Finite Group Databasebeta

← #11774665

Subgroups — #11774665

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×#3832C211yesmaximal, characteristic
2#10590340C211yesmaximal, characteristic, complemented
3C4×#3712C211yesmaximal, characteristic, complemented
4C3×#4535C212yesmaximal, complemented
5#10589082C212yesmaximal, complemented
6#68238—32nomaximal
7C2×#3832C411yescharacteristic
8C32×C4×Q8C2211yescharacteristic
9Q8×C3⋊Dic3C411yescharacteristic
10#64102—21no—
11#64102C411yescharacteristic
12#64101—21no—
13#64101C411yescharacteristic
14C4×C4⋊(C3⋊D3)—21no—
15C4×C4⋊(C3⋊D3)C2211yescharacteristic
16C42×C3⋊D3—21no—
17C42×C3⋊D3C2211yescharacteristic
18C2×#3712—21no—
19C2×#3712—21no—
20C3×C4⋊(C3⋊C8)C2212yes—
21C3×C4×C3⋊C8C2212yes—
22#63572—21no—
23#63555—21no—
24C4×#239—32no—
25C4×#239—61no—
26#4837—32no—
27C4×ΓL(1,𝔽25)—32no—
28C3×Q8⋊C8—32no—
29#4535D312yescomplemented
30#4483—32no—
31#4459—32no—
32C2×C32×Q8C2×C411yescharacteristic
33#3832—22no—
34#3832—22no—
35C2×C4⋊(C3⋊D3)—21no—
36C2×C4⋊(C3⋊D3)C2×C411yescharacteristic
37C2×C4×C3⋊D3—21no—
38C2×C4×C3⋊D3—21no—
39C2×C4×C3⋊D3—21no—
40C2×C4×C3⋊D3C2×C411yescharacteristic
41C2×C4×C3⋊D3D411yescharacteristic
42C2×C4×C3⋊D3D411yescharacteristic
43C32×C4⋊C4—21no—
44C32×C4⋊C4C2×C411yescharacteristic
45C32×C42—21no—
46C32×C42C2311yescharacteristic
47#3752—21no—
48#3752—21no—
49#3752—21no—
50C4⋊(C3⋊Dic3)—21no—
51C4⋊(C3⋊Dic3)C2×C411yescharacteristic
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×#239—32no—
63C2×#239—61no—
64C3×C4×Q8—21no—
65C3×C4×Q8C2×D312yes—
66Q8×Dic3—32no—
67Q8×Dic3—61no—
68C2×ΓL(1,𝔽25)—62no—
69C2×ΓL(1,𝔽25)—62no—
70#893—32no—
71#893—61no—
72#893—62no—
73#893—62no—
74#892—32no—
75#892—61no—
76#892—62no—
77#892—62no—
78C4×D12—32no—
79C4×D12—61no—
80C4×D12—62no—
81C4×D12—62no—
82C42×D3—32no—
83C42×D3—61no—
84C42×D3—62no—
85C42×D3—62no—
86C3×C4⋊C8—32no—
87C3×C4×C8—32no—
88#815—62no—
89Dic3⋊C8—62no—
90C4⋊(C3⋊C8)C2×D312yes—
91C4×C3⋊C8C2×D312yes—
92C22×C3⋊D3—21no—
93C22×C3⋊D3C22⋊C411yescharacteristic
94C32×Q8—21no—
95C32×Q8M1612yescomplemented
96C2×C32×C4—21no—
97C2×C32×C4—21no—
98C2×C32×C4C22×C411yescharacteristic
99C2×C32×C4C2×D411yescharacteristic
100C2×C32×C4C2×D411yescharacteristic
101C2×C3⋊Dic3—21no—
102C2×C3⋊Dic3C22⋊C411yescharacteristic
103C2×C3⋊Dic3C22⋊C411yescharacteristic
104C2×C3⋊Dic3C22⋊C411yescharacteristic
105C4⋊(C3⋊D3)—24no—
106C4⋊(C3⋊D3)—21no—
107C4⋊(C3⋊D3)M1612yescomplemented
108C4×C3⋊D3—24no—
109C4×C3⋊D3—21no—
110C4×C3⋊D3—21no—
111C4×C3⋊D3—21no—
112C4×C3⋊D3—21no—
113C4×C3⋊D3—22no—
114C4×C3⋊D3—41no—
115C4×C3⋊D3—41no—
116C4×C3⋊D3—42no—
117C3×C3⋊C8—24no—
118C3×C3⋊C8—42no—
119C4×#41—91no—
120#423—91no—
121C4×M16—91no—
122Q8⋊C8—32no—
123D4⋊C8—92no—
124C2×C3×Q8—21no—
125C2×C3×Q8C4×D312yes—
126#239—64no—
127#239—64no—
128#239—64no—
129#239—64no—
130C2×D12—32no—
131C2×D12—61no—
132C2×D12—62no—
133C2×D12—62no—
134C2×C4×D3—32no—
135C2×C4×D3—32no—
136C2×C4×D3—32no—
137C2×C4×D3—61no—
138C2×C4×D3—61no—
139C2×C4×D3—61no—
140C2×C4×D3—62no—
141C2×C4×D3—62no—
142C2×C4×D3—62no—
143C2×C4×D3—62no—
144C2×C4×D3—62no—
145C2×C4×D3—62no—
146C2×C3×C8—62no—
147C2×C3×C8—62no—
148C3×C4⋊C4—22no—
149C3×C4⋊C4—22no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4C4×D312yes—
152C3×C42—22no—
153C3×C42—22no—
154C3×C42—21no—
155C3×C42C22×D312yes—
156(C2×D3)⋊C4—62no—
157(C2×D3)⋊C4—62no—
158(C2×D3)⋊C4—62no—
159(C2×D3)⋊C4—62no—
160(C2×D3)⋊C4—62no—
161(C2×D3)⋊C4—62no—
162C4⋊Dic3—32no—
163C4⋊Dic3—61no—
164C4⋊Dic3—62no—
165C4⋊Dic3—62no—
166C4×Dic3—32no—
167C4×Dic3—32no—
168C4×Dic3—32no—
169C4×Dic3—61no—
170C4×Dic3—61no—
171C4×Dic3—61no—
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×C32C2×C22⋊C411yescharacteristic
181C2×C3⋊D3—21no—
182C2×C3⋊D3—22no—
183C2×C3⋊D3—21no—
184C2×C3⋊D3—21no—
185C2×C3⋊D3—41no—
186C2×C3⋊D3—41no—
187C32×C4—22no—
188C32×C4—21no—
189C32×C4—41no—
190C32×C4C2×M1611yescharacteristic
191C32×C4C2×M1611yescharacteristic
192C32×C4AGL(1,ℤ/8)11yescharacteristic
193C32×C4#13711yescharacteristic
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×Q8D3211yescharacteristic
203C4×D4—91no—
204C4×D4—91no—
205C4×D4—181no—
206C4×D4—181no—
207#117—91no—
208#117—181no—
209C2×C42—91no—
210C2×C42—181no—
211C4⋊C8—32no—
212C4⋊C8—181no—
213#97—181no—
214C4×C8—32no—
215C22×D3—32no—
216C22×D3—61no—
217C22×D3—62no—
218C22×D3—62no—
219C3×Q8—22no—
220C3×Q8—22no—
221C3×Q8—22no—
222C3×Q8ΓL(1,𝔽25)14yescomplemented
223C2×C3×C4—22no—
224C2×C3×C4—22no—
225C2×C3×C4—22no—
226C2×C3×C4—22no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—21no—
230C2×C3×C4C2×C4×D312yes—
231C2×C3×C4D3×D412yes—
232C2×C3×C4D3×D412yes—
233C2×Dic3—32no—
234C2×Dic3—32no—
235C2×Dic3—32no—
236C2×Dic3—61no—
237C2×Dic3—61no—
238C2×Dic3—61no—
239C2×Dic3—62no—
240C2×Dic3—62no—
241D12—34no—
242D12—62no—
243D12—68no—
244D12—68no—
245D12—62no—
246D12—62no—
247C4×D3—64no—
248C4×D3—68no—
249C4×D3—68no—
250C4×D3—62no—
251C4×D3—62no—
252C4×D3—62no—
253C4×D3—62no—
254C4×D3—62no—
255C4×D3—62no—
256C4×D3—122no—
257C4×D3—121no—
258C4×D3—121no—
259C4×D3—124no—
260C4×D3—124no—
261C4×D3—122no—
262C4×D3—122no—
263C4×D3—122no—
264C4×D3—122no—
265C3×C8—64no—
266C3×C8—122no—
267C3⋊C8—24no—
268C3⋊C8—42no—
269C2×C32#40711yescharacteristic
270C2×C32#41711yescharacteristic
271C2×C32C2×#10411yescharacteristic
272C3⋊D3—41no—
273C3⋊D3—42no—
274#41—182no—
275#41—182no—
276C2×Q8#62411yescharacteristic
277C2×D4—91no—
278C2×D4—181no—
279C22×C4—91no—
280C22×C4—91no—
281C22×C4—91no—
282C22×C4—181no—
283C22×C4—181no—
284C22×C4—181no—
285M16—184no—
286M16—362no—
287C2×C8—62no—
288C2×C8—62no—
289C4⋊C4—21no—
290C4⋊C4—91no—
291C4⋊C4—181no—
292C4⋊C4#62411yescharacteristic
293C22⋊C4—181no—
294C22⋊C4—181no—
295C22⋊C4—181no—
296C42—21no—
297C42—91no—
298C42—91no—
299C42—91no—
300C42—181no—
301C42—181no—
302C42C2×D3211yescharacteristic
303C22×C3—21no—
304C22×C3D3×C22⋊C412yes—
305C2×D3—64no—
306C2×D3—62no—
307C2×D3—64no—
308C2×D3—62no—
309C2×D3—62no—
310C2×D3—62no—
311C2×D3—121no—
312C2×D3—121no—
313C2×D3—122no—
314C2×D3—122no—
315C2×D3—122no—
316C2×D3—122no—
317C3×C4—24no—
318C3×C4—24no—
319C3×C4—22no—
320C3×C4—21no—
321C3×C4—21no—
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—41no—
325C3×C4—42no—
326C3×C4—42no—
327C3×C4C2×ΓL(1,𝔽25)12yes—
328C3×C4D3×M1612yes—
329C3×C4#91412yes—
330C3×C4#91812yes—
331Dic3—34no—
332Dic3—62no—
333Dic3—62no—
334Dic3—62no—
335Dic3—62no—
336Dic3—121no—
337Dic3—122no—
338Dic3—122no—
339C32#148911yescharacteristic, complemented, Sylow
340C23—91no—
341C23—181no—
342Q8—21no—
343Q8#371212yescomplemented
344D4—92no—
345D4—184no—
346D4—181no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—91no—
350C2×C4—91no—
351C2×C4—91no—
352C2×C4—182no—
353C2×C4—184no—
354C2×C4—181no—
355C2×C4—181no—
356C2×C4—181no—
357C2×C4—181no—
358C2×C4—181no—
359C2×C4—362no—
360C2×C4—361no—
361C2×C4—361no—
362C2×C4C2×#62411yescharacteristic
363C2×C4#381111yescharacteristic
364C2×C4#381111yescharacteristic
365C8—64no—
366C8—122no—
367C2×C3—21no—
368C2×C3—21no—
369C2×C3—21no—
370C2×C3#472612yes—
371C2×C3#480312yes—
372C2×C3D3×#10412yes—
373D3—122no—
374D3—124no—
375D3—122no—
376D3—124no—
377C22—181no—
378C22—182no—
379C22—181no—
380C22—181no—
381C22—361no—
382C22—361no—
383C22#6397411yescharacteristic
384C4—22no—
385C4—21no—
386C4—41no—
387C4—92no—
388C4—181no—
389C4—181no—
390C4—361no—
391C4#6381811yescharacteristic
392C4C2×#371211yescharacteristic
393C4#6394111yescharacteristic
394C4#6394511yescharacteristic
395C3—21no—
396C3#6823812yescomplemented
397C2—361no—
398C2—362no—
399C2#1059115911yescharacteristic
400C2#1059155411yescharacteristic
401C2#1059167111yescharacteristic
subgroupG/H
centerC2×C4#3811
commutatorC2×C32×C4C22×C4
FrattiniC42C2×D32
FittingC32×C4×Q8C22
socleC22×C32C2×C22⋊C4
prime radicalC2×#3832C4
cosocleC32×C42C23
hypercenterC4×Q8D32
hypocenterC32#1489
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
2. #10590340C2semidirect
3. C4×#3712C2semidirect
4. C3×#4535C2semidirect
5. #10589082C2semidirect
29. #4535D3semidirect
95. C32×Q8M16semidirect
107. C4⋊(C3⋊D3)M16semidirect
222. C3×Q8ΓL(1,𝔽25)semidirect
339. C32#1489semidirect
343. Q8#3712semidirect
396. C3#68238semidirect

Maximal subgroups

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

subgroupindex
C4×#38322
#105903402
C4×#37122
C3×#45352
#105890822
#682383