Finite Group Databasebeta

← #34001325

Subgroups — #34001325

Order 18688.

596 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C22⋊#10596833C211yesmaximal, characteristic
2C2×#17989453C211yesmaximal, characteristic
3#22624751C212yesmaximal, complemented
4#22624749C212yesmaximal, complemented
5C22⋊Q8×Dic73C211yesmaximal, characteristic
6C23×#10596833—21no—
7C2×C22⋊#10596833—21no—
8C2×C4×#10596833—21no—
9#17989453—21no—
10#17989453C2212yes—
11#17989453C2212yes—
12C2×Q8×Dic73C411yescharacteristic
13C22×C4×Dic73C2211yescharacteristic
14#17988494C2211yescharacteristic
15#17988486C2211yescharacteristic
16#17988473—21no—
17#17988462—21no—
18#17988447C411yescharacteristic
19C4⋊C4×Dic73—21no—
20C4⋊C4×Dic73C411yescharacteristic
21C22⋊C4×Dic73—21no—
22C2×C22⋊Q8×C73C411yescharacteristic
23C22×#10596833—24no—
24C22×#10596833—22no—
25C22×#10596833—22no—
26C22×#10596833—42no—
27C22×#10596833—42no—
28C22⋊#10596833—24no—
29C4×#10596833—42no—
30C23×Dic73D411yescharacteristic
31C23×Dic73D411yescharacteristic
32Q8×Dic73—42no—
33C2×C4⋊Dic73—21no—
34C2×C4⋊Dic73—21no—
35C2×C4⋊Dic73C2×C411yescharacteristic
36C2×C4⋊Dic73C2×C411yescharacteristic
37C2×C22⋊Dic73—21no—
38C2×C4×Dic73—21no—
39C2×C4×Dic73—21no—
40C2×C4×Dic73—42no—
41C2×C4×Dic73C2×C411yescharacteristic
42C2×C4×Dic73D411yescharacteristic
43C2×C4×Dic73D411yescharacteristic
44C2×C4×Dic73C2311yescharacteristic
45#14556498—21no—
46#14556498—21no—
47#14556498—21no—
48#14556498—21no—
49#14556498—21no—
50C22×Q8×C73C2×C411yescharacteristic
51C23×C4×C73C2×C411yescharacteristic
52C22⋊Q8×C73—22no—
53C22⋊Q8×C73C814yescomplemented
54C2×C4⋊C4×C73—21no—
55C2×C4⋊C4×C73C2×C411yescharacteristic
56C2×C22⋊C4×C73—21no—
57C2×#10596833—24no—
58C2×#10596833—24no—
59C2×#10596833—42no—
60C2×#10596833—48no—
61C2×#10596833—42no—
62C2×#10596833—42no—
63C2×#10596833—42no—
64C22×Dic73—22no—
65C22×Dic73—22no—
66C22×Dic73—22no—
67C22×Dic73—22no—
68C22×Dic73—21no—
69C22×Dic73—21no—
70C22×Dic73—41no—
71C22×Dic73C22⋊C411yescharacteristic
72C22×Dic73C22⋊C411yescharacteristic
73C22×Dic73SD1612yes—
74C22×Dic73Q1612yes—
75C22×Dic73C2×D411yescharacteristic
76C22×Dic73C2×D411yescharacteristic
77C4⋊Dic73—42no—
78C4⋊Dic73—42no—
79C4⋊Dic73—41no—
80C4⋊Dic73—41no—
81C22⋊Dic73—42no—
82C4×Dic73—22no—
83C4×Dic73—22no—
84C4×Dic73—41no—
85C4×Dic73—41no—
86C4×Dic73—42no—
87C4×Dic73—42no—
88C24×C73C22⋊C411yescharacteristic
89C2×Q8×C73—21no—
90C2×Q8×C73—41no—
91C2×Q8×C73—41no—
92C2×Q8×C73C2×C812yescomplemented
93C22×C4×C73—21no—
94C22×C4×C73—21no—
95C22×C4×C73—41no—
96C22×C4×C73—41no—
97C22×C4×C73C22⋊C411yescharacteristic
98C22×C4×C73C2×C812yes—
99C22×C4×C73M1612yes—
100C22×C4×C73C22×C411yescharacteristic
101C4⋊C4×C73—24no—
102C4⋊C4×C73—21no—
103C4⋊C4×C73C2×C812yescomplemented
104C22⋊C4×C73—24no—
105C23×C73—21no—
106C23×C73—21no—
107C23×C73—21no—
108C23×C73—21no—
109C23×C73C22⋊C812yes—
110C23×C73#10312yes—
111C23×C73#10412yes—
112C23×C73C2×C22⋊C411yescharacteristic
113Q8×C73—42no—
114Q8×C73—42no—
115C2×C4×C73—21no—
116C2×C4×C73—21no—
117C2×C4×C73—21no—
118C2×C4×C73—21no—
119C2×C4×C73—22no—
120C2×C4×C73—22no—
121C2×C4×C73—42no—
122C2×C4×C73—42no—
123C2×C4×C73—41no—
124C2×C4×C73—41no—
125C2×C4×C73—41no—
126C2×C4×C73—41no—
127C2×C4×C73C22⋊C812yes—
128C2×C4×C73C22×C811yescharacteristic
129C2×C4×C73C2×M1611yescharacteristic
130C2×Dic73—21no—
131C2×Dic73—21no—
132C2×Dic73—21no—
133C2×Dic73—21no—
134C2×Dic73—24no—
135C2×Dic73—24no—
136C2×Dic73—21no—
137C2×Dic73—21no—
138C2×Dic73—21no—
139C2×Dic73—44no—
140C2×Dic73—41no—
141C2×Dic73—41no—
142C2×Dic73—41no—
143C2×Dic73—41no—
144C2×Dic73—41no—
145C2×Dic73—41no—
146C2×Dic73#9912yes—
147C2×Dic73#10312yes—
148C2×Dic73#10412yes—
149C2×Dic73C2×SD1611yescharacteristic
150C2×Dic73C2×Q1611yescharacteristic
151C2×Dic73#13512yes—
152#10596833—44no—
153#10596833—44no—
154C22×C73—22no—
155C22×C73—22no—
156C22×C73—22no—
157C22×C73—22no—
158C22×C73—22no—
159C22×C73—22no—
160C22×C73Q8⋊C814yes—
161C22×C73C2×C22⋊C811yescharacteristic
162C22×C73C2×#9911yescharacteristic
163C22×C73#41311yescharacteristic
164C22×C73C2×#10311yescharacteristic
165C22×C73#41611yescharacteristic
166C22×C73C2×#10411yescharacteristic
167C22×C73C2×#10411yescharacteristic
168C4×C73—22no—
169C4×C73—21no—
170C4×C73—21no—
171C4×C73—42no—
172C4×C73—42no—
173Dic73—21no—
174Dic73—21no—
175Dic73—22no—
176Dic73—41no—
177Dic73—81no—
178Dic73—81no—
179Dic73#32814yes—
180#7566—731nomaximal, Sylow
181C2×C73—24no—
182C2×C73#145911yescharacteristic
183C2×C73C2×Q8⋊C811yescharacteristic
184C2×C73#147611yescharacteristic
185C2×C73C2×#32811yescharacteristic
186C2×C73#150111yescharacteristic
187C2×C73#152411yescharacteristic
188C2×C73#154211yescharacteristic
189C4×C22⋊Q8—731no—
190C4×C22⋊C8—731no—
191C2×#336—731no—
192#1272—732no—
193#1271—732no—
194C73#756611yescharacteristic, complemented, Sylow
195C23×C8—1461no—
196C2×C22⋊Q8#6440811yescharacteristic
197C2×C4×Q8—731no—
198C22×C42—731no—
199C2×C22⋊C8—1461no—
200C2×C4×C8—1461no—
201#391—731no—
202#389—731no—
203#387—1461no—
204#385—1461no—
205#380—731no—
206C4×C4⋊C4—731no—
207C4×C4⋊C4—1461no—
208C4×C22⋊C4—1461no—
209#336—732no—
210#336—732no—
211#336—1461no—
212C22×Q8C2×#6440811yescharacteristic
213C23×C4—731no—
214C23×C4—731no—
215C23×C4C2×#6440811yescharacteristic
216C22×C8—2922no—
217C22×C8—2922no—
218C22×C8—1462no—
219C22×C8—1462no—
220C22×C8—1464no—
221C22⋊Q8—22no—
222C22⋊Q8#1059683314yescomplemented
223C4×Q8—2922no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—731no—
226C2×C4⋊C4—731no—
227C2×C4⋊C4—1461no—
228C2×C4⋊C4—1461no—
229C2×C4⋊C4C2×#6440811yescharacteristic
230C2×C22⋊C4—21no—
231C2×C22⋊C4—1461no—
232C2×C42—2922no—
233C2×C42—731no—
234C2×C42—731no—
235C2×C42—731no—
236C2×C42—731no—
237C2×C42—1461no—
238C2×C42—1461no—
239C22⋊C8—1464no—
240C4×C8—2922no—
241#95—1461no—
242#95—1461no—
243#95—1461no—
244#95—1461no—
245#95—1461no—
246C24C22⋊#6440811yescharacteristic
247C2×Q8—21no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8C2×#1059683312yescomplemented
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—2921no—
256C22×C4—731no—
257C22×C4—732no—
258C22×C4—732no—
259C22×C4—731no—
260C22×C4—731no—
261C22×C4—731no—
262C22×C4—1461no—
263C22×C4—1461no—
264C22×C4—1462no—
265C22×C4—1462no—
266C22×C4—1462no—
267C22×C4—1462no—
268C22×C4C22⋊#6440811yescharacteristic
269C22×C4C2×#1059683312yes—
270C22×C4#1192403712yes—
271C22×C4C22×#6440811yescharacteristic
272C2×C8—2922no—
273C2×C8—2922no—
274C2×C8—2922no—
275C2×C8—2928no—
276C2×C8—2922no—
277C2×C8—1464no—
278C2×C8—1464no—
279C4⋊C4—21no—
280C4⋊C4—24no—
281C4⋊C4—2921no—
282C4⋊C4—2921no—
283C4⋊C4—2922no—
284C4⋊C4—2922no—
285C4⋊C4C2×#1059683312yescomplemented
286C22⋊C4—24no—
287C22⋊C4—2922no—
288C42—2921no—
289C42—2921no—
290C42—2922no—
291C42—2922no—
292C42—1462no—
293C42—1462no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23C22⋊#1059683312yes—
299C23#1455665412yes—
300C23#1455665612yes—
301C23C2×C22⋊#6440811yescharacteristic
302Q8—42no—
303Q8—42no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—2924no—
317C2×C4—2921no—
318C2×C4—2921no—
319C2×C4—2921no—
320C2×C4—2921no—
321C2×C4—2921no—
322C2×C4—2921no—
323C2×C4—732no—
324C2×C4—732no—
325C2×C4—732no—
326C2×C4—731no—
327C2×C4—731no—
328C2×C4—732no—
329C2×C4—1461no—
330C2×C4—1461no—
331C2×C4—1461no—
332C2×C4—1464no—
333C2×C4—1464no—
334C2×C4—1461no—
335C2×C4—1461no—
336C2×C4—1461no—
337C2×C4—1461no—
338C2×C4C22⋊#1059683312yes—
339C2×C4C22×#1059683311yescharacteristic
340C2×C4C2×#1192403711yescharacteristic
341C8—2924no—
342C8—2924no—
343C22—22no—
344C22—22no—
345C22—22no—
346C22—22no—
347C22—22no—
348C22—22no—
349C22Q8⋊#1059683314yes—
350C22C2×C22⋊#1059683311yescharacteristic
351C22D73⋊#9911yescharacteristic
352C22#1798957411yescharacteristic
353C22C2×#1455665411yescharacteristic
354C22#1798957711yescharacteristic
355C22D73⋊#10411yescharacteristic
356C22C2×#1455665611yescharacteristic
357C4—21no—
358C4—21no—
359C4—22no—
360C4—42no—
361C4—42no—
362C4—2921no—
363C4—5841no—
364C4—5841no—
365C4—734no—
366C4—1461no—
367C4—1461no—
368C4—1462no—
369C2—24no—
370C2#2262516711yescharacteristic
371C2C2×Q8⋊#1059683311yescharacteristic
372C2#2262522211yescharacteristic
373C2D73⋊#32811yescharacteristic
374C2#2262530911yescharacteristic
375C2#2262538411yescharacteristic
376C2#2262543311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#64408
commutatorC2×C4×C73C22×C8
FrattiniC22×C4C22×#64408
FittingC2×C22⋊Q8×C73C4
socleC23×C73C2×C22⋊C4
prime radicalC24×C73C22⋊C4
cosocleC2×C4×Dic73C23
hypercenterC2×C22⋊Q8#64408
hypocenterC73#7566
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #22624751C2semidirect
4. #22624749C2semidirect
53. C22⋊Q8×C73C8semidirect
92. C2×Q8×C73C2×C8semidirect
103. C4⋊C4×C73C2×C8semidirect
194. C73#7566semidirect
222. C22⋊Q8#10596833semidirect
250. C2×Q8C2×#10596833semidirect
285. C4⋊C4C2×#10596833semidirect

Maximal subgroups

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

subgroupindex
C4×C22⋊#105968332
C2×#179894532
#226247512
#226247492
C22⋊Q8×Dic732
#756673