Finite Group Databasebeta

← #33114099

Subgroups — #33114099

Order 18688.

1074 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22618520C212yesmaximal
2#22612555C211yesmaximal, characteristic
3#22611278C214yesmaximal, complemented
4#22611273C214yesmaximal, complemented
5C2×#17988356C211yesmaximal, characteristic
6#22609640C212yesmaximal
7C2×#544×C73C211yesmaximal, characteristic
8C22×#11924012C2212yes—
9C2×#14556550C2211yescharacteristic
10C2×#14556550C2211yescharacteristic
11#17988630C2214yes—
12#17988630C2214yes—
13#17988627—22no—
14#17988627—22no—
15#17988623—24no—
16#17988483—22no—
17#17988460—22no—
18#17988456C2212yes—
19#17988452C2212yes—
20#17988356C2214yes—
21#17988346C2218yes—
22#544×C73—24no—
23#544×C73—22no—
24#544×C73C2214yes—
25C2×C4⋊Q8×C73—22no—
26C2×#125×C73—22no—
27C2×C22⋊Q8×C73—22no—
28C2×#117×C73C2212yes—
29C22×C4⋊C4×C73C2211yescharacteristic
30C2×#11924012—22no—
31C2×#11924012—24no—
32C2×#11924012Q818yescomplemented
33C2×#11924012C2314yes—
34C22×#10596831—22no—
35#14556550—22no—
36#14556550—22no—
37#14556550—22no—
38#14556550Q814yescomplemented
39C2×C4⋊Dic73—21no—
40C2×C4⋊Dic73C2311yescharacteristic
41C2×C4⋊Dic73C2311yescharacteristic
42C2×#11924007—24no—
43C2×C22⋊Dic73—21no—
44C2×C22⋊Dic73—21no—
45C2×C4×Dic73—21no—
46C2×C4×Dic73C2312yes—
47#14556524—24no—
48#14556524—24no—
49#14556522—24no—
50#14556522—24no—
51#14556498—24no—
52C22×Q8×C73—22no—
53C23×C4×C73D412yes—
54C23×C4×C73C2311yescharacteristic
55C4⋊Q8×C73—28no—
56C4⋊Q8×C73—28no—
57#125×C73—28no—
58#125×C73—28no—
59C22⋊Q8×C73—28no—
60C22⋊Q8×C73—28no—
61#117×C73—22no—
62#117×C73—22no—
63#117×C73D414yes—
64#117×C73C2314yes—
65C2×C4⋊C4×C73—22no—
66C2×C4⋊C4×C73—24no—
67C2×C4⋊C4×C73—22no—
68C2×C4⋊C4×C73—22no—
69C2×C4⋊C4×C73D412yes—
70C2×C4⋊C4×C73D412yes—
71C2×C4⋊C4×C73D412yes—
72C2×C4⋊C4×C73C2312yes—
73C2×C22⋊C4×C73—22no—
74C2×C42×C73—22no—
75C22×Dic73—22no—
76C22×Dic73—21no—
77C22×Dic73—21no—
78#11924012—28no—
79#11924012—24no—
80#11924012C2×Q818yescomplemented
81C2×#10596831—28no—
82C2×#10596831—24no—
83C4⋊Dic73—24no—
84C4⋊Dic73—22no—
85C4⋊Dic73—22no—
86#11924007—84no—
87C22⋊Dic73—24no—
88C22⋊Dic73—41no—
89C22⋊Dic73—41no—
90C4×Dic73—22no—
91C4×Dic73—42no—
92C4×Dic73C2×Q814yescomplemented
93C24×C73C2×D411yescharacteristic
94C2×Q8×C73—28no—
95C2×Q8×C73—84no—
96C22×C4×C73—22no—
97C22×C4×C73—22no—
98C22×C4×C73—22no—
99C22×C4×C73—22no—
100C22×C4×C73—82no—
101C22×C4×C73C2×D414yes—
102C22×C4×C73C2×D411yescharacteristic
103C22×C4×C73C2×D412yes—
104C22×C4×C73C2×D412yes—
105C22×C4×C73C2×Q814yes—
106C22×C4×C73C2×Q812yes—
107C22×C4×C73#4112yes—
108C22×C4×C73C2411yescharacteristic
109C4⋊C4×C73—28no—
110C4⋊C4×C73—216no—
111C4⋊C4×C73—24no—
112C4⋊C4×C73—22no—
113C4⋊C4×C73—22no—
114C4⋊C4×C73—24no—
115C4⋊C4×C73C2×D414yes—
116C4⋊C4×C73C2×D414yes—
117C22⋊C4×C73—24no—
118C22⋊C4×C73—24no—
119C42×C73—24no—
120C42×C73—24no—
121C23×C73—41no—
122C23×C73—41no—
123C23×C73#12011yescharacteristic
124C23×C73#12011yescharacteristic
125C23×C73C22⋊Q812yes—
126C23×C73C4⋊Q812yes—
127C23×C73C22×D411yescharacteristic
128Q8×C73—88no—
129C2×C4×C73—24no—
130C2×C4×C73—24no—
131C2×C4×C73—24no—
132C2×C4×C73—44no—
133C2×C4×C73—44no—
134C2×C4×C73—44no—
135C2×C4×C73—44no—
136C2×C4×C73—82no—
137C2×C4×C73#12012yes—
138C2×C4×C73C22⋊Q814yes—
139C2×C4×C73C22⋊Q814yes—
140C2×C4×C73C22⋊Q812yes—
141C2×C4×C73C4⋊Q814yes—
142C2×C4×C73C4⋊Q814yes—
143C2×C4×C73C4⋊Q812yes—
144C2×C4×C73C22×D412yes—
145C2×C4×C73C22×Q812yes—
146C2×C4×C73C22×Q811yescharacteristic
147C2×C4×C73C2×#4111yescharacteristic
148C2×Dic73—24no—
149C2×Dic73—42no—
150C2×Dic73—42no—
151C2×Dic73—42no—
152C2×Dic73—42no—
153C2×Dic73—41no—
154C2×Dic73—41no—
155#10596831—28no—
156C22×C73—21no—
157C22×C73—21no—
158C22×C73—21no—
159C22×C73—21no—
160C22×C73—21no—
161C22×C73—21no—
162C22×C73—42no—
163C22×C73—42no—
164C22×C73#39512yes—
165C22×C73#39512yes—
166C22×C73C2×#12011yescharacteristic
167C22×C73C2×C22⋊Q811yescharacteristic
168C22×C73C2×C4⋊Q811yescharacteristic
169C4×C73—44no—
170C4×C73—44no—
171C4×C73—44no—
172C4×C73#39514yes—
173C4×C73C2×C22⋊Q812yes—
174C4×C73C2×C4⋊Q812yes—
175Dic73—44no—
176Dic73—81no—
177Dic73—81no—
178#23561—731nomaximal, Sylow
179C2×C73—21no—
180C2×C73—21no—
181C2×C73—22no—
182C2×C73—22no—
183C2×C73C2×#39511yescharacteristic
184C2×C73#301911yescharacteristic
185C2×C73#302011yescharacteristic
186C2×#544D7311yescharacteristic
187#2911—732no—
188#2317—731no—
189#2061—734no—
190#2060—734no—
191C2×#343—731no—
192#1733—732no—
193C73#2356111yescharacteristic, complemented, Sylow
194C22×M16—732no—
195#544—24no—
196#544—22no—
197#544C2×D7314yes—
198C2×C4⋊Q8—22no—
199C2×#125—22no—
200C2×C22⋊Q8—22no—
201C2×#117—731no—
202C2×#117—731no—
203C2×#117C2×D7312yes—
204C22×C4⋊C4C2×D7311yescharacteristic
205#428—734no—
206#428—734no—
207#428—1462no—
208#428—1462no—
209#427—1464no—
210#389—1462no—
211#385—1462no—
212#384—732no—
213#382—732no—
214#343—734no—
215#339—738no—
216C22×Q8—22no—
217C23×C4C22⋊D7312yes—
218C23×C4C22×D7311yescharacteristic
219C2×M16—738no—
220C2×M16—734no—
221C2×M16—1462no—
222C2×M16—1464no—
223C22×C8—1462no—
224C4⋊Q8—28no—
225C4⋊Q8—28no—
226#125—28no—
227#125—28no—
228C22⋊Q8—28no—
229C22⋊Q8—28no—
230#117—22no—
231#117—22no—
232#117—734no—
233#117—1462no—
234#117—1462no—
235#117—1462no—
236#117C22⋊D7314yes—
237#117C22×D7314yes—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—731no—
243C2×C4⋊C4—731no—
244C2×C4⋊C4—1464no—
245C2×C4⋊C4—1461no—
246C2×C4⋊C4C22⋊D7312yes—
247C2×C4⋊C4C22⋊D7312yes—
248C2×C4⋊C4C22⋊D7312yes—
249C2×C4⋊C4C22×D7312yes—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—1461no—
252C2×C22⋊C4—1461no—
253C2×C42—22no—
254C2×C42—732no—
255C2×C42—1461no—
256#107—1464no—
257#107—1464no—
258#106—1464no—
259#106—1464no—
260#95—1464no—
261C24C2×C22⋊D7311yescharacteristic
262C2×Q8—28no—
263C2×Q8—84no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—82no—
269C22×C4—1461no—
270C22×C4—1461no—
271C22×C4—1462no—
272C22×C4C2×C22⋊D7314yes—
273C22×C4C2×C22⋊D7311yescharacteristic
274C22×C4C2×C22⋊D7312yes—
275C22×C4C2×C22⋊D7312yes—
276C22×C4Q8×D7314yes—
277C22×C4Q8×D7312yes—
278C22×C4#1192402912yes—
279C22×C4C23×D7311yescharacteristic
280M16—738no—
281M16—1468no—
282M16—1464no—
283C2×C8—1468no—
284C2×C8—1464no—
285C4⋊C4—28no—
286C4⋊C4—216no—
287C4⋊C4—24no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—24no—
291C4⋊C4—5844no—
292C4⋊C4—1462no—
293C4⋊C4—1462no—
294C4⋊C4—1464no—
295C4⋊C4C2×C22⋊D7314yes—
296C4⋊C4C2×C22⋊D7314yes—
297C22⋊C4—24no—
298C22⋊C4—24no—
299C22⋊C4—2921no—
300C22⋊C4—2921no—
301C22⋊C4—1464no—
302C42—24no—
303C42—24no—
304C42—2922no—
305C42—734no—
306C42—1462no—
307C23—41no—
308C23—41no—
309C23#1455656011yescharacteristic
310C23#1455656011yescharacteristic
311C23#1455656712yes—
312C23#1455658912yes—
313C23C22×C22⋊D7311yescharacteristic
314Q8—88no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—82no—
323C2×C4—2922no—
324C2×C4—2922no—
325C2×C4—2922no—
326C2×C4—2922no—
327C2×C4—2921no—
328C2×C4—2921no—
329C2×C4—1464no—
330C2×C4#1455656012yes—
331C2×C4#1455656712yes—
332C2×C4#1455656714yes—
333C2×C4#1455656714yes—
334C2×C4#1455658914yes—
335C2×C4#1455658914yes—
336C2×C4#1455658912yes—
337C2×C4C22×C22⋊D7312yes—
338C2×C4C2×Q8×D7311yescharacteristic
339C2×C4C2×Q8×D7312yes—
340C2×C4C2×#1192402911yescharacteristic
341C8—1468no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—42no—
349C22—42no—
350C22#1798850612yes—
351C22#1798850612yes—
352C22C2×#1455656011yescharacteristic
353C22C2×#1455656711yescharacteristic
354C22C2×#1455658911yescharacteristic
355C4—44no—
356C4—44no—
357C4—44no—
358C4—2924no—
359C4—5841no—
360C4—5841no—
361C4#1798850614yes—
362C4C2×#1455656712yes—
363C4C2×#1455658912yes—
364C2—21no—
365C2—21no—
366C2—22no—
367C2—22no—
368C2C2×#1798850611yescharacteristic
369C2#2261928811yescharacteristic
370C2#2261929511yescharacteristic
subgroupG/H
centerC22C2×#14556560
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC2×#544×C73C2
socleC22×C73C2×#120
prime radicalC24×C73C2×D4
hypercenterC2×#544D73
hypocenterC73#23561
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#544

Complements

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

NH ≅ G/Ntype
3. #22611278C2semidirect
4. #22611273C2semidirect
32. C2×#11924012Q8semidirect
38. #14556550Q8semidirect
80. #11924012C2×Q8semidirect
92. C4×Dic73C2×Q8semidirect
193. C73#23561semidirect

Maximal subgroups

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

subgroupindex
#226185202
#226125552
#226112782
#226112732
C2×#179883562
#226096402
C2×#544×C732
#2356173