Finite Group Databasebeta

← #41014232

Subgroups — #41014232

Order 27008.

408 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 85 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×#20269949C211yesmaximal, characteristic
2C2×#20269892C211yesmaximal, characteristic, complemented
3C2×#20269891C211yesmaximal, characteristic, complemented
4C2×#20269891C211yesmaximal, characteristic, complemented
5#27213004C211yesmaximal, characteristic, complemented
6C22⋊C4×Dic211C211yesmaximal, characteristic, complemented
7C2×#99×C211C211yesmaximal, characteristic, complemented
8C22×C22⋊D211C411yescharacteristic, complemented
9C22×C22⋊D211C2211yescharacteristic
10C23×Dic211C2211yescharacteristic
11#20269949—21no—
12#20269949—21no—
13#20269949—21no—
14#20269949—21no—
15C2×Dic211⋊C4—21no—
16C2×Dic211⋊C4C411yescharacteristic, complemented
17C2×C22⋊Dic211C411yescharacteristic, complemented
18C2×C22⋊Dic211C2211yescharacteristic
19C2×C22⋊Dic211C2211yescharacteristic
20C2×(C2×D211)⋊C4C411yescharacteristic, complemented
21C2×C4×Dic211—21no—
22#20269892—22no—
23#20269891—22no—
24#20269891—22no—
25C211⋊#95—21no—
26C211⋊#95—21no—
27C22×D4×C211C2211yescharacteristic
28C2×C22⋊C4×C211C2211yescharacteristic
29C2×C22⋊C4×C211C2211yescharacteristic
30#99×C211—22no—
31C23×D211C2×C411yescharacteristic
32C2×C22⋊D211—21no—
33C2×C22⋊D211—21no—
34C2×C22⋊D211—21no—
35C2×C22⋊D211—21no—
36C2×C22⋊D211—21no—
37C2×C22⋊D211—21no—
38C2×C22⋊D211—21no—
39C2×C22⋊D211—21no—
40C2×C22⋊D211—42no—
41C22×Dic211—21no—
42C22×Dic211—21no—
43C22×Dic211—42no—
44C22×Dic211C2×C411yescharacteristic
45C22×Dic211C2×C411yescharacteristic
46C22×Dic211C2×C411yescharacteristic
47C22×Dic211D411yescharacteristic
48C22×Dic211D411yescharacteristic
49C22×Dic211D411yescharacteristic
50C22×Dic211D411yescharacteristic
51Dic211⋊C4—21no—
52Dic211⋊C4—21no—
53Dic211⋊C4—42no—
54C22⋊Dic211—21no—
55C22⋊Dic211—21no—
56C22⋊Dic211—21no—
57C22⋊Dic211—21no—
58C22⋊Dic211—21no—
59C22⋊Dic211D412yescomplemented
60(C2×D211)⋊C4—21no—
61(C2×D211)⋊C4—21no—
62C4×Dic211—42no—
63C24×C211C2×C411yescharacteristic
64C24×C211C2311yescharacteristic
65C2×D4×C211—21no—
66C2×D4×C211—21no—
67C2×D4×C211—21no—
68C2×D4×C211—21no—
69C2×D4×C211—21no—
70C2×D4×C211—21no—
71C22×C4×C211—21no—
72C22×C4×C211—21no—
73C22×C4×C211C2×C411yescharacteristic
74C22⋊C4×C211—21no—
75C22⋊C4×C211—21no—
76C22⋊C4×C211—21no—
77C22⋊C4×C211D412yescomplemented
78C22×D211—21no—
79C22×D211—21no—
80C22×D211—21no—
81C22×D211—41no—
82C22×D211—41no—
83C22⋊D211—42no—
84C22⋊D211—41no—
85C22⋊D211—41no—
86C22⋊D211—41no—
87C22⋊D211—41no—
88C22⋊D211—42no—
89C2×Dic211—22no—
90C2×Dic211—22no—
91C2×Dic211—22no—
92C2×Dic211—21no—
93C2×Dic211—21no—
94C2×Dic211—21no—
95C2×Dic211—22no—
96C2×Dic211—41no—
97C2×Dic211—41no—
98C2×Dic211—41no—
99C2×Dic211—41no—
100C2×Dic211—41no—
101C2×Dic211—41no—
102C2×Dic211—41no—
103C2×Dic211—41no—
104C2×Dic211C22⋊C411yescharacteristic
105C2×Dic211C22⋊C411yescharacteristic
106C2×Dic211C22⋊C411yescharacteristic
107C2×Dic211C22⋊C411yescharacteristic
108C23×C211—21no—
109C23×C211—21no—
110C23×C211—21no—
111C23×C211—41no—
112C23×C211—41no—
113C23×C211—41no—
114C23×C211—41no—
115C23×C211C22×C411yescharacteristic
116C23×C211C2×D411yescharacteristic
117C23×C211C2×D411yescharacteristic
118C23×C211C2×D411yescharacteristic
119C23×C211C2×D411yescharacteristic
120C23×C211#4111yescharacteristic
121C23×C211#4111yescharacteristic
122D4×C211—41no—
123D4×C211—41no—
124D4×C211—41no—
125D4×C211—41no—
126C2×C4×C211—21no—
127C2×C4×C211—21no—
128C2×C4×C211—21no—
129C2×C4×C211—22no—
130C2×C4×C211—41no—
131C2×C4×C211—41no—
132C2×C4×C211—41no—
133C2×C4×C211—41no—
134C2×C4×C211—41no—
135C22×C211—21no—
136C22×C211—21no—
137C22×C211—21no—
138C22×C211—21no—
139C22×C211—21no—
140C22×C211—21no—
141C22×C211—41no—
142C22×C211—41no—
143C22×C211—41no—
144C22×C211—41no—
145C22×C211—41no—
146C22×C211—41no—
147C22×C211—41no—
148C22×C211—81no—
149C22×C211—81no—
150C22×C211C2×C22⋊C411yescharacteristic
151C22×C211C4×D411yescharacteristic
152C22×C211C4×D411yescharacteristic
153C22×C211#12111yescharacteristic
154C22×C211#12111yescharacteristic
155C22×C211#12411yescharacteristic
156C22×C211#12711yescharacteristic
157C2×D211—41no—
158C2×D211—41no—
159C2×D211—41no—
160C2×D211—81no—
161C2×D211—81no—
162C4×C211—42no—
163C4×C211—41no—
164C4×C211—81no—
165Dic211—21no—
166Dic211—21no—
167Dic211—21no—
168Dic211—42no—
169Dic211—41no—
170Dic211—81no—
171Dic211#9912yescomplemented
172C2×C211—21no—
173C2×C211—21no—
174C2×C211—21no—
175C2×C211—21no—
176C2×C211—21no—
177C2×C211—21no—
178C2×C211—81no—
179C2×C211#39011yescharacteristic
180C2×C211C2×#9911yescharacteristic
181C2×C211#41011yescharacteristic
182D211—81no—
183C211#196411yescharacteristic, complemented, Sylow
184#1964—2111nomaximal, Sylow
185C2×#121—2111no—
186C2×#99—2111no—
187C2×#99—2111no—
188C2×#99—2111no—
189C2×#99D21111yescharacteristic, complemented
190#380—2111no—
191C4×C22⋊C4—2111no—
192C22×D4—2111no—
193C22×D4—2111no—
194C22×D4C2×D21111yescharacteristic
195C23×C4—2111no—
196#121—4221no—
197#121—4221no—
198#121—4221no—
199#121—4221no—
200C2×C4⋊C4—4221no—
201C2×C4⋊C4—2111no—
202C2×C22⋊C4—2111no—
203C2×C22⋊C4—2111no—
204C2×C22⋊C4—2111no—
205C2×C22⋊C4—2111no—
206C2×C22⋊C4C2×D21111yescharacteristic
207C2×C22⋊C4C2×D21111yescharacteristic
208C2×C42—4221no—
209#99—22no—
210#99—4222no—
211#99—4222no—
212#99—4222no—
213#95—4221no—
214#95—4221no—
215C24—2111no—
216C24C4×D21111yescharacteristic
217C24C22×D21111yescharacteristic
218C2×D4—21no—
219C2×D4—21no—
220C2×D4—21no—
221C2×D4—21no—
222C2×D4—21no—
223C2×D4—21no—
224C2×D4—8442no—
225C2×D4—4221no—
226C2×D4—4221no—
227C2×D4—4221no—
228C2×D4—4221no—
229C2×D4—4221no—
230C2×D4—4221no—
231C2×D4—4221no—
232C2×D4—4221no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—8442no—
236C22×C4—4221no—
237C22×C4—4221no—
238C22×C4—2111no—
239C22×C4—2111no—
240C22×C4—2111no—
241C22×C4—2111no—
242C22×C4—2111no—
243C22×C4—2111no—
244C22×C4—2111no—
245C22×C4C4×D21111yescharacteristic
246C4⋊C4—8442no—
247C4⋊C4—4221no—
248C4⋊C4—4221no—
249C22⋊C4—21no—
250C22⋊C4—21no—
251C22⋊C4—21no—
252C22⋊C4—4221no—
253C22⋊C4—4221no—
254C22⋊C4—4221no—
255C22⋊C4—4221no—
256C22⋊C4—4221no—
257C22⋊C4—4221no—
258C22⋊C4—4221no—
259C22⋊C4—2112no—
260C22⋊C4C22⋊D21112yescomplemented
261C42—8442no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—8441no—
270C23—8441no—
271C23—4221no—
272C23—4221no—
273C23—4221no—
274C23C2×C4×D21111yescharacteristic
275C23D4×D21111yescharacteristic
276C23D4×D21111yescharacteristic
277C23D4×D21111yescharacteristic
278C23C2×C22⋊D21111yescharacteristic
279C23#1680996111yescharacteristic
280C23#1680996211yescharacteristic
281D4—41no—
282D4—41no—
283D4—41no—
284D4—41no—
285D4—8441no—
286D4—8441no—
287D4—8441no—
288D4—8441no—
289D4—8442no—
290D4—8442no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—8441no—
301C2×C4—8441no—
302C2×C4—8441no—
303C2×C4—8441no—
304C2×C4—8441no—
305C2×C4—8441no—
306C2×C4—8441no—
307C2×C4—8441no—
308C2×C4—4221no—
309C2×C4—4221no—
310C2×C4—4221no—
311C2×C4—4222no—
312C2×C4—4222no—
313C2×C4—4222no—
314C2×C4—4222no—
315C2×C4—2111no—
316C2×C4—2111no—
317C2×C4—2111no—
318C2×C4—2111no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—81no—
333C22—81no—
334C22—8441no—
335C22—8441no—
336C22—8441no—
337C22—16881no—
338C22—16881no—
339C22C22⋊C4×D21111yescharacteristic
340C22(C22⋊D211)⋊C411yescharacteristic
341C22C4×C22⋊D21111yescharacteristic
342C22#2026994911yescharacteristic
343C22#2026995011yescharacteristic
344C22#2026996311yescharacteristic
345C22#2026997311yescharacteristic
346C4—42no—
347C4—41no—
348C4—81no—
349C4—8442no—
350C4—8441no—
351C4—16881no—
352C4—4221no—
353C4—4221no—
354C4—4221no—
355C4—2112no—
356C2—21no—
357C2—21no—
358C2—21no—
359C2—21no—
360C2—21no—
361C2—21no—
362C2—81no—
363C2—16881no—
364C2#2721304611yescharacteristic
365C2#99×D21111yescharacteristic
366C2#2721311811yescharacteristic
subgroupG/H
centerC22C22⋊C4×D211
commutatorC23×C211C22×C4
FrattiniC24C22×D211
FittingC2×#99×C211C2
socleC22×C211C2×C22⋊C4
prime radicalC2×#20269949C2
cosocleC24×C211C23
hypercenterC2×#99D211
hypocenterC211#1964
LCSG ▷ C23×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
2. C2×#20269892C2semidirect
3. C2×#20269891C2semidirect
4. C2×#20269891C2semidirect
5. #27213004C2semidirect
6. C22⋊C4×Dic211C2semidirect
7. C2×#99×C211C2semidirect
8. C22×C22⋊D211C4semidirect
16. C2×Dic211⋊C4C4semidirect
17. C2×C22⋊Dic211C4semidirect
20. C2×(C2×D211)⋊C4C4semidirect
59. C22⋊Dic211D4semidirect
77. C22⋊C4×C211D4semidirect
171. Dic211#99semidirect
183. C211#1964semidirect
189. C2×#99D211semidirect
260. C22⋊C4C22⋊D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202699492
C2×#202698922
C2×#202698912
#272130042
C22⋊C4×Dic2112
C2×#99×C2112
#1964211