Finite Group Databasebeta

← #11806917

Subgroups — #11806917

Order 1152.

1258 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 1258 conjugacy classes of subgroups — too many to draw. Only its 175 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
1C2×#64231C211yesmaximal, characteristic
2C2×C4×GO+(4,𝔽2)C212yesmaximal, complemented
3#10594605C212yesmaximal, complemented
4C4⋊C4×#157C211yesmaximal, characteristic, complemented
5#10593454C212yesmaximal, complemented
6#10593453C214yesmaximal, complemented
7C4×#3773C211yesmaximal, characteristic, complemented
8#10593405C212yesmaximal, complemented
9C22×GO+(4,𝔽2)C2212yes—
10C22×GO+(4,𝔽2)C2212yes—
11C2×#3802C2212yes—
12C2×C4×D32C2214yes—
13C2×C4⋊#157C2211yescharacteristic
14C2×C4×#157C2212yes—
15C2×C4×#157C2211yescharacteristic
16C2×#3773C2211yescharacteristic
17C2×#3773C2211yescharacteristic
18C2×D32⋊C4C2214yes—
19C2×D32⋊C4C2214yes—
20#64231C418yescomplemented
21C4×GO+(4,𝔽2)—28no—
22C4⋊C4×C3⋊D3C2211yescharacteristic
23C3×#893—22no—
24#63921C2214yes—
25Dic3×D12—22no—
26C4×#624C2212yes—
27#63879—24no—
28#63781C2212yes—
29#63737C2212yes—
30C4×D3×D4—62no—
31C22×D32C2314yes—
32C22×#157C2312yes—
33C22×#157C2311yescharacteristic
34C22×#157C2311yescharacteristic
35C2×GO+(4,𝔽2)—24no—
36C2×GO+(4,𝔽2)—24no—
37C2×GO+(4,𝔽2)—42no—
38C2×GO+(4,𝔽2)—42no—
39C2×GO+(4,𝔽2)C2×C418yescomplemented
40C2×GO+(4,𝔽2)C2×C418yescomplemented
41C2×C4×C3⋊D3C2312yes—
42C2×C4×C3⋊D3C2311yescharacteristic
43C2×C3×D12—22no—
44C2×C3×C4×D3—24no—
45C2×#625—22no—
46C2×#624C2312yes—
47C2×#624C2312yes—
48C2×D3×Dic3—24no—
49#3802—24no—
50#3802C2×C418yescomplemented
51C4×D32—28no—
52C4×D32—44no—
53C4⋊#157—22no—
54C4×#157—24no—
55C4×#157C2×C414yescomplemented
56#3773—22no—
57#3773D414yescomplemented
58D32⋊C4—28no—
59D32⋊C4—44no—
60C32×C4⋊C4D411yescharacteristic, complemented
61C4⋊(C3⋊Dic3)D411yescharacteristic, complemented
62#3750D412yescomplemented
63C3×(C2×D3)⋊C4—24no—
64C3×C4×Dic3—22no—
65(C2×D3)⋊Dic3—24no—
66Dic32—22no—
67#2366—91nomaximal, Sylow
68C2×D3×D4—62no—
69C22×C4×D3—64no—
70C3×C4×D4—62no—
71D4×Dic3—62no—
72C4×#67—64no—
73#893—22no—
74D3×C4⋊C4—62no—
75#67⋊C4—64no—
76D3×C22⋊C4—64no—
77C4×D12—62no—
78C42×D3—62no—
79C22×C3⋊D3C2411yescharacteristic
80C22×C3×D3—24no—
81C2×D32—24no—
82C2×D32—24no—
83C2×D32—24no—
84C2×D32—24no—
85C2×D32—24no—
86C2×D32C22×C418yes—
87C2×#157—21no—
88C2×#157—21no—
89C2×#157—22no—
90C2×#157—22no—
91C2×#157—22no—
92C2×#157C22×C412yes—
93C2×#157C22×C412yes—
94C2×#157C2×D412yes—
95C2×#157#4112yes—
96GO+(4,𝔽2)—216no—
97GO+(4,𝔽2)—48no—
98C2×C32×C4C2×D412yes—
99C2×C32×C4C2×D411yescharacteristic
100C2×C3⋊Dic3C2×D412yes—
101C2×C3⋊Dic3C2×D411yescharacteristic
102C4×C3⋊D3—22no—
103C4×C3⋊D3—22no—
104C4×C3⋊D3C22×C412yes—
105C4×C3⋊D3#4112yes—
106C2×C3×Dic3—22no—
107C2×C3×Dic3—22no—
108C3×D12—28no—
109C3×C4×D3—48no—
110#625—28no—
111#624—22no—
112#624—22no—
113#624—22no—
114#624C2×D414yes—
115D3×Dic3—48no—
116C2×#127—91no—
117C2×C4×D4—92no—
118C2×C4×D4—92no—
119#390—92no—
120#390—92no—
121#386—94no—
122C4×C4⋊C4—91no—
123C4×C4⋊C4—91no—
124C23×D3—64no—
125C2×C3×D4—62no—
126C22×C3×C4—64no—
127C2×#67—64no—
128C22×Dic3—64no—
129D3×D4—68no—
130D3×D4—68no—
131C2×D12—22no—
132C2×C4×D3—24no—
133C2×C4×D3—62no—
134C2×C4×D3—62no—
135C2×C4×D3—62no—
136C2×C4×D3—64no—
137C2×C4×D3—64no—
138C2×C4×D3—128no—
139C2×C4×D3—128no—
140C3×C4⋊C4—22no—
141C3×C22⋊C4—64no—
142C3×C42—62no—
143#217—64no—
144(C2×D3)⋊C4—24no—
145(C2×D3)⋊C4—64no—
146C4⋊Dic3—62no—
147Dic3⋊C4—64no—
148C4×Dic3—22no—
149C4×Dic3—62no—
150C4×Dic3—62no—
151C22×C32C22×D411yescharacteristic
152C2×C3⋊D3C23×C411yescharacteristic
153C2×C3⋊D3C22×D411yescharacteristic
154C2×C3⋊D3C2×#4111yescharacteristic
155C2×C3⋊D3C2×#4111yescharacteristic
156C2×C3⋊D3H(5,𝔽2)11yescharacteristic
157C2×C3⋊D3H(5,𝔽2)11yescharacteristic
158C2×C3×D3—28no—
159C2×C3×D3—44no—
160C2×C3×D3—44no—
161D32—28no—
162D32—28no—
163D32—44no—
164D32—44no—
165#157—22no—
166#157—42no—
167#157C4×D414yescomplemented
168C32×C4—22no—
169C32×C4C4×D412yescomplemented
170C3⋊Dic3—22no—
171C3⋊Dic3C4×D412yescomplemented
172C3×Dic3—24no—
173C3×Dic3—42no—
174C22×D4—92no—
175C22×D4—92no—
176C22×D4—92no—
177C23×C4—94no—
178#127—98no—
179C4×D4—62no—
180C4×D4—182no—
181C4×D4—184no—
182C4×D4—188no—
183C2×C4⋊C4—91no—
184C2×C4⋊C4—91no—
185C2×C4⋊C4—91no—
186C2×C4⋊C4—91no—
187C2×C22⋊C4—94no—
188C2×C22⋊C4—94no—
189C2×C22⋊C4—94no—
190C2×C42—92no—
191C2×C42—91no—
192C2×C42—92no—
193#95—92no—
194#95—92no—
195C23×C3—64no—
196C22×D3—24no—
197C22×D3—68no—
198C22×D3—62no—
199C22×D3—68no—
200C22×D3—124no—
201C22×D3—124no—
202C22×D3—124no—
203C22×D3—124no—
204C3×D4—68no—
205C2×C3×C4—22no—
206C2×C3×C4—24no—
207C2×C3×C4—62no—
208C2×C3×C4—62no—
209C2×C3×C4—128no—
210#67—616no—
211C2×Dic3—22no—
212C2×Dic3—22no—
213C2×Dic3—62no—
214C2×Dic3—64no—
215C2×Dic3—128no—
216D12—28no—
217C4×D3—48no—
218C4×D3—64no—
219C4×D3—64no—
220C4×D3—64no—
221C4×D3—64no—
222C4×D3—122no—
223C4×D3—122no—
224C4×D3—124no—
225C4×D3—124no—
226C4×D3—128no—
227C2×C32C2×C4×D411yescharacteristic
228C2×C32D4211yescharacteristic
229C2×C32#55011yescharacteristic
230C3⋊D3C2×C4×D411yescharacteristic
231C3⋊D3#51811yescharacteristic
232C3⋊D3#55011yescharacteristic
233C3⋊D3#55911yescharacteristic
234C3×D3—48no—
235C24—94no—
236C2×D4—62no—
237C2×D4—98no—
238C2×D4—98no—
239C2×D4—98no—
240C2×D4—184no—
241C2×D4—182no—
242C2×D4—184no—
243C2×D4—184no—
244C2×D4—362no—
245C2×D4—362no—
246C22×C4—64no—
247C22×C4—92no—
248C22×C4—92no—
249C22×C4—91no—
250C22×C4—91no—
251C22×C4—91no—
252C22×C4—92no—
253C22×C4—92no—
254C22×C4—184no—
255C22×C4—188no—
256C22×C4—364no—
257C4⋊C4—94no—
258C4⋊C4—92no—
259C4⋊C4—91no—
260C4⋊C4—182no—
261C4⋊C4—182no—
262C4⋊C4GO+(4,𝔽2)11yescharacteristic, complemented
263C22⋊C4—64no—
264C22⋊C4—184no—
265C22⋊C4—188no—
266C22⋊C4—364no—
267C42—62no—
268C42—94no—
269C42—182no—
270C42—184no—
271C22×C3—22no—
272C22×C3—68no—
273C22×C3—124no—
274C22×C3—124no—
275C2×D3—28no—
276C2×D3—44no—
277C2×D3—44no—
278C2×D3—62no—
279C2×D3—62no—
280C2×D3—62no—
281C2×D3—62no—
282C2×D3—62no—
283C2×D3—62no—
284C2×D3—128no—
285C2×D3—128no—
286C2×D3—128no—
287C2×D3—128no—
288C3×C4—24no—
289C3×C4—44no—
290C3×C4—64no—
291C3×C4—122no—
292Dic3—24no—
293Dic3—42no—
294Dic3—64no—
295Dic3—124no—
296C32#236611yescharacteristic, complemented, Sylow
297C23—64no—
298C23—98no—
299C23—91no—
300C23—184no—
301C23—184no—
302C23—184no—
303C23—184no—
304C23—184no—
305D4—68no—
306D4—188no—
307D4—1816no—
308D4—368no—
309C2×C4—62no—
310C2×C4—62no—
311C2×C4—94no—
312C2×C4—92no—
313C2×C4—92no—
314C2×C4—92no—
315C2×C4—91no—
316C2×C4—92no—
317C2×C4—92no—
318C2×C4—92no—
319C2×C4—92no—
320C2×C4—128no—
321C2×C4—182no—
322C2×C4—182no—
323C2×C4—182no—
324C2×C4—182no—
325C2×C4—182no—
326C2×C4—181no—
327C2×C4—181no—
328C2×C4—182no—
329C2×C4—182no—
330C2×C4—182no—
331C2×C4—368no—
332C2×C4C2×GO+(4,𝔽2)12yes—
333C2×C4C2×GO+(4,𝔽2)11yescharacteristic
334C2×C3—22no—
335C2×C3—22no—
336C2×C3—22no—
337C2×C3—128no—
338D3—48no—
339D3—62no—
340D3—62no—
341D3—62no—
342D3—62no—
343C22—68no—
344C22—91no—
345C22—91no—
346C22—91no—
347C22—91no—
348C22—91no—
349C22—91no—
350C22—124no—
351C22—124no—
352C22—188no—
353C22—188no—
354C22—364no—
355C22—364no—
356C22C22×GO+(4,𝔽2)11yescharacteristic
357C4—22no—
358C4—64no—
359C4—94no—
360C4—92no—
361C4—122no—
362C4—182no—
363C4—182no—
364C4—362no—
365C4C4×GO+(4,𝔽2)12yescomplemented
366C3—22no—
367C2—91no—
368C2—91no—
369C2—91no—
370C2—91no—
371C2—128no—
372C2C2×C4×GO+(4,𝔽2)11yescharacteristic
373C2D4×GO+(4,𝔽2)11yescharacteristic
374C2#1059652511yescharacteristic
subgroupG/H
center, FrattiniC22C22×GO+(4,𝔽2)
commutatorC2×C3⋊D3C23×C4
FittingC32×C4⋊C4D4
socleC22×C32C22×D4
prime radicalC2×#64231C2
cosocleC22×C3⋊D3C24
hypercenterC4⋊C4GO+(4,𝔽2)
hypocenterC32#2366
LCSG ▷ C2×C3⋊D3 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×GO+(4,𝔽2)C2semidirect
3. #10594605C2semidirect
4. C4⋊C4×#157C2semidirect
5. #10593454C2semidirect
6. #10593453C2semidirect
7. C4×#3773C2semidirect
8. #10593405C2semidirect
20. #64231C4semidirect
39. C2×GO+(4,𝔽2)C2×C4semidirect
40. C2×GO+(4,𝔽2)C2×C4semidirect
50. #3802C2×C4semidirect
55. C4×#157C2×C4semidirect
57. #3773D4semidirect
60. C32×C4⋊C4D4semidirect
61. C4⋊(C3⋊Dic3)D4semidirect
62. #3750D4semidirect
167. #157C4×D4semidirect
169. C32×C4C4×D4semidirect
171. C3⋊Dic3C4×D4semidirect
262. C4⋊C4GO+(4,𝔽2)semidirect
296. C32#2366semidirect
365. C4C4×GO+(4,𝔽2)semidirect

Maximal subgroups

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

subgroupindex
C2×#642312
C2×C4×GO+(4,𝔽2)2
#105946052
C4⋊C4×#1572
#105934542
#105934532
C4×#37732
#105934052
#23669