Finite Group Databasebeta

← #11774657

Subgroups — #11774657

Order 1152.

687 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 687 conjugacy classes of subgroups — too many to draw. Only its 112 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
1C4×#3830C211yesmaximal, characteristic
2#10590340C211yesmaximal, characteristic, complemented
3C4×#3712C211yesmaximal, characteristic, complemented
4C3×D4⋊(C3⋊C8)C212yesmaximal, complemented
5#10589085C212yesmaximal, complemented
6#68174—32nomaximal
7C2×#3830C411yescharacteristic
8C32×C4×D4C2211yescharacteristic
9D4×C3⋊Dic3C411yescharacteristic
10C4×#638—21no—
11C2×C4×C3⋊Dic3—21no—
12#64101C411yescharacteristic
13#64096C411yescharacteristic
14#64090—21no—
15#64086—21no—
16C42×C3⋊D3C2211yescharacteristic
17C4×#634C2211yescharacteristic
18C2×#3712—21no—
19C2×#3712—21no—
20C3×C4⋊(C3⋊C8)C2212yes—
21C3×C4×C3⋊C8C2212yes—
22#63572—21no—
23#63555—21no—
24C4×#237—32no—
25C4×#237—61no—
26#4837—32no—
27C4×ΓL(1,𝔽25)—32no—
28C3×D4⋊C8—32no—
29D4⋊(C3⋊C8)D312yescomplemented
30#4484—32no—
31#4456—32no—
32C2×C32×D4C2×C411yescharacteristic
33C22×C32×C4—21no—
34C2×#638—21no—
35C22×C3⋊Dic3—21no—
36C22×C3⋊Dic3—21no—
37#3830—22no—
38#3830—22no—
39C2×C4×C3⋊D3C2×C411yescharacteristic
40C2×C4×C3⋊D3D411yescharacteristic
41C2×C4×C3⋊D3D411yescharacteristic
42C2×#634C2×C411yescharacteristic
43C32×C4⋊C4C2×C411yescharacteristic
44C32×C22⋊C4—21no—
45C32×C42C2311yescharacteristic
46#3757—21no—
47#3752—21no—
48C4⋊(C3⋊Dic3)C2×C411yescharacteristic
49#3750—21no—
50#3750—21no—
51C4×C3⋊Dic3—21no—
52C4×C3⋊Dic3—21no—
53C4×C3⋊Dic3—21no—
54C4×C3⋊Dic3C2×C411yescharacteristic
55C4×C3⋊Dic3D411yescharacteristic
56C4×C3⋊Dic3D411yescharacteristic
57C2×C3×C3⋊C8—22no—
58C2×C3×C3⋊C8—22no—
59#3712—24no—
60#3712—42no—
61#1489—91noSylow
62C2×#237—32no—
63C2×#237—61no—
64C3×C4×D4—21no—
65C3×C4×D4C2×D312yes—
66D4×Dic3—32no—
67D4×Dic3—61no—
68C4×#67—62no—
69C4×#67—62no—
70C2×C4×Dic3—62no—
71C2×C4×Dic3—62no—
72C2×ΓL(1,𝔽25)—62no—
73C2×ΓL(1,𝔽25)—62no—
74#892—32no—
75#892—61no—
76#887—32no—
77#887—61no—
78#67⋊C4—62no—
79#67⋊C4—62no—
80#877—62no—
81#877—62no—
82C42×D3—32no—
83C42×D3—61no—
84C4×Dic6—32no—
85C4×Dic6—61no—
86C3×C4⋊C8—32no—
87C3×C4×C8—32no—
88#815—62no—
89Dic3⋊C8—62no—
90C4⋊(C3⋊C8)C2×D312yes—
91C4×C3⋊C8C2×D312yes—
92C23×C32—21no—
93C22×C3⋊D3C22⋊C411yescharacteristic
94C32×D4—21no—
95C32×D4M1612yescomplemented
96C2×C32×C4—21no—
97C2×C32×C4—41no—
98C2×C32×C4—41no—
99C2×C32×C4C22×C411yescharacteristic
100C2×C32×C4C2×D411yescharacteristic
101C2×C32×C4C2×D411yescharacteristic
102#638—24no—
103C2×C3⋊Dic3—24no—
104C2×C3⋊Dic3—21no—
105C2×C3⋊Dic3—21no—
106C2×C3⋊Dic3—42no—
107C2×C3⋊Dic3C22⋊C411yescharacteristic
108C2×C3⋊Dic3C22⋊C411yescharacteristic
109C2×C3⋊Dic3C22⋊C411yescharacteristic
110C4×C3⋊D3—21no—
111C4×C3⋊D3—21no—
112C4×C3⋊D3—21no—
113C4×C3⋊D3—21no—
114C4×C3⋊D3—22no—
115#634—21no—
116#634M1612yescomplemented
117C3×C3⋊C8—24no—
118C3×C3⋊C8—42no—
119C4×#41—91no—
120#423—91no—
121C4×M16—91no—
122Q8⋊C8—92no—
123D4⋊C8—32no—
124C2×C3×D4—21no—
125C2×C3×D4C4×D312yes—
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C2×#67—62no—
129C2×#67—62no—
130C22×Dic3—62no—
131C22×Dic3—62no—
132C22×Dic3—62no—
133C22×Dic3—62no—
134#237—64no—
135#237—64no—
136#237—64no—
137#237—64no—
138C2×C4×D3—32no—
139C2×C4×D3—32no—
140C2×C4×D3—32no—
141C2×C4×D3—61no—
142C2×C4×D3—61no—
143C2×C4×D3—61no—
144C2×Dic6—32no—
145C2×Dic6—61no—
146C2×C3×C8—62no—
147C2×C3×C8—62no—
148C3×C4⋊C4—21no—
149C3×C4⋊C4C4×D312yes—
150C3×C22⋊C4—22no—
151C3×C22⋊C4—22no—
152C3×C42—21no—
153C3×C42C22×D312yes—
154#217—62no—
155#217—62no—
156(C2×D3)⋊C4—62no—
157(C2×D3)⋊C4—62no—
158C4⋊Dic3—32no—
159C4⋊Dic3—61no—
160Dic3⋊C4—62no—
161Dic3⋊C4—62no—
162Dic3⋊C4—62no—
163Dic3⋊C4—62no—
164C4×Dic3—32no—
165C4×Dic3—32no—
166C4×Dic3—32no—
167C4×Dic3—61no—
168C4×Dic3—61no—
169C4×Dic3—61no—
170C4×Dic3—62no—
171C4×Dic3—62no—
172C4×Dic3—62no—
173C4×Dic3—62no—
174C4×Dic3—62no—
175C4×Dic3—62no—
176C2×C3⋊C8—22no—
177C2×C3⋊C8—22no—
178ΓL(1,𝔽25)—68no—
179ΓL(1,𝔽25)—124no—
180C22×C32—22no—
181C22×C32—41no—
182C22×C32—41no—
183C22×C32C2×C22⋊C411yescharacteristic
184C2×C3⋊D3—21no—
185C2×C3⋊D3—21no—
186C2×C3⋊D3—21no—
187C32×C4—21no—
188C32×C4—41no—
189C32×C4C2×M1611yescharacteristic
190C32×C4C2×M1611yescharacteristic
191C32×C4AGL(1,ℤ/8)11yescharacteristic
192C32×C4#13711yescharacteristic
193C3⋊Dic3—22no—
194C3⋊Dic3—21no—
195C3⋊Dic3—21no—
196C3⋊Dic3—41no—
197C3⋊Dic3#10412yes—
198C2×#41—91no—
199C2×M16—181no—
200C2×M16—181no—
201C4×Q8—91no—
202C4×Q8—91no—
203C4×D4—91no—
204C4×D4—181no—
205C4×D4—181no—
206C4×D4D3211yescharacteristic
207#117—91no—
208#117—181no—
209C2×C42—91no—
210C2×C42—181no—
211C4⋊C8—32no—
212C4⋊C8—181no—
213#97—181no—
214C4×C8—32no—
215C23×C3—22no—
216C23×C3—22no—
217C22×D3—32no—
218C22×D3—61no—
219C3×D4—22no—
220C3×D4—22no—
221C3×D4—22no—
222C3×D4ΓL(1,𝔽25)14yescomplemented
223C2×C3×C4—22no—
224C2×C3×C4—22no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—42no—
229C2×C3×C4—42no—
230C2×C3×C4—42no—
231C2×C3×C4—42no—
232C2×C3×C4C2×C4×D312yes—
233C2×C3×C4D3×D412yes—
234C2×C3×C4D3×D412yes—
235#67—68no—
236#67—68no—
237C2×Dic3—32no—
238C2×Dic3—32no—
239C2×Dic3—32no—
240C2×Dic3—61no—
241C2×Dic3—61no—
242C2×Dic3—61no—
243C2×Dic3—62no—
244C2×Dic3—68no—
245C2×Dic3—68no—
246C2×Dic3—62no—
247C2×Dic3—62no—
248C2×Dic3—62no—
249C2×Dic3—124no—
250C2×Dic3—124no—
251C4×D3—64no—
252C4×D3—62no—
253C4×D3—62no—
254C4×D3—62no—
255C4×D3—62no—
256C4×D3—62no—
257C4×D3—62no—
258C4×D3—122no—
259C4×D3—121no—
260C4×D3—121no—
261Dic6—34no—
262Dic6—62no—
263Dic6—62no—
264Dic6—62no—
265C3×C8—64no—
266C3×C8—122no—
267C3⋊C8—24no—
268C3⋊C8—42no—
269C2×C32—42no—
270C2×C32#40711yescharacteristic
271C2×C32#41711yescharacteristic
272C2×C32C2×#10411yescharacteristic
273C3⋊D3—41no—
274#41—182no—
275#41—182no—
276C2×Q8—91no—
277C2×D4—181no—
278C2×D4#62411yescharacteristic
279C22×C4—21no—
280C22×C4—91no—
281C22×C4—91no—
282C22×C4—91no—
283C22×C4—181no—
284C22×C4—181no—
285M16—184no—
286M16—362no—
287C2×C8—62no—
288C2×C8—62no—
289C4⋊C4—91no—
290C4⋊C4—181no—
291C4⋊C4—181no—
292C4⋊C4#62411yescharacteristic
293C22⋊C4—21no—
294C22⋊C4—181no—
295C22⋊C4—181no—
296C42—91no—
297C42—91no—
298C42—91no—
299C42—181no—
300C42—181no—
301C42—181no—
302C42C2×D3211yescharacteristic
303C22×C3—24no—
304C22×C3—24no—
305C22×C3—21no—
306C22×C3—42no—
307C22×C3—42no—
308C22×C3—42no—
309C22×C3—42no—
310C22×C3D3×C22⋊C412yes—
311C2×D3—62no—
312C2×D3—62no—
313C2×D3—62no—
314C2×D3—62no—
315C2×D3—121no—
316C2×D3—121no—
317C3×C4—22no—
318C3×C4—21no—
319C3×C4—21no—
320C3×C4—21no—
321C3×C4—21no—
322C3×C4—41no—
323C3×C4—42no—
324C3×C4—42no—
325C3×C4C2×ΓL(1,𝔽25)12yes—
326C3×C4D3×M1612yes—
327C3×C4#91112yes—
328C3×C4#91512yes—
329Dic3—34no—
330Dic3—62no—
331Dic3—64no—
332Dic3—62no—
333Dic3—64no—
334Dic3—62no—
335Dic3—62no—
336Dic3—121no—
337Dic3—122no—
338Dic3—122no—
339C32#148911yescharacteristic, complemented, Sylow
340C23—21no—
341C23—91no—
342Q8—92no—
343Q8—181no—
344D4—21no—
345D4—184no—
346D4#371212yescomplemented
347C2×C4—21no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—91no—
351C2×C4—91no—
352C2×C4—91no—
353C2×C4—182no—
354C2×C4—184no—
355C2×C4—181no—
356C2×C4—181no—
357C2×C4—181no—
358C2×C4—181no—
359C2×C4—181no—
360C2×C4—181no—
361C2×C4—362no—
362C2×C4C2×#62411yescharacteristic
363C2×C4#381111yescharacteristic
364C2×C4#381111yescharacteristic
365C8—64no—
366C8—122no—
367C2×C3—21no—
368C2×C3—21no—
369C2×C3—21no—
370C2×C3—44no—
371C2×C3—44no—
372C2×C3#472612yes—
373C2×C3#477112yes—
374C2×C3D3×#10412yes—
375D3—122no—
376D3—122no—
377C22—22no—
378C22—41no—
379C22—41no—
380C22—181no—
381C22—181no—
382C22—181no—
383C22#6397411yescharacteristic
384C4—21no—
385C4—41no—
386C4—92no—
387C4—181no—
388C4—182no—
389C4—181no—
390C4—361no—
391C4#6381811yescharacteristic
392C4C2×#371211yescharacteristic
393C4#6392711yescharacteristic
394C4#6393111yescharacteristic
395C3—21no—
396C3#6817412yescomplemented
397C2—42no—
398C2—361no—
399C2#1059115911yescharacteristic
400C2#1059141811yescharacteristic
401C2#1059167111yescharacteristic
subgroupG/H
centerC2×C4#3811
commutatorC2×C32×C4C22×C4
FrattiniC42C2×D32
FittingC32×C4×D4C22
socleC22×C32C2×C22⋊C4
prime radicalC2×#3830C4
cosocleC32×C42C23
hypercenterC4×D4D32
hypocenterC32#1489
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
2. #10590340C2semidirect
3. C4×#3712C2semidirect
4. C3×D4⋊(C3⋊C8)C2semidirect
5. #10589085C2semidirect
29. D4⋊(C3⋊C8)D3semidirect
95. C32×D4M16semidirect
116. #634M16semidirect
222. C3×D4ΓL(1,𝔽25)semidirect
339. C32#1489semidirect
346. D4#3712semidirect
396. C3#68174semidirect

Maximal subgroups

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

subgroupindex
C4×#38302
#105903402
C4×#37122
C3×D4⋊(C3⋊C8)2
#105890852
#681743