Finite Group Databasebeta

← #41014647

Subgroups — #41014647

Order 27008.

382 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 67 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
1#27213705C211yesmaximal, characteristic
2C2×#20269892C211yesmaximal, characteristic, complemented
3C2×#20269891C211yesmaximal, characteristic, complemented
4#27213025C211yesmaximal, characteristic, complemented
5#27213019C211yesmaximal, characteristic, complemented
6#27212912C211yesmaximal, characteristic, complemented
7#386×C211C211yesmaximal, characteristic, complemented
8C22×C22⋊D211C2211yescharacteristic
9#20269960—21no—
10#20269958—21no—
11#20269949—21no—
12#20269949—21no—
13#20269946—21no—
14#20269945—21no—
15C2×Dic211⋊C4—21no—
16C2×C22⋊Dic211C2211yescharacteristic
17C2×C22⋊Dic211C2211yescharacteristic
18C2×C22⋊Dic211C2211yescharacteristic
19C2×(C2×D211)⋊C4—21no—
20#20269892—22no—
21#20269891—22no—
22C211⋊#95—21no—
23C211⋊#95—21no—
24C22×D4×C211C2211yescharacteristic
25C23×C4×C211C2211yescharacteristic
26C2×C22⋊C4×C211—21no—
27C2×C22⋊C4×C211C2211yescharacteristic
28#95×C211—21no—
29C23×D211D411yescharacteristic
30C2×C22⋊D211—21no—
31C2×C22⋊D211—21no—
32C2×C22⋊D211—21no—
33C2×C22⋊D211—21no—
34C2×C22⋊D211—21no—
35C2×C22⋊D211—21no—
36C2×C22⋊D211—21no—
37C2×C22⋊D211—21no—
38C22×Dic211—21no—
39C22×Dic211—21no—
40C22×Dic211D411yescharacteristic
41C22×Dic211D411yescharacteristic
42C22×Dic211D411yescharacteristic
43Dic211⋊C4—21no—
44Dic211⋊C4—21no—
45Dic211⋊C4—42no—
46C22⋊Dic211—21no—
47C22⋊Dic211—21no—
48C22⋊Dic211—21no—
49C22⋊Dic211—21no—
50C22⋊Dic211—21no—
51C22⋊Dic211—21no—
52C22⋊Dic211—21no—
53C22⋊Dic211D412yescomplemented
54(C2×D211)⋊C4—21no—
55(C2×D211)⋊C4—21no—
56(C2×D211)⋊C4—42no—
57C24×C211D411yescharacteristic
58C24×C211C2311yescharacteristic
59C2×D4×C211—21no—
60C2×D4×C211—21no—
61C2×D4×C211—21no—
62C2×D4×C211—21no—
63C2×D4×C211—21no—
64C2×D4×C211—21no—
65C22×C4×C211—21no—
66C22×C4×C211—21no—
67C22×C4×C211—21no—
68C22×C4×C211—21no—
69C22×C4×C211—42no—
70C22×C4×C211D411yescharacteristic
71C22⋊C4×C211—21no—
72C22⋊C4×C211—21no—
73C22⋊C4×C211—42no—
74C22×D211—21no—
75C22×D211—21no—
76C22×D211—21no—
77C22×D211—41no—
78C22×D211—41no—
79C22⋊D211—41no—
80C22⋊D211—41no—
81C22⋊D211—41no—
82C22⋊D211—41no—
83C22⋊D211—81no—
84C22⋊D211—81no—
85C2×Dic211—21no—
86C2×Dic211—21no—
87C2×Dic211—21no—
88C2×Dic211—22no—
89C2×Dic211—21no—
90C2×Dic211—21no—
91C2×Dic211—41no—
92C2×Dic211—41no—
93C2×Dic211—41no—
94C2×Dic211—41no—
95C2×Dic211—41no—
96C2×Dic211—41no—
97C2×Dic211—41no—
98C23×C211—21no—
99C23×C211—21no—
100C23×C211—21no—
101C23×C211—41no—
102C23×C211—41no—
103C23×C211—41no—
104C23×C211—41no—
105C23×C211C2×D411yescharacteristic
106C23×C211C2×D411yescharacteristic
107C23×C211C2×D411yescharacteristic
108C23×C211C2×D411yescharacteristic
109C23×C211#4111yescharacteristic
110C23×C211#4111yescharacteristic
111C23×C211#4111yescharacteristic
112D4×C211—41no—
113D4×C211—41no—
114D4×C211—41no—
115D4×C211—41no—
116C2×C4×C211—22no—
117C2×C4×C211—22no—
118C2×C4×C211—22no—
119C2×C4×C211—21no—
120C2×C4×C211—21no—
121C2×C4×C211—21no—
122C2×C4×C211—41no—
123C2×C4×C211—41no—
124C2×C4×C211—41no—
125C2×C4×C211—41no—
126C2×C4×C211—41no—
127C2×C4×C211—41no—
128C2×C4×C211—41no—
129C22×C211—21no—
130C22×C211—21no—
131C22×C211—21no—
132C22×C211—21no—
133C22×C211—21no—
134C22×C211—21no—
135C22×C211—41no—
136C22×C211—41no—
137C22×C211—41no—
138C22×C211—41no—
139C22×C211—42no—
140C22×C211—41no—
141C22×C211—41no—
142C22×C211—41no—
143C22×C211—81no—
144C22×C211#12011yescharacteristic
145C22×C211#12111yescharacteristic
146C22×C211#12111yescharacteristic
147C22×C211#12111yescharacteristic
148C22×C211#12311yescharacteristic
149C22×C211#12311yescharacteristic
150C22×C211#12411yescharacteristic
151C2×D211—42no—
152C2×D211—41no—
153C2×D211—41no—
154C2×D211—41no—
155C2×D211—81no—
156C4×C211—41no—
157C4×C211—42no—
158C4×C211—81no—
159Dic211—42no—
160Dic211—41no—
161Dic211—41no—
162Dic211—41no—
163Dic211—81no—
164C2×C211—21no—
165C2×C211—21no—
166C2×C211—21no—
167C2×C211—21no—
168C2×C211—21no—
169C2×C211—21no—
170C2×C211—81no—
171C2×C211#39411yescharacteristic
172C2×C211B(4,𝔽2)11yescharacteristic
173C2×C211#45811yescharacteristic
174D211—81no—
175C211#204411yescharacteristic, complemented, Sylow
176#2044—2111nomaximal, Sylow
177#534—2111no—
178C2×#99—2111no—
179C2×#99—2111no—
180#386—2111no—
181#386D21111yescharacteristic, complemented
182#385—2111no—
183#342—2111no—
184C22×D4—2111no—
185C22×D4C2×D21111yescharacteristic
186C23×C4C2×D21111yescharacteristic
187#123—4221no—
188#123—4221no—
189#121—4221no—
190#121—4221no—
191#120—4221no—
192#120—4221no—
193C2×C4⋊C4—4221no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—4221no—
196C2×C22⋊C4—2111no—
197C2×C22⋊C4—2111no—
198C2×C22⋊C4—2111no—
199C2×C22⋊C4C2×D21111yescharacteristic
200#99—4222no—
201#99—4222no—
202#95—21no—
203#95—4221no—
204#95—4221no—
205C24—2111no—
206C24C22⋊D21111yescharacteristic
207C24C22×D21111yescharacteristic
208C2×D4—21no—
209C2×D4—21no—
210C2×D4—21no—
211C2×D4—21no—
212C2×D4—21no—
213C2×D4—21no—
214C2×D4—4221no—
215C2×D4—4221no—
216C2×D4—4221no—
217C2×D4—4221no—
218C2×D4—4221no—
219C2×D4—4221no—
220C2×D4—4221no—
221C2×D4—4221no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—42no—
227C22×C4—4221no—
228C22×C4—4221no—
229C22×C4—2111no—
230C22×C4—2111no—
231C22×C4—2111no—
232C22×C4C22⋊D21111yescharacteristic
233C4⋊C4—8442no—
234C4⋊C4—4221no—
235C4⋊C4—4221no—
236C22⋊C4—21no—
237C22⋊C4—21no—
238C22⋊C4—42no—
239C22⋊C4—8442no—
240C22⋊C4—4221no—
241C22⋊C4—4221no—
242C22⋊C4—4221no—
243C22⋊C4—4221no—
244C22⋊C4—4221no—
245C22⋊C4—4221no—
246C22⋊C4—4221no—
247C22⋊C4—4221no—
248C22⋊C4—4221no—
249C22⋊C4—2112no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—8441no—
258C23—8441no—
259C23—4221no—
260C23—4221no—
261C23—4221no—
262C23D4×D21111yescharacteristic
263C23D4×D21111yescharacteristic
264C23D4×D21111yescharacteristic
265C23C2×C22⋊D21111yescharacteristic
266C23#1680996111yescharacteristic
267C23#1680996211yescharacteristic
268C23#1680996211yescharacteristic
269D4—41no—
270D4—41no—
271D4—41no—
272D4—41no—
273D4—8441no—
274D4—8441no—
275D4—8441no—
276D4—8441no—
277D4—16881no—
278D4—16881no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—8441no—
293C2×C4—8441no—
294C2×C4—8441no—
295C2×C4—8441no—
296C2×C4—8441no—
297C2×C4—8441no—
298C2×C4—8441no—
299C2×C4—4221no—
300C2×C4—4221no—
301C2×C4—4221no—
302C2×C4—4222no—
303C2×C4—4221no—
304C2×C4—4221no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—42no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—81no—
320C22—8442no—
321C22—8441no—
322C22—8441no—
323C22—8441no—
324C22—16881no—
325C22#2026994511yescharacteristic
326C22#2026994911yescharacteristic
327C22#2026995011yescharacteristic
328C22#2026995011yescharacteristic
329C22#2026995911yescharacteristic
330C22#2026996011yescharacteristic
331C22#2026996311yescharacteristic
332C4—41no—
333C4—42no—
334C4—81no—
335C4—8442no—
336C4—8441no—
337C4—8441no—
338C4—8441no—
339C4—16881no—
340C2—21no—
341C2—21no—
342C2—21no—
343C2—21no—
344C2—21no—
345C2—21no—
346C2—81no—
347C2—16881no—
348C2#2721306111yescharacteristic
349C2#2721333211yescharacteristic
350C2#2721333511yescharacteristic
subgroupG/H
centerC22#20269945
commutator, cosocleC24×C211C23
FrattiniC24C22×D211
Fitting#386×C211C2
socleC22×C211#120
prime radical#27213705C2
hypercenter#386D211
hypocenterC211#2044
LCSG ▷ C24×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C24 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×#20269892C2semidirect
3. C2×#20269891C2semidirect
4. #27213025C2semidirect
5. #27213019C2semidirect
6. #27212912C2semidirect
7. #386×C211C2semidirect
53. C22⋊Dic211D4semidirect
175. C211#2044semidirect
181. #386D211semidirect

Maximal subgroups

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

subgroupindex
#272137052
C2×#202698922
C2×#202698912
#272130252
#272130192
#272129122
#386×C2112
#2044211