Finite Group Databasebeta

← C4×Q8×Dic14 #13513800

Subgroups — C4×Q8×Dic14 #13513800

Order 1792.

2324 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×Dic14C211yesmaximal, characteristic
2C42×C7×Q8C211yesmaximal, characteristic
3Q8×Dic7⋊C4C212yesmaximal
4Q8×C4⋊Dic7C211yesmaximal, characteristic
5C4×Q8×Dic7C212yesmaximal
6#11731102C216yesmaximal
7#11731101C213yesmaximal
8C4⋊C4×Dic14C213yesmaximal
9C4×#6451C216yesmaximal
10C4×#6432C213yesmaximal
11C42×Dic14C213yesmaximal
12C2×C4×C7×Q8C2211yescharacteristic
13C2×C4×C7×Q8C2211yescharacteristic
14C2×C4×C7×Q8C2211yescharacteristic
15C2×Q8×Dic7C2212yes—
16C2×Q8×Dic7C2212yes—
17Q8×Dic14C4116yesdirect product factor
18C2×#6451C2216yes—
19C2×#6432C2213yes—
20C2×C4×Dic14C2213yes—
21C2×C4×Dic14C2216yes—
22C2×C4×Dic14C2213yes—
23C2×C4×Dic14C2211yescharacteristic
24C4×C7×C4⋊C4C2213yes—
25C4×C7×C4⋊C4C2213yes—
26C4×C7×C4⋊C4C2213yes—
27C43×C7C2213yes—
28#89757C2216yes—
29#89748C2216yes—
30C4⋊C4×Dic7C2216yes—
31C4⋊C4×Dic7C2216yes—
32#89745C2216yes—
33#89707C2213yes—
34C4×C4⋊Dic7C2213yes—
35C4×C4⋊Dic7C2213yes—
36C4×C4⋊Dic7C2213yes—
37C4×C4⋊Dic7C2216yes—
38C4×Dic7⋊C4C2216yes—
39C4×Dic7⋊C4C2216yes—
40C4×Dic7⋊C4C2216yes—
41C42×Dic7C2216yes—
42#89701C2216yes—
43#89700C2216yes—
44#89700C2216yes—
45#89419C22112yes—
46#89418C22112yes—
47C4×Q82—71nomaximal, Sylow
48C22×C7×Q8C2311yescharacteristic
49C22×Dic14C2316yes—
50C22×Dic14C2311yescharacteristic
51C4×C7×Q8—24no—
52C4×C7×Q8C2×C418yes—
53C4×C7×Q8Q818yescomplemented
54C2×C7×C4⋊C4C2313yes—
55C2×C7×C4⋊C4C2313yes—
56C2×C7×C4⋊C4C2313yes—
57C2×C42×C7C2313yes—
58C2×C42×C7C2313yes—
59C2×C42×C7C2313yes—
60C2×C42×C7C2313yes—
61C2×C42×C7C2313yes—
62C2×C42×C7C2311yescharacteristic
63C2×C42×C7C2313yes—
64Q8×Dic7—28no—
65Q8×Dic7C2×C4116yes—
66C2×C4⋊Dic7C2313yes—
67C2×C4⋊Dic7C2316yes—
68C2×C4⋊Dic7C2316yes—
69C2×C4⋊Dic7C2313yes—
70C2×C4⋊Dic7C2316yes—
71C2×C4⋊Dic7C2311yescharacteristic
72C2×Dic7⋊C4C2316yes—
73C2×Dic7⋊C4C23112yes—
74C2×Dic7⋊C4C2316yes—
75C2×Dic7⋊C4C2312yes—
76C2×C4×Dic7C2316yes—
77C2×C4×Dic7C2316yes—
78C2×C4×Dic7C2316yes—
79C2×C4×Dic7C2316yes—
80C2×C4×Dic7C2316yes—
81C2×C4×Dic7C2312yes—
82#6451C2×C4148yes—
83#6432C2×C4124yes—
84C4×Dic14—212no—
85C4×Dic14—412no—
86C4×Dic14C2×C4124yes—
87C4×Dic14Q818yesdirect product factor
88C7×#95C2316yes—
89C7⋊#95C2316yes—
90C7⋊#95C2316yes—
91C7⋊#95C23112yes—
92C7⋊#95C2316yes—
93C2×Q82—71no—
94#2369—73no—
95#2369—76no—
96Q8×C4⋊C4—73no—
97Q8×C4⋊C4—72no—
98Q8×C4⋊C4—71no—
99C4×C4⋊Q8—76no—
100C4×C4⋊Q8—73no—
101C42×Q8—73no—
102C42×Q8—72no—
103C42×Q8D711yescharacteristic
104C2×C7×Q8C22×C414yes—
105C2×C7×Q8C2×Q814yes—
106C2×C7×Q8#4114yes—
107C22×C4×C7C2413yes—
108C22×C4×C7C2413yes—
109C22×C4×C7C2411yescharacteristic
110C22×C4×C7C2413yes—
111C22×C4×C7C2413yes—
112C22×C4×C7C2411yescharacteristic
113C22×C4×C7C2411yescharacteristic
114C22×Dic7C2416yes—
115C22×Dic7C2416yes—
116C22×Dic7C2412yes—
117C22×Dic7C2412yes—
118C2×Dic14—46no—
119C2×Dic14—46no—
120C2×Dic14C22×C4124yes—
121C2×Dic14C22×C414yes—
122C2×Dic14C2×Q814yes—
123C2×Dic14#4114yes—
124C7×C4⋊C4—26no—
125C7×C4⋊C4C22×C4112yes—
126C7×C4⋊C4C2×Q8112yes—
127C42×C7—26no—
128C42×C7—26no—
129C42×C7—26no—
130C42×C7—43no—
131C42×C7C22×C4112yes—
132C42×C7C2×Q8112yes—
133C42×C7C2×Q814yes—
134C4⋊Dic7—26no—
135C4⋊Dic7—212no—
136C4⋊Dic7—46no—
137C4⋊Dic7C22×C4112yes—
138C4⋊Dic7C22×C4124yes—
139C4⋊Dic7C2×Q814yes—
140Dic7⋊C4—212no—
141Dic7⋊C4—412no—
142Dic7⋊C4C22×C4124yes—
143Dic7⋊C4C2×Q818yes—
144C4×Dic7—212no—
145C4×Dic7—212no—
146C4×Dic7—212no—
147C4×Dic7—46no—
148C4×Dic7C22×C4124yes—
149C4×Dic7C2×Q818yes—
150Q82—716no—
151C2×C4⋊Q8—76no—
152C2×C4⋊Q8—73no—
153C2×C4×Q8—72no—
154C2×C4×Q8—72no—
155C2×C4×Q8—73no—
156C2×C4×Q8—71no—
157C2×C4×Q8—73no—
158C2×C4×Q8—76no—
159C2×C4×Q8C2×D711yescharacteristic
160C2×C4×Q8C2×D711yescharacteristic
161C2×C4×Q8C2×D711yescharacteristic
162#391—76no—
163#391—712no—
164#389—76no—
165#389—76no—
166#389—76no—
167#389—76no—
168#389—712no—
169#384—76no—
170#384—73no—
171C4×C4⋊C4—76no—
172C4×C4⋊C4—76no—
173C4×C4⋊C4—76no—
174C4×C4⋊C4—76no—
175C4×C4⋊C4—73no—
176C4×C4⋊C4—76no—
177C4×C4⋊C4—73no—
178C4×C4⋊C4—76no—
179C4×C4⋊C4—73no—
180C4×C4⋊C4C2×D713yes—
181C4×C4⋊C4C2×D713yes—
182C4×C4⋊C4C2×D713yes—
183C43—76no—
184C43C2×D713yes—
185C23×C7C2511yescharacteristic
186C7×Q8C4×Q8116yescomplemented
187C2×C4×C7—23no—
188C2×C4×C7—21no—
189C2×C4×C7—23no—
190C2×C4×C7—23no—
191C2×C4×C7—21no—
192C2×C4×C7—23no—
193C2×C4×C7—23no—
194C2×C4×C7—23no—
195C2×C4×C7—23no—
196C2×C4×C7C23×C416yes—
197C2×C4×C7C23×C416yes—
198C2×C4×C7C23×C412yes—
199C2×C4×C7C22×Q816yes—
200C2×C4×C7C22×Q816yes—
201C2×C4×C7C22×Q812yes—
202C2×C4×C7C22×Q812yes—
203C2×C4×C7C22×Q812yes—
204C2×C4×C7C22×Q812yes—
205C2×C4×C7C2×#4116yes—
206C2×C4×C7C2×#4112yes—
207C2×C4×C7H(5,𝔽2)12yes—
208C2×Dic7—22no—
209C2×Dic7—26no—
210C2×Dic7—22no—
211C2×Dic7—26no—
212C2×Dic7—26no—
213C2×Dic7—26no—
214C2×Dic7—26no—
215C2×Dic7C23×C4112yes—
216C2×Dic7C23×C414yes—
217C2×Dic7C22×Q814yes—
218C2×Dic7C22×Q814yes—
219C2×Dic7C2×#4114yes—
220Dic14—424no—
221Dic14C4×Q8116yesdirect product factor
222C22×Q8—71no—
223C22×Q8—76no—
224C22×Q8C22×D711yescharacteristic
225C4⋊Q8—748no—
226C4⋊Q8—724no—
227C4×Q8—24no—
228C4×Q8—716no—
229C4×Q8—724no—
230C4×Q8—78no—
231C4×Q8—1412no—
232C4×Q8—148no—
233C4×Q8—2812no—
234C4×Q8Dic1418yesdirect product factor
235C4×Q8C4×D718yes—
236C2×C4⋊C4—76no—
237C2×C4⋊C4—76no—
238C2×C4⋊C4—73no—
239C2×C4⋊C4—72no—
240C2×C4⋊C4—71no—
241C2×C4⋊C4—76no—
242C2×C4⋊C4—76no—
243C2×C4⋊C4—73no—
244C2×C4⋊C4—712no—
245C2×C4⋊C4—76no—
246C2×C4⋊C4C22×D713yes—
247C2×C4⋊C4C22×D713yes—
248C2×C4⋊C4C22×D713yes—
249C2×C42—76no—
250C2×C42—72no—
251C2×C42—76no—
252C2×C42—76no—
253C2×C42—76no—
254C2×C42—76no—
255C2×C42C22×D713yes—
256C2×C42C22×D713yes—
257C2×C42C22×D713yes—
258C2×C42C22×D713yes—
259C2×C42C22×D713yes—
260C2×C42C22×D713yes—
261C2×C42C22×D711yescharacteristic
262#95—76no—
263#95—76no—
264#95—712no—
265#95—76no—
266#95C22×D716yes—
267C22×C7C24×C411yescharacteristic
268C22×C7C23×Q811yescharacteristic
269C22×C7C23×Q811yescharacteristic
270C22×C7C22×#4111yescharacteristic
271C22×C7C22×#4111yescharacteristic
272C22×C7C2×H(5,𝔽2)11yescharacteristic
273C22×C7#58511yescharacteristic
274C4×C7—26no—
275C4×C7—22no—
276C4×C7—26no—
277C4×C7—43no—
278C4×C7C2×C4×Q8112yes—
279C4×C7C2×C4×Q814yes—
280C4×C7Q8214yescomplemented
281Dic7—212no—
282Dic7—24no—
283Dic7—46no—
284Dic7C2×C4×Q818yes—
285C2×Q8—74no—
286C2×Q8—724no—
287C2×Q8—74no—
288C2×Q8—74no—
289C2×Q8—286no—
290C2×Q8—286no—
291C2×Q8C2×Dic1414yes—
292C2×Q8C2×C4×D714yes—
293C2×Q8#115714yes—
294C22×C4—72no—
295C22×C4—72no—
296C22×C4—76no—
297C22×C4—76no—
298C22×C4C23×D713yes—
299C22×C4C23×D713yes—
300C22×C4C23×D713yes—
301C22×C4C23×D711yescharacteristic
302C22×C4C23×D713yes—
303C22×C4C23×D711yescharacteristic
304C22×C4C23×D711yescharacteristic
305C4⋊C4—26no—
306C4⋊C4—724no—
307C4⋊C4—724no—
308C4⋊C4—712no—
309C4⋊C4—78no—
310C4⋊C4—74no—
311C4⋊C4—1412no—
312C4⋊C4—1412no—
313C4⋊C4—146no—
314C4⋊C4—2812no—
315C4⋊C4—286no—
316C4⋊C4C2×Dic14112yes—
317C4⋊C4C2×C4×D7112yes—
318C42—26no—
319C42—26no—
320C42—26no—
321C42—43no—
322C42—724no—
323C42—78no—
324C42—1412no—
325C42—1412no—
326C42—1412no—
327C42—286no—
328C42C2×Dic14112yes—
329C42C2×C4×D7112yes—
330C42Q8×D714yes—
331C2×C7C22×C4×Q811yescharacteristic
332C2×C7C22×C4×Q811yescharacteristic
333C2×C7C4×H(5,𝔽2)11yescharacteristic
334C2×C7C2×Q8211yescharacteristic
335C2×C7Q8×#4111yescharacteristic
336C2×C7Q8×#4111yescharacteristic
337C2×C7#348311yescharacteristic
338C23C24×D711yescharacteristic
339Q8—716no—
340Q8—2824no—
341Q8C4×Dic14116yesdirect product factor
342C2×C4—21no—
343C2×C4—23no—
344C2×C4—23no—
345C2×C4—21no—
346C2×C4—23no—
347C2×C4—23no—
348C2×C4—23no—
349C2×C4—23no—
350C2×C4—23no—
351C2×C4—74no—
352C2×C4—74no—
353C2×C4—74no—
354C2×C4—712no—
355C2×C4—74no—
356C2×C4—142no—
357C2×C4—146no—
358C2×C4—142no—
359C2×C4—146no—
360C2×C4—146no—
361C2×C4—146no—
362C2×C4—146no—
363C2×C4C22×Dic1416yes—
364C2×C4C22×Dic1416yes—
365C2×C4C22×Dic1412yes—
366C2×C4C22×C4×D716yes—
367C2×C4C22×C4×D716yes—
368C2×C4C22×C4×D712yes—
369C2×C4C2×#115716yes—
370C2×C4C2×Q8×D712yes—
371C2×C4C2×Q8×D712yes—
372C2×C4C2×Q8×D712yes—
373C2×C4D7×#4112yes—
374C2×C4#655312yes—
375C7C4×Q8211yescharacteristic, complemented, Sylow
376C22C23×Dic1411yescharacteristic
377C22C23×C4×D711yescharacteristic
378C22C22×#115711yescharacteristic
379C22C22×Q8×D711yescharacteristic
380C22C2×D7×#4111yescharacteristic
381C22C2×#655311yescharacteristic
382C22#9061611yescharacteristic
383C4—22no—
384C4—26no—
385C4—26no—
386C4—43no—
387C4—78no—
388C4—144no—
389C4—1412no—
390C4—286no—
391C4C2×C4×Dic14112yes—
392C4Q8×Dic1414yesdirect product factor
393C4C4×Q8×D714yes—
394C2C22×C4×Dic1411yescharacteristic
395C2C2×Q8×Dic1411yescharacteristic
396C2C2×C4×Q8×D711yescharacteristic
397C2Q8×#115711yescharacteristic
398C2#41×Dic1411yescharacteristic
399C2C4×#655311yescharacteristic
400C2#1174291311yescharacteristic
subgroupG/H
centerC22×C4C23×D7
commutatorC22×C7C24×C4
FrattiniC23C24×D7
FittingC42×C7×Q8C2
socle, prime radical, cosocleC23×C7C25
hypercenterC42×Q8D7
hypocenterC7C4×Q82
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22×C4 ◁ C42×Q8

Complements

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

NH ≅ G/Ntype
17. Q8×Dic14C4direct (N × H)
53. C4×C7×Q8Q8semidirect
87. C4×Dic14Q8direct (N × H)
186. C7×Q8C4×Q8semidirect
221. Dic14C4×Q8direct (N × H)
234. C4×Q8Dic14direct (N × H)
280. C4×C7Q82semidirect
341. Q8C4×Dic14direct (N × H)
375. C7C4×Q82semidirect
392. C4Q8×Dic14direct (N × H)

Maximal subgroups

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

subgroupindex
C2×Q8×Dic142
C42×C7×Q82
Q8×Dic7⋊C42
Q8×C4⋊Dic72
C4×Q8×Dic72
#117311022
#117311012
C4⋊C4×Dic142
C4×#64512
C4×#64322
C42×Dic142
C4×Q827