Finite Group Databasebeta

← #2715×C211 #41010335

Subgroups — #2715×C211 #41010335

Order 27008.

498 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#126×C211C211yesmaximal, characteristic, complemented
2C2×#124×C211C211yesmaximal, characteristic
3#397×C211C211yesmaximal, characteristic, complemented
4#397×C211C211yesmaximal, characteristic, complemented
5#396×C211C211yesmaximal, characteristic, complemented
6#396×C211C211yesmaximal, characteristic, complemented
7#395×C211C211yesmaximal, characteristic, complemented
8#394×C211C211yesmaximal, characteristic
9#391×C211C211yesmaximal, characteristic, complemented
10#389×C211C211yesmaximal, characteristic, complemented
11#388×C211C211yesmaximal, characteristic, complemented
12#387×C211C211yesmaximal, characteristic, complemented
13#386×C211C211yesmaximal, characteristic
14#385×C211C211yesmaximal, characteristic, complemented
15#380×C211C211yesmaximal, characteristic, complemented
16C22×Q8×C211C2211yescharacteristic
17C22×D4×C211C2211yescharacteristic
18C23×C4×C211C2211yescharacteristic
19#126×C211—24no—
20#124×C211—24no—
21C2×C4⋊C4×C211C2211yescharacteristic
22C2×C4⋊C4×C211C2211yescharacteristic
23C2×C4⋊C4×C211C2211yescharacteristic
24C2×C4⋊C4×C211C2211yescharacteristic
25C2×C4⋊C4×C211C2211yescharacteristic
26C2×C4⋊C4×C211C2211yescharacteristic
27C2×C4⋊C4×C211C2211yescharacteristic
28C2×C4⋊C4×C211C2211yescharacteristic
29C2×C4⋊C4×C211C2211yescharacteristic
30C2×C22⋊C4×C211C2211yescharacteristic
31C2×C22⋊C4×C211C2211yescharacteristic
32C2×C22⋊C4×C211C2211yescharacteristic
33C2×C22⋊C4×C211C2211yescharacteristic
34C2×C22⋊C4×C211C2211yescharacteristic
35C2×C22⋊C4×C211C2211yescharacteristic
36C2×C22⋊C4×C211C2211yescharacteristic
37C2×C22⋊C4×C211C2211yescharacteristic
38C2×C22⋊C4×C211C2211yescharacteristic
39C2×C22⋊C4×C211C2211yescharacteristic
40C2×C22⋊C4×C211C2211yescharacteristic
41C2×C42×C211C2211yescharacteristic
42C2×C42×C211C2211yescharacteristic
43#95×C211C2211yescharacteristic
44#95×C211C2211yescharacteristic
45#95×C211C2211yescharacteristic
46#95×C211C2211yescharacteristic
47#95×C211C2211yescharacteristic
48#95×C211C2211yescharacteristic
49#95×C211C2211yescharacteristic
50#95×C211C2211yescharacteristic
51#95×C211C2211yescharacteristic
52#95×C211C2211yescharacteristic
53C24×C211C2311yescharacteristic
54C24×C211C2311yescharacteristic
55C2×Q8×C211—22no—
56C2×Q8×C211—22no—
57C2×Q8×C211—41no—
58C2×D4×C211—22no—
59C2×D4×C211—22no—
60C2×D4×C211—41no—
61C22×C4×C211—22no—
62C22×C4×C211—41no—
63C22×C4×C211—41no—
64C22×C4×C211C2311yescharacteristic
65C22×C4×C211C2311yescharacteristic
66C22×C4×C211C2311yescharacteristic
67C22×C4×C211C2311yescharacteristic
68C22×C4×C211C2311yescharacteristic
69C22×C4×C211C2311yescharacteristic
70C22×C4×C211C2311yescharacteristic
71C22×C4×C211C2311yescharacteristic
72C22×C4×C211C2311yescharacteristic
73C22×C4×C211C2311yescharacteristic
74C22×C4×C211C2311yescharacteristic
75C22×C4×C211C2311yescharacteristic
76C22×C4×C211C2311yescharacteristic
77C4⋊C4×C211—22no—
78C4⋊C4×C211—22no—
79C4⋊C4×C211—22no—
80C4⋊C4×C211—22no—
81C4⋊C4×C211—22no—
82C4⋊C4×C211—41no—
83C4⋊C4×C211—41no—
84C4⋊C4×C211—41no—
85C4⋊C4×C211—41no—
86C22⋊C4×C211—22no—
87C22⋊C4×C211—22no—
88C22⋊C4×C211—22no—
89C22⋊C4×C211—22no—
90C22⋊C4×C211—22no—
91C22⋊C4×C211—22no—
92C22⋊C4×C211—41no—
93C22⋊C4×C211—41no—
94C22⋊C4×C211—41no—
95C22⋊C4×C211—41no—
96C22⋊C4×C211D414yes—
97C42×C211—22no—
98C42×C211—22no—
99C23×C211—21no—
100C23×C211—21no—
101C23×C211—21no—
102C23×C211—21no—
103C23×C211—21no—
104C23×C211—21no—
105C23×C211—21no—
106C23×C211—21no—
107C23×C211—21no—
108C23×C211—21no—
109C23×C211—21no—
110C23×C211—21no—
111C23×C211—21no—
112C23×C211C2×D412yes—
113C23×C211C2411yescharacteristic
114Q8×C211—44no—
115D4×C211—44no—
116C2×C4×C211—21no—
117C2×C4×C211—21no—
118C2×C4×C211—21no—
119C2×C4×C211—21no—
120C2×C4×C211—21no—
121C2×C4×C211—21no—
122C2×C4×C211—21no—
123C2×C4×C211—21no—
124C2×C4×C211—21no—
125C2×C4×C211—21no—
126C2×C4×C211—21no—
127C2×C4×C211—21no—
128C2×C4×C211—21no—
129C2×C4×C211—21no—
130C2×C4×C211—21no—
131C2×C4×C211—21no—
132C2×C4×C211—21no—
133C2×C4×C211—21no—
134C2×C4×C211—21no—
135C2×C4×C211—21no—
136C2×C4×C211—21no—
137C2×C4×C211—21no—
138C2×C4×C211—21no—
139C2×C4×C211—21no—
140C2×C4×C211—21no—
141C2×C4×C211—21no—
142C2×C4×C211—21no—
143C2×C4×C211—21no—
144C2×C4×C211—21no—
145C2×C4×C211—21no—
146C2×C4×C211—21no—
147C2×C4×C211—21no—
148C2×C4×C211—21no—
149C2×C4×C211—21no—
150C2×C4×C211—21no—
151C2×C4×C211—44no—
152C2×C4×C211C2×D412yes—
153C2×C4×C211C2×D412yes—
154C2×C4×C211#4112yes—
155C2×C4×C211#4112yes—
156C22×C211—22no—
157C22×C211—22no—
158C22×C211—22no—
159C22×C211—41no—
160C22×C211—41no—
161C22×C211—41no—
162C22×C211—41no—
163C22×C211—41no—
164C22×C211—41no—
165C22×C211—41no—
166C22×C211—41no—
167C22×C211—41no—
168C22×C211—41no—
169C22×C211—41no—
170C22×C211C22×D411yescharacteristic
171C22×C211C2×#4111yescharacteristic
172C22×C211C2×#4111yescharacteristic
173C22×C211H(5,𝔽2)11yescharacteristic
174C22×C211H(5,𝔽2)11yescharacteristic
175C22×C2112−1+411yescharacteristic
176C22×C2112−1+411yescharacteristic
177C4×C211—22no—
178C4×C211—22no—
179C4×C211—22no—
180C4×C211—22no—
181C4×C211—41no—
182C4×C211—41no—
183C4×C211—41no—
184C4×C211—41no—
185C4×C211—41no—
186C4×C211—41no—
187C4×C211—41no—
188C4×C211—41no—
189C4×C211—41no—
190C2×C211—42no—
191C2×C211—81no—
192C2×C211#53711yescharacteristic
193C2×C211#53811yescharacteristic
194C2×C211#54211yescharacteristic
195C2×C211#54611yescharacteristic
196C2×C211#54711yescharacteristic
197C2×C211#55611yescharacteristic
198C2×C211#56311yescharacteristic
199C211#271511yescharacteristic, direct product factor, Sylow
200#2715C21111yesmaximal, characteristic, direct product factor, Sylow
201C2×#126C2×C21111yescharacteristic, complemented
202C2×#124C2×C21111yescharacteristic
203#397C2×C21111yescharacteristic, complemented
204#397C2×C21111yescharacteristic, complemented
205#396C2×C21111yescharacteristic, complemented
206#396C2×C21111yescharacteristic, complemented
207#395C2×C21111yescharacteristic, complemented
208#394C2×C21111yescharacteristic
209#391C2×C21111yescharacteristic, complemented
210#389C2×C21111yescharacteristic, complemented
211#388C2×C21111yescharacteristic, complemented
212#387C2×C21111yescharacteristic, complemented
213#386C2×C21111yescharacteristic
214#385C2×C21111yescharacteristic, complemented
215#380C2×C21111yescharacteristic, complemented
216C22×Q8C22×C21111yescharacteristic
217C22×D4C22×C21111yescharacteristic
218C23×C4C22×C21111yescharacteristic
219#126—24no—
220#124—24no—
221C2×C4⋊C4C22×C21111yescharacteristic
222C2×C4⋊C4C22×C21111yescharacteristic
223C2×C4⋊C4C22×C21111yescharacteristic
224C2×C4⋊C4C22×C21111yescharacteristic
225C2×C4⋊C4C22×C21111yescharacteristic
226C2×C4⋊C4C22×C21111yescharacteristic
227C2×C4⋊C4C22×C21111yescharacteristic
228C2×C4⋊C4C22×C21111yescharacteristic
229C2×C4⋊C4C22×C21111yescharacteristic
230C2×C22⋊C4C22×C21111yescharacteristic
231C2×C22⋊C4C22×C21111yescharacteristic
232C2×C22⋊C4C22×C21111yescharacteristic
233C2×C22⋊C4C22×C21111yescharacteristic
234C2×C22⋊C4C22×C21111yescharacteristic
235C2×C22⋊C4C22×C21111yescharacteristic
236C2×C22⋊C4C22×C21111yescharacteristic
237C2×C22⋊C4C22×C21111yescharacteristic
238C2×C22⋊C4C22×C21111yescharacteristic
239C2×C22⋊C4C22×C21111yescharacteristic
240C2×C22⋊C4C22×C21111yescharacteristic
241C2×C42C22×C21111yescharacteristic
242C2×C42C22×C21111yescharacteristic
243#95C22×C21111yescharacteristic
244#95C22×C21111yescharacteristic
245#95C22×C21111yescharacteristic
246#95C22×C21111yescharacteristic
247#95C22×C21111yescharacteristic
248#95C22×C21111yescharacteristic
249#95C22×C21111yescharacteristic
250#95C22×C21111yescharacteristic
251#95C22×C21111yescharacteristic
252#95C22×C21111yescharacteristic
253C24C23×C21111yescharacteristic
254C24C23×C21111yescharacteristic
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—41no—
258C2×D4—22no—
259C2×D4—22no—
260C2×D4—41no—
261C22×C4—22no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4C23×C21111yescharacteristic
265C22×C4C23×C21111yescharacteristic
266C22×C4C23×C21111yescharacteristic
267C22×C4C23×C21111yescharacteristic
268C22×C4C23×C21111yescharacteristic
269C22×C4C23×C21111yescharacteristic
270C22×C4C23×C21111yescharacteristic
271C22×C4C23×C21111yescharacteristic
272C22×C4C23×C21111yescharacteristic
273C22×C4C23×C21111yescharacteristic
274C22×C4C23×C21111yescharacteristic
275C22×C4C23×C21111yescharacteristic
276C22×C4C23×C21111yescharacteristic
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C22⋊C4—22no—
287C22⋊C4—22no—
288C22⋊C4—22no—
289C22⋊C4—22no—
290C22⋊C4—22no—
291C22⋊C4—22no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C22⋊C4D4×C21114yes—
297C42—22no—
298C42—22no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23C2×D4×C21112yes—
313C23C24×C21111yescharacteristic
314Q8—44no—
315D4—44no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—44no—
352C2×C4C2×D4×C21112yes—
353C2×C4C2×D4×C21112yes—
354C2×C4#41×C21112yes—
355C2×C4#41×C21112yes—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22C22×D4×C21111yescharacteristic
371C22C2×#41×C21111yescharacteristic
372C22C2×#41×C21111yescharacteristic
373C22H(5,𝔽2)×C21111yescharacteristic
374C22H(5,𝔽2)×C21111yescharacteristic
375C222−1+4×C21111yescharacteristic
376C222−1+4×C21111yescharacteristic
377C4—22no—
378C4—22no—
379C4—22no—
380C4—22no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—41no—
386C4—41no—
387C4—41no—
388C4—41no—
389C4—41no—
390C2—42no—
391C2—81no—
392C2#537×C21111yescharacteristic
393C2#538×C21111yescharacteristic
394C2#542×C21111yescharacteristic
395C2#546×C21111yescharacteristic
396C2#547×C21111yescharacteristic
397C2#556×C21111yescharacteristic
398C2#563×C21111yescharacteristic
subgroupG/H
center, socleC23×C211C24
commutator, Frattini, cosocleC23C24×C211
prime radicalC22×D4×C211C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C211 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#126×C211C2semidirect
3. #397×C211C2semidirect
4. #397×C211C2semidirect
5. #396×C211C2semidirect
6. #396×C211C2semidirect
7. #395×C211C2semidirect
9. #391×C211C2semidirect
10. #389×C211C2semidirect
11. #388×C211C2semidirect
12. #387×C211C2semidirect
14. #385×C211C2semidirect
15. #380×C211C2semidirect
199. C211#2715direct (N × H)
200. #2715C211direct (N × H)
201. C2×#126C2×C211semidirect
203. #397C2×C211semidirect
204. #397C2×C211semidirect
205. #396C2×C211semidirect
206. #396C2×C211semidirect
207. #395C2×C211semidirect
209. #391C2×C211semidirect
210. #389C2×C211semidirect
211. #388C2×C211semidirect
212. #387C2×C211semidirect
214. #385C2×C211semidirect
215. #380C2×C211semidirect

Maximal subgroups

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

subgroupindex
C2×#126×C2112
C2×#124×C2112
#397×C2112
#396×C2112
#395×C2112
#394×C2112
#391×C2112
#389×C2112
#388×C2112
#387×C2112
#386×C2112
#385×C2112
#380×C2112
#2715211