237499152
#237499052
#237497562
#237497252
C4×#145587242
#554×C1492
#3385149