Finite Group Databasebeta

← #11740208

Subgroups — #11740208

Order 896.

608 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×C22⋊Q8C211yesmaximal, characteristic
2C2×#6507C211yesmaximal, characteristic, complemented
3C22×Dic7⋊C4C211yesmaximal, characteristic
4C2×#6441C211yesmaximal, characteristic
5#90001C211yesmaximal, characteristic, complemented
6#89982C211yesmaximal, characteristic
7#89755C211yesmaximal, characteristic, complemented
8#89752C211yesmaximal, characteristic, complemented
9#89751C211yesmaximal, characteristic, complemented
10#89748C211yesmaximal, characteristic, complemented
11#89744C211yesmaximal, characteristic, complemented
12#89744C211yesmaximal, characteristic, complemented
13#89718C211yesmaximal, characteristic
14#89714C211yesmaximal, characteristic
15C22⋊C4×Dic7C211yesmaximal, characteristic
16C22×C7×Q8C2211yescharacteristic
17C23×C4×C7C2211yescharacteristic
18C23×Dic7C2211yescharacteristic
19C23×Dic7C2211yescharacteristic
20C22×Dic14C2211yescharacteristic
21C7×C22⋊Q8—24no—
22C2×C7×C4⋊C4C2211yescharacteristic
23C2×C7×C4⋊C4C2211yescharacteristic
24C2×C7×C4⋊C4C2211yescharacteristic
25C2×C7×C22⋊C4C2211yescharacteristic
26C2×C7×C22⋊C4C2211yescharacteristic
27C2×C22⋊Dic7C2211yescharacteristic
28C2×C22⋊Dic7C2211yescharacteristic
29#6507—24no—
30C2×C4⋊Dic7C2211yescharacteristic
31C2×Dic7⋊C4—24no—
32C2×Dic7⋊C4C2211yescharacteristic
33C2×Dic7⋊C4C2211yescharacteristic
34C2×Dic7⋊C4C2211yescharacteristic
35C2×Dic7⋊C4C2211yescharacteristic
36C2×Dic7⋊C4C2211yescharacteristic
37C2×Dic7⋊C4C2211yescharacteristic
38C2×Dic7⋊C4C2211yescharacteristic
39C2×Dic7⋊C4C2211yescharacteristic
40C2×Dic7⋊C4C2211yescharacteristic
41C2×Dic7⋊C4C2211yescharacteristic
42C2×Dic7⋊C4C2211yescharacteristic
43C2×Dic7⋊C4C2211yescharacteristic
44C2×C4×Dic7C2211yescharacteristic
45C2×C4×Dic7C2211yescharacteristic
46#6441—24no—
47C7⋊#95C2211yescharacteristic
48C7⋊#95C2211yescharacteristic
49C7⋊#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51C7⋊#95C2211yescharacteristic
52C7⋊#95C2211yescharacteristic
53C7⋊#95C2211yescharacteristic
54C7⋊#95C2211yescharacteristic
55#2440—71nomaximal, Sylow
56C24×C7C2311yescharacteristic
57C2×C7×Q8—22no—
58C2×C7×Q8—22no—
59C2×C7×Q8—41no—
60C22×C4×C7—22no—
61C22×C4×C7—22no—
62C22×C4×C7—41no—
63C22×C4×C7C2311yescharacteristic
64C22×C4×C7C2311yescharacteristic
65C22×C4×C7C2311yescharacteristic
66C22×C4×C7C2311yescharacteristic
67C22×C4×C7C2311yescharacteristic
68C22×C4×C7C2311yescharacteristic
69C22×Dic7—22no—
70C22×Dic7—22no—
71C22×Dic7—22no—
72C22×Dic7—22no—
73C22×Dic7—41no—
74C22×Dic7Q814yescomplemented
75C22×Dic7C2311yescharacteristic
76C22×Dic7C2311yescharacteristic
77C22×Dic7C2311yescharacteristic
78C22×Dic7C2311yescharacteristic
79C22×Dic7C2311yescharacteristic
80C22×Dic7C2311yescharacteristic
81C22×Dic7C2311yescharacteristic
82C22×Dic7C2311yescharacteristic
83C2×Dic14—22no—
84C2×Dic14—22no—
85C2×Dic14—41no—
86C7×C4⋊C4—22no—
87C7×C4⋊C4—22no—
88C7×C4⋊C4—22no—
89C7×C22⋊C4—22no—
90C7×C22⋊C4D414yes—
91C22⋊Dic7—22no—
92C22⋊Dic7Q814yescomplemented
93C4⋊Dic7—22no—
94Dic7⋊C4—22no—
95Dic7⋊C4—22no—
96Dic7⋊C4—22no—
97Dic7⋊C4—22no—
98Dic7⋊C4—22no—
99Dic7⋊C4—22no—
100Dic7⋊C4—22no—
101Dic7⋊C4—22no—
102Dic7⋊C4—22no—
103Dic7⋊C4—41no—
104Dic7⋊C4—41no—
105Dic7⋊C4D414yes—
106C4×Dic7—22no—
107C4×Dic7—22no—
108C2×C4⋊Q8—71no—
109C2×C22⋊Q8—71no—
110C2×C22⋊Q8D711yescharacteristic
111C22×C4⋊C4—71no—
112#398—71no—
113#398—71no—
114#395—71no—
115#395—71no—
116#389—71no—
117#389—71no—
118#385—71no—
119#385—71no—
120#384—71no—
121#380—71no—
122C4×C22⋊C4—71no—
123C23×C7—21no—
124C23×C7—21no—
125C23×C7—21no—
126C23×C7—21no—
127C23×C7C2×D412yes—
128C23×C7C2×Q812yes—
129C23×C7C2×Q812yes—
130C23×C7C2411yescharacteristic
131C7×Q8—44no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—21no—
141C2×C4×C7—21no—
142C2×C4×C7—21no—
143C2×C4×C7—21no—
144C2×C4×C7—21no—
145C2×C4×C7—21no—
146C2×C4×C7—21no—
147C2×C4×C7—44no—
148C2×C4×C7C2×D412yes—
149C2×C4×C7C2×D412yes—
150C2×C4×C7C2×D412yes—
151C2×Dic7—21no—
152C2×Dic7—21no—
153C2×Dic7—21no—
154C2×Dic7—21no—
155C2×Dic7—21no—
156C2×Dic7—21no—
157C2×Dic7—21no—
158C2×Dic7—21no—
159C2×Dic7—28no—
160C2×Dic7—21no—
161C2×Dic7—21no—
162C2×Dic7—21no—
163C2×Dic7—21no—
164C2×Dic7—21no—
165C2×Dic7—21no—
166C2×Dic7—21no—
167C2×Dic7—21no—
168C2×Dic7—21no—
169C2×Dic7—44no—
170C2×Dic7C2×D412yes—
171C2×Dic7C2×D412yes—
172C2×Dic7C2×Q812yes—
173C2×Dic7C2×Q812yes—
174C2×Dic7C2×Q812yes—
175C2×Dic7C2×Q812yes—
176C2×Dic7#4112yes—
177Dic14—44no—
178C22×Q8—71no—
179C22×Q8C2×D711yescharacteristic
180C23×C4—71no—
181C23×C4—71no—
182C23×C4C2×D711yescharacteristic
183C4⋊Q8—144no—
184C22⋊Q8—24no—
185C22⋊Q8—144no—
186C2×C4⋊C4—71no—
187C2×C4⋊C4—71no—
188C2×C4⋊C4—71no—
189C2×C4⋊C4—71no—
190C2×C4⋊C4—71no—
191C2×C4⋊C4—71no—
192C2×C4⋊C4—71no—
193C2×C4⋊C4—71no—
194C2×C4⋊C4—71no—
195C2×C4⋊C4—71no—
196C2×C4⋊C4—71no—
197C2×C4⋊C4—71no—
198C2×C4⋊C4—71no—
199C2×C4⋊C4—144no—
200C2×C4⋊C4C2×D711yescharacteristic
201C2×C4⋊C4C2×D711yescharacteristic
202C2×C4⋊C4C2×D711yescharacteristic
203C2×C22⋊C4—71no—
204C2×C22⋊C4—71no—
205C2×C22⋊C4C2×D711yescharacteristic
206C2×C22⋊C4C2×D711yescharacteristic
207C2×C42—71no—
208C2×C42—71no—
209#95—71no—
210#95—71no—
211#95—71no—
212#95—71no—
213#95—71no—
214#95—71no—
215#95—71no—
216#95—71no—
217C22×C7—22no—
218C22×C7—22no—
219C22×C7—22no—
220C22×C7—22no—
221C22×C7—22no—
222C22×C7—22no—
223C22×C7C4⋊Q814yescomplemented
224C22×C7C22×D411yescharacteristic
225C22×C7C22×D411yescharacteristic
226C22×C7C22×Q811yescharacteristic
227C22×C7C22×Q811yescharacteristic
228C22×C7C2×#4111yescharacteristic
229C22×C7H(5,𝔽2)11yescharacteristic
230C22×C72−1+411yescharacteristic
231C4×C7—22no—
232C4×C7—22no—
233C4×C7—22no—
234C4×C7—41no—
235C4×C7—41no—
236C4×C7—41no—
237Dic7—22no—
238Dic7—22no—
239Dic7—22no—
240Dic7—22no—
241Dic7—41no—
242Dic7—41no—
243Dic7—41no—
244Dic7C22⋊Q814yescomplemented
245C24C22×D711yescharacteristic
246C2×Q8—22no—
247C2×Q8—22no—
248C2×Q8—41no—
249C2×Q8—142no—
250C2×Q8—142no—
251C2×Q8—281no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—41no—
255C22×C4—71no—
256C22×C4—74no—
257C22×C4—71no—
258C22×C4—71no—
259C22×C4—71no—
260C22×C4—71no—
261C22×C4—71no—
262C22×C4—71no—
263C22×C4—71no—
264C22×C4—142no—
265C22×C4—142no—
266C22×C4—142no—
267C22×C4—142no—
268C22×C4—281no—
269C22×C4C22×D711yescharacteristic
270C22×C4C22×D711yescharacteristic
271C22×C4C22×D711yescharacteristic
272C22×C4C22×D711yescharacteristic
273C22×C4C22×D711yescharacteristic
274C22×C4C22×D711yescharacteristic
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—74no—
279C4⋊C4—142no—
280C4⋊C4—142no—
281C4⋊C4—142no—
282C4⋊C4—142no—
283C4⋊C4—142no—
284C4⋊C4—142no—
285C4⋊C4—142no—
286C4⋊C4—142no—
287C4⋊C4—142no—
288C4⋊C4—142no—
289C4⋊C4—281no—
290C4⋊C4—281no—
291C22⋊C4—22no—
292C22⋊C4—74no—
293C22⋊C4—142no—
294C22⋊C4C22⋊D714yes—
295C42—142no—
296C42—142no—
297C2×C7—24no—
298C2×C7C2×C22⋊Q811yescharacteristic
299C2×C7C2×C4⋊Q811yescharacteristic
300C2×C7#54411yescharacteristic
301C2×C7D4211yescharacteristic
302C2×C7#54711yescharacteristic
303C2×C7D4×Q811yescharacteristic
304C2×C7#55411yescharacteristic
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23Q8×D712yes—
310C23Q8×D712yes—
311C23C2×C22⋊D712yes—
312C23C23×D711yescharacteristic
313Q8—44no—
314Q8—284no—
315C2×C4—21no—
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—44no—
331C2×C4—72no—
332C2×C4—72no—
333C2×C4—72no—
334C2×C4—72no—
335C2×C4—72no—
336C2×C4—72no—
337C2×C4—72no—
338C2×C4—148no—
339C2×C4—141no—
340C2×C4—141no—
341C2×C4—141no—
342C2×C4—141no—
343C2×C4—141no—
344C2×C4—141no—
345C2×C4—141no—
346C2×C4—141no—
347C2×C4—141no—
348C2×C4—141no—
349C2×C4—141no—
350C2×C4—141no—
351C2×C4—141no—
352C2×C4—141no—
353C2×C4—141no—
354C2×C4—141no—
355C2×C4—141no—
356C2×C4—284no—
357C2×C4D4×D712yes—
358C2×C4C2×C22⋊D712yes—
359C2×C4C2×C22⋊D712yes—
360C7#244011yescharacteristic, complemented, Sylow
361C22—22no—
362C22—22no—
363C22—22no—
364C22—22no—
365C22—22no—
366C22—22no—
367C22#650714yescomplemented
368C22C2×D4×D711yescharacteristic
369C22#654811yescharacteristic
370C22C2×Q8×D711yescharacteristic
371C22C2×Q8×D711yescharacteristic
372C22D7×#4111yescharacteristic
373C22#655411yescharacteristic
374C22C22×C22⋊D711yescharacteristic
375C4—22no—
376C4—22no—
377C4—22no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—74no—
382C4—142no—
383C4—142no—
384C4—142no—
385C4—142no—
386C4—281no—
387C4—281no—
388C4—281no—
389C2—24no—
390C2#9031411yescharacteristic
391C2D7×C22⋊Q811yescharacteristic
392C2#9032311yescharacteristic
393C2#9032711yescharacteristic
394C2D4×C22⋊D711yescharacteristic
395C2C2×#650711yescharacteristic
396C2#9051511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC2×C7×C22⋊Q8C2
prime radicalC24×C7C23
hypercenterC2×C22⋊Q8D7
hypocenterC7#2440
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#6507C2semidirect
5. #90001C2semidirect
7. #89755C2semidirect
8. #89752C2semidirect
9. #89751C2semidirect
10. #89748C2semidirect
11. #89744C2semidirect
12. #89744C2semidirect
74. C22×Dic7Q8semidirect
92. C22⋊Dic7Q8semidirect
223. C22×C7C4⋊Q8semidirect
244. Dic7C22⋊Q8semidirect
360. C7#2440semidirect
367. C22#6507semidirect

Maximal subgroups

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

subgroupindex
C2×C7×C22⋊Q82
C2×#65072
C22×Dic7⋊C42
C2×#64412
#900012
#899822
#897552
#897522
#897512
#897482
#897442
#897182
#897142
C22⋊C4×Dic72
#24407