Finite Group Databasebeta

← #34006128

Subgroups — #34006128

Order 18688.

610 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989603C211yesmaximal, characteristic
2#22625785C211yesmaximal, characteristic
3C2×Q8⋊#10596833C212yesmaximal, complemented
4#22624767C212yesmaximal, complemented
5C22⋊Q8×Dic73C211yesmaximal, characteristic
6#17989603—24no—
7C2×C4⋊#10596833—21no—
8C2×C4⋊#10596833C2212yes—
9C2×C22⋊#10596833—21no—
10C2×C22⋊#10596833—21no—
11C2×C4×#10596833C2212yes—
12#17989453—21no—
13Q8⋊#10596833—24no—
14C2×Q8×Dic73C2211yescharacteristic
15C22×C4×Dic73C2211yescharacteristic
16#17988494C411yescharacteristic
17#17988486C411yescharacteristic
18#17988473—21no—
19#17988462—21no—
20#17988447C411yescharacteristic
21C4⋊C4×Dic73—21no—
22C4⋊C4×Dic73C2211yescharacteristic
23C22⋊C4×Dic73—21no—
24C2×C22⋊Q8×C73C411yescharacteristic
25C22×#10596833—22no—
26C22×#10596833—22no—
27C4⋊#10596833—24no—
28C4⋊#10596833—22no—
29C4⋊#10596833—22no—
30C22⋊#10596833—24no—
31C22⋊#10596833—42no—
32C4×#10596833—22no—
33C4×#10596833D414yes—
34C23×Dic73D411yescharacteristic
35C23×Dic73D411yescharacteristic
36Q8×Dic73—22no—
37Q8×Dic73—22no—
38C2×C4⋊Dic73—21no—
39C2×C4⋊Dic73—21no—
40C2×C4⋊Dic73C2×C411yescharacteristic
41C2×C4⋊Dic73C2×C411yescharacteristic
42C2×C22⋊Dic73—21no—
43C2×C4×Dic73—22no—
44C2×C4×Dic73—22no—
45C2×C4×Dic73—21no—
46C2×C4×Dic73—21no—
47C2×C4×Dic73C2×C411yescharacteristic
48C2×C4×Dic73D411yescharacteristic
49C2×C4×Dic73D411yescharacteristic
50C2×C4×Dic73C2311yescharacteristic
51#14556498—21no—
52#14556498—21no—
53#14556498—21no—
54#14556498—21no—
55#14556498—21no—
56C22×Q8×C73C2×C411yescharacteristic
57C23×C4×C73C2×C411yescharacteristic
58C22⋊Q8×C73—22no—
59C22⋊Q8×C73C814yescomplemented
60C2×C4⋊C4×C73—21no—
61C2×C4⋊C4×C73C2×C411yescharacteristic
62C2×C22⋊C4×C73—21no—
63C2×#10596833—24no—
64C2×#10596833—42no—
65C2×#10596833—42no—
66C2×#10596833—42no—
67C2×#10596833—42no—
68C2×#10596833—42no—
69C22×Dic73—21no—
70C22×Dic73—22no—
71C22×Dic73—21no—
72C22×Dic73—21no—
73C22×Dic73—21no—
74C22×Dic73—22no—
75C22×Dic73—21no—
76C22×Dic73—21no—
77C22×Dic73—41no—
78C22×Dic73—41no—
79C22×Dic73C22⋊C411yescharacteristic
80C22×Dic73C22⋊C411yescharacteristic
81C22×Dic73C2×D411yescharacteristic
82C22×Dic73C2×D411yescharacteristic
83C4⋊Dic73—22no—
84C4⋊Dic73—22no—
85C4⋊Dic73—42no—
86C4⋊Dic73—41no—
87C4⋊Dic73—41no—
88C22⋊Dic73—42no—
89C4×Dic73—22no—
90C4×Dic73—21no—
91C4×Dic73—22no—
92C4×Dic73—21no—
93C4×Dic73—42no—
94C4×Dic73—41no—
95C4×Dic73—41no—
96C4×Dic73C2×D412yes—
97C4×Dic73#4112yes—
98C24×C73C22⋊C411yescharacteristic
99C2×Q8×C73—21no—
100C2×Q8×C73—22no—
101C2×Q8×C73—22no—
102C2×Q8×C73C2×C812yes—
103C22×C4×C73—22no—
104C22×C4×C73—22no—
105C22×C4×C73—21no—
106C22×C4×C73—21no—
107C22×C4×C73C22⋊C411yescharacteristic
108C22×C4×C73C2×C812yes—
109C22×C4×C73M1612yes—
110C22×C4×C73C22×C411yescharacteristic
111C4⋊C4×C73—24no—
112C4⋊C4×C73—21no—
113C4⋊C4×C73C2×C812yes—
114C22⋊C4×C73—24no—
115C23×C73—21no—
116C23×C73—21no—
117C23×C73—21no—
118C23×C73—21no—
119C23×C73—21no—
120C23×C73—21no—
121C23×C73C22⋊C812yes—
122C23×C73C2×C22⋊C411yescharacteristic
123Q8×C73—24no—
124Q8×C73—42no—
125C2×C4×C73—21no—
126C2×C4×C73—21no—
127C2×C4×C73—24no—
128C2×C4×C73—24no—
129C2×C4×C73—22no—
130C2×C4×C73—22no—
131C2×C4×C73—41no—
132C2×C4×C73—41no—
133C2×C4×C73—41no—
134C2×C4×C73—41no—
135C2×C4×C73C22⋊C812yes—
136C2×C4×C73C4×D412yes—
137C2×C4×C73C22×C811yescharacteristic
138C2×C4×C73C2×M1611yescharacteristic
139C2×C4×C73#13112yes—
140C2×Dic73—21no—
141C2×Dic73—21no—
142C2×Dic73—21no—
143C2×Dic73—21no—
144C2×Dic73—21no—
145C2×Dic73—21no—
146C2×Dic73—21no—
147C2×Dic73—21no—
148C2×Dic73—42no—
149C2×Dic73—42no—
150C2×Dic73—42no—
151C2×Dic73—42no—
152C2×Dic73—41no—
153C2×Dic73—41no—
154C2×Dic73—41no—
155C2×Dic73—41no—
156C2×Dic73—41no—
157C2×Dic73—41no—
158C2×Dic73#12011yescharacteristic
159C2×Dic73#12112yes—
160C2×Dic73#12311yescharacteristic
161C2×Dic73AGL(1,ℤ/8)11yescharacteristic
162C2×Dic73#13712yes—
163C2×Dic73#13711yescharacteristic
164#10596833—44no—
165#10596833—82no—
166C22×C73—22no—
167C22×C73—22no—
168C22×C73—22no—
169C22×C73—41no—
170C22×C73—41no—
171C22×C73—41no—
172C22×C73—41no—
173C22×C73#38611yescharacteristic
174C22×C73C2×C22⋊C811yescharacteristic
175C22×C73#40811yescharacteristic
176C22×C73#41711yescharacteristic
177C22×C73#41911yescharacteristic
178C22×C73#42111yescharacteristic
179C22×C73#42111yescharacteristic
180C4×C73—22no—
181C4×C73—42no—
182C4×C73—42no—
183C4×C73C8×D412yes—
184C4×C73#43512yes—
185Dic73—21no—
186Dic73—21no—
187Dic73—41no—
188Dic73—41no—
189Dic73—81no—
190Dic73—81no—
191Dic73#46812yes—
192Dic73#47012yes—
193#9073—731nomaximal, Sylow
194C2×C73—42no—
195C2×C73#147711yescharacteristic
196C2×C73#147911yescharacteristic
197C2×C73#149511yescharacteristic
198C2×C73#149711yescharacteristic
199C2×C73#186911yescharacteristic
200C2×C73#189411yescharacteristic
201C2×C73#189711yescharacteristic
202C2×#431—731no—
203C4×C22⋊Q8—731no—
204#1797—731no—
205C2×Q8⋊C8—732no—
206#1279—732no—
207C73#907311yescharacteristic, complemented, Sylow
208C2×C22⋊Q8#6440811yescharacteristic
209C2×C4×Q8—731no—
210C22×C42—731no—
211#431—1464no—
212C2×C4⋊C8—732no—
213C2×C4⋊C8—1461no—
214C2×C22⋊C8—1461no—
215C2×C22⋊C8—1461no—
216C2×C4×C8—732no—
217#391—731no—
218#389—731no—
219#387—1461no—
220#385—1461no—
221#380—731no—
222C4×C4⋊C4—731no—
223C4×C4⋊C4—1461no—
224C4×C22⋊C4—1461no—
225#336—1461no—
226Q8⋊C8—1464no—
227C22×Q8C2×#6440811yescharacteristic
228C23×C4—731no—
229C23×C4—731no—
230C23×C4C2×#6440811yescharacteristic
231C22×C8—1462no—
232C22×C8—1462no—
233C22⋊Q8—22no—
234C22⋊Q8#1059683314yescomplemented
235C4×Q8—1462no—
236C4×Q8—1462no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—731no—
239C2×C4⋊C4—731no—
240C2×C4⋊C4—1461no—
241C2×C4⋊C4—1461no—
242C2×C4⋊C4C2×#6440811yescharacteristic
243C2×C22⋊C4—21no—
244C2×C22⋊C4—1461no—
245C2×C42—731no—
246C2×C42—731no—
247C2×C42—731no—
248C2×C42—731no—
249C2×C42—1461no—
250C2×C42—1461no—
251C2×C42—1462no—
252C2×C42—1462no—
253C4⋊C8—1462no—
254C4⋊C8—1462no—
255C4⋊C8—1464no—
256C22⋊C8—2922no—
257C22⋊C8—1464no—
258C4×C8—734no—
259C4×C8—1462no—
260#95—1461no—
261#95—1461no—
262#95—1461no—
263#95—1461no—
264#95—1461no—
265C24C22⋊#6440811yescharacteristic
266C2×Q8—21no—
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8C2×#1059683312yes—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—2921no—
275C22×C4—2921no—
276C22×C4—731no—
277C22×C4—731no—
278C22×C4—731no—
279C22×C4—731no—
280C22×C4—1461no—
281C22×C4—1461no—
282C22×C4—1461no—
283C22×C4—1461no—
284C22×C4—1462no—
285C22×C4—1461no—
286C22×C4—1461no—
287C22×C4—1462no—
288C22×C4C22⋊#6440811yescharacteristic
289C22×C4C2×#1059683312yes—
290C22×C4#1192403712yes—
291C22×C4C22×#6440811yescharacteristic
292C2×C8—2922no—
293C2×C8—2922no—
294C2×C8—2922no—
295C2×C8—2922no—
296C2×C8—2922no—
297C2×C8—1464no—
298C4⋊C4—21no—
299C4⋊C4—24no—
300C4⋊C4—2921no—
301C4⋊C4—2921no—
302C4⋊C4—2922no—
303C4⋊C4—1462no—
304C4⋊C4—1462no—
305C4⋊C4C2×#1059683312yes—
306C22⋊C4—24no—
307C22⋊C4—2922no—
308C42—2921no—
309C42—2921no—
310C42—2922no—
311C42—732no—
312C42—732no—
313C42—1461no—
314C42—1462no—
315C42—1461no—
316C42—1462no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23C22⋊#1059683312yes—
324C23C2×C22⋊#6440811yescharacteristic
325Q8—24no—
326Q8—42no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—2922no—
338C2×C4—2922no—
339C2×C4—2922no—
340C2×C4—2922no—
341C2×C4—2921no—
342C2×C4—2921no—
343C2×C4—2921no—
344C2×C4—2921no—
345C2×C4—2921no—
346C2×C4—2921no—
347C2×C4—732no—
348C2×C4—731no—
349C2×C4—731no—
350C2×C4—732no—
351C2×C4—731no—
352C2×C4—731no—
353C2×C4—1461no—
354C2×C4—1461no—
355C2×C4—1461no—
356C2×C4—1461no—
357C2×C4—1461no—
358C2×C4—1461no—
359C2×C4—1461no—
360C2×C4—1461no—
361C2×C4C22⋊#1059683312yes—
362C2×C4D4×#6440812yes—
363C2×C4C22×#1059683311yescharacteristic
364C2×C4C2×#1192403711yescharacteristic
365C2×C4#1455667812yes—
366C8—2924no—
367C8—5842no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22#1798953511yescharacteristic
376C22C2×C22⋊#1059683311yescharacteristic
377C22#1798956011yescharacteristic
378C22#1798957811yescharacteristic
379C22#1798958011yescharacteristic
380C22#1798958411yescharacteristic
381C22#1798958611yescharacteristic
382C4—22no—
383C4—42no—
384C4—42no—
385C4—2921no—
386C4—2921no—
387C4—5841no—
388C4—5841no—
389C4—732no—
390C4—732no—
391C4—1461no—
392C4—1461no—
393C4D4×#1059683312yes—
394C4#1798961812yes—
395C2—42no—
396C2#2262522611yescharacteristic
397C2#2262523311yescharacteristic
398C2#2262528811yescharacteristic
399C2#2262529311yescharacteristic
400C2#2262590411yescharacteristic
401C2#2262594611yescharacteristic
402C2#2262595011yescharacteristic
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#9073
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. C2×Q8⋊#10596833C2semidirect
4. #22624767C2semidirect
59. C22⋊Q8×C73C8semidirect
207. C73#9073semidirect
234. C22⋊Q8#10596833semidirect

Maximal subgroups

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

subgroupindex
C2×#179896032
#226257852
C2×Q8⋊#105968332
#226247672
C22⋊Q8×Dic732
#907373